Theodoros Katsaounis
email: thodoros.katsaounis@uoc.gr

Julier Olivier
email: julien.olivier@univ-amu.fr

AND Athanasios E Tzavaras
email: athanasios.tzavaras@kaust.edu.sa

EMERGENCE OF COHERENT LOCALIZED STRUCTURES IN SHEAR DEFORMATIONS OF TEMPERATURE DEPENDENT FLUIDS *

.

Introduction

The phenomenon of shear localization during high strain-rate deformations of metals [START_REF] Zener | Effect of strain rate upon plastic flow of steel[END_REF][START_REF] Clifton | On critical conditions for shear band formation at high strain rates[END_REF][START_REF] Hartley | Measurement of the temperature profile during shear band formation in steels deforming at high-strain rates[END_REF] is a striking instance of material instability in mechanics, that has attracted considerable attention in the mechanics literature (e.g. [START_REF] Clifton | On critical conditions for shear band formation at high strain rates[END_REF][START_REF] Fressengeas | Instability and localization of plastic flow in shear at high strain rates[END_REF][START_REF] Molinari | Analytical characterization of shear localization in thermoviscoplastic materials[END_REF][START_REF] Wright | On stress collapse in adiabatic shear bands[END_REF][START_REF] Wright | The Physics and Mathematics of Shear Bands[END_REF]) as well as a number of mathematical studies ([4, 15, 2, 20, 17, 5, 10] and references therein). Material instability is typically associated with illposedness of an underlying initial value problem, what has coined the term Hadamard instability for its description in the mechanics literature. It should however be emphasized that while Hadamard instability indicates the catastrophic growth of oscillations around a mean state, what is observed in localization is the orderly albeit extremely fast development of coherent structures: the shear bands. The latter is a nonlinear phenomenon and a linearized analysis can only serve as a precursor to such behavior.

We work with the simplest model capturing the mechanism of shear band formation, a simple shear motion of a non-Newtonian fluid with temperature decreasing viscosity µ (θ) < 0, (1.1)

v t = σ x , θ t = κθ xx + σv x σ = µ(θ)v n x ,
where n > 0 describes the strain-rate sensitivity which typically in these problems satisfies n 1. Mathematical studies of (2.1), for power law viscosities µ(θ) = θ -α , α > 0, show that the uniform shear is asymptotically stable for n > α [START_REF] Dafermos | Adiabatic shearing of incompressible fluids with temperature dependent viscosity Quart[END_REF][START_REF] Tzavaras | Effect of thermal softening in shearing of strain-rate dependent materials Arch[END_REF], but indicate unstable response in the complementary region n < α, [START_REF] Tzavaras | Effect of thermal softening in shearing of strain-rate dependent materials Arch[END_REF][START_REF] Bertsch | The effect of temperature dependent viscosity on shear flow of incompressible fluids[END_REF][START_REF] Tzavaras | Nonlinear analysis techniques for shear band formation at high strain rates[END_REF]. The mathematical analysis has not so far characterized the precise behavior in the instability region; however, detailed and resolved numerical studies (e.g. [START_REF] Wright | On stress collapse in adiabatic shear bands[END_REF][START_REF] Estep | Analysis of shear layers in a fluid with temperature-dependent viscosity[END_REF][START_REF] Th | Adaptive finite element computations of shear band formation Math[END_REF]) show that instability is followed by formation of shear bands.

The difficulty is that in order to analyze the unstable regime one needs to account for the combined effect of parabolic regularizations in conjunction to ill-posed initial value problems. An asymptotic theory for calculating an effective equation for the compound effect is proposed in [START_REF] Th | Effective equations for localization and shear band formation[END_REF][START_REF] Th | Localization and shear bands in high strain-rate plasticity[END_REF]. It is based on the theory of relaxation systems and the Chapman-Enskog expansion and produces an effective asymptotic equation. For a power law the effective equation changes type from forward to backward parabolic, across the threshold -α + n = 0 of the parameter space. The criteria of [START_REF] Th | Effective equations for localization and shear band formation[END_REF][START_REF] Th | Localization and shear bands in high strain-rate plasticity[END_REF] are based on asymptotic analysis and as such are formal in nature.

Our objective, here, is to develop a mathematical theory for the onset of localisation and the emergence of a nonlinear localized state, thus providing a rigorous justification to the behavior conjectured by the effective equation in [START_REF] Th | Effective equations for localization and shear band formation[END_REF]. We employ the constitutive hypothesis of an exponential law for the viscosity coefficient, (H exp ) µ(θ) = e -αθ , α > 0 and hence σ = e -αθ v n x . The model (2.1) for the exponential law (H exp ) admits a special class of solutions describing uniform shearing (1.2)

v s (x) = x, θ s (t) = 1 α log (αt + c 0 ), c 0 = e αθ 0 , σ s (t) = 1 αt + c 0 .
The graph σ s -t describing the average stress vs. average strain response is monotonically decreasing, and thus the deformation of uniform shear is characterized by strain softening response. We set the following goals: (i) To study the linearized problem around the uniform shearing solution and to assess the effects of the various parameters of the problem (ii) To inquire whether there are special solutions of the nonlinear problem that indicate the emergence of localized deformations and formation of shear bands. Since the base solution (1.2) around which we linearize is time-dependent the linearized analysis necessarily involves analysis of non-autonomous systems and also perturbations have to be compared against the base solution. We perform a linearization using ideas of relative perturbations (1.3)

u := v x = 1 + δ ū(x, t) + O(δ 2 ) θ = θ s (t) + δ θ(x, t) + O(δ 2 ) σ = σ s (t) + δσ s (t)σ(x, t) + O(δ 2 ) leading to linearized system (1.4)          ūt = σ s (t) nū xx -α θxx θt -κ θxx = σ s (t) (n + 1)ū -α θ ūx (0, t) = ūx (1, t) = 0 , θx (0, t) = θx (1, t) = 0 .
with σ computed by σ = nū -α θ. This problem turns out to have the following behavior:

(a) For the case without thermal diffusion κ = 0:

When n = 0 the high frequency modes grow exponentially fast with the exponent of the order of the frequency what is characteristic of ill-posedness and Hadamard instability.

When n > 0 the modes are still unstable but the rate of growth is bounded independently of the frequency. In fact, the behavior in that range is characteristic of Turing instability. (b) The effect of thermal diffusion (κ > 0) for the non-autonomous linearized model (1.4) can also be assessed: Perturbations do grow initially in the early stages of deformation, but over time the effect of the diffusion intensifies and eventually the system gets stabilized. For a quantitative description of the above qualitative statement we refer to lemmas 3.2 and 3.3. In addition, this behavior can also occur for the full solution of the nonlinear problem (1.1) subject to simple shear boundary conditions as indicated in numerical simulations presented in Figure 2. This type of metastable response was already conjectured (for a somewhat different model) in numerical experiments of [START_REF] Wright | On stress collapse in adiabatic shear bands[END_REF][START_REF] Walter | Numerical experiments on adiabatic shear band in one space dimension[END_REF]. The linearized analysis presented here offers a theoretical explanation of this interesting behavior.

In the second part of this article we consider the adiabatic variant of the model (1.1) with exponential consitutive law (H exp ). The system is expressed for u = v x in the form (1.5)

u t = ∂ xx e -αθ u n θ t = e -αθ u n+1
and is now considered on the whole space (x, t) ∈ R × [0, ∞). For the system (1.5), we construct a class of localizing solutions

(1.6) u(x, t) = φ λ (t) U λ √ λxφ λ (t) σ(x, t) = σ s (t) 1 φ λ (t) Σ λ √ λxφ λ (t) θ(x, t) = θ s (t) + λ n+1 α (θ s (t) -θ 0 ) + Θ λ √ λxφ λ (t)
where φ λ (t) = α c 0 t + 1 λ α depends on the parameter λ > 0, and (1.6) is determined via the solution (U λ , Σ λ , Θ λ ) of the singular initial value problem (5.1), (5.2) with ν = λ. The existence of the solution (U λ , Σ λ , Θ λ ) is established in Theorem 5.1 where most importantly its behavior at the origin and at infinity is precised. The emerging solution (1.6) depends on three parameters: θ 0 linked to the uniform shear in (1.2), a parameter λ which can be viewed as a length scale of initial data, and a Σ 0 determining the size of the initial profile.

To our knowledge this is the first instance that the compound effect of Hadamard instability and parabolic regularizations is mathematically analyzed in a nonlinear context. The reader should contrast the nonlinear response captured by the exact solution (1.6) to the response of the associated linearized problem (1.4). As precised in Theorem 6.1, the response of (1.6) indicates that the competition among Hadamard instability and momentum diffusion in a nonlinear context can lead to a nonlinear coherent structure that localizes in a narrow band. By contrast, for a linear model, the combined effect of Hadamard instability and parabolicity leads to unstable oscillatory modes growing in amplitude (see (3.22)). This justifies the scenario that nonlinearity can kill the oscillations caused by Hadamard instability so that the nonuniformity of deformation produced by the instability merges into a single localized zone. It would be interesting to explore this type of response for other models related to material instability in solid mechanics.

The exposition is organized as follows: Section 2 presents a description of the problem and the formulation via relative perturbations. The linearized analysis is presented in section 3, first the part that can be done via spectral analysis for autonomous systems, and then the part concerning the effect of thermal diffusion, which is based on energy estimates for a non-autonomous linear system. Detailed asymptotics for the eigenvalues and the connection with Hadamard and Turing instabilities are presented there. The presentation of the localized solutions is split in sections 4-6: Section 4 presents the invariance properties, the ansatz of localized solutions, and also introduces the relevant auxiliary problem (5.1), (5.2). The construction of solutions to the singular initial-value problem (5.1), (5.2) and the proof of their properties is the centerpiece of our analysis and forms the objective of section 5. The construction is based on a series of steps, involving desingularization, a nonlinear transformation, and eventually the existence of a heteroclinic orbit via the Poincaré-Bendixson theorem. The construction of the localized solutions is then effected in section 6.

Modeling of Shear Bands -Relative perturbations

In this work we study the onset of localization and formation of shear bands for the model (1.1) describing shear motions of a non-Newtonian fluid with viscosity µ(θ) decreasing with temperature. For concreteness we take a viscosity satisfying the exponential law (H exp ) and are interested in the range where the rate sensitivity n is positive and small. The model (1.1) then becomes (2.1)

v t = σ x , θ t = κθ xx + σv x σ = e -αθ v n x .
and describes shear motions between two parallel plates located at x = 0 and x = 1 with v the velocity in the shear direction, σ the shear stress, θ the temperature, and

(2.2) u = v x
depicting the strain rate. The boundary conditions are taken

v(0, t) = 0 , v(1, t) = 1 , (2.3) θ x (0, t) = 0 , θ x (1, t) = 0 . (2.4) Equation (2.
3) reflects prescribed velocities at the boundaries, while (2.4) manifests that the bounding plates are adiabatic. Note that (2.3) implies that the stress satisfies (2.5)

σ x (0, t) = σ x (1, t) = 0 ,
and that (2.6)

1 0 u(x, t)dx = 1 .
The initial conditions are

(2.7) v(x, 0) = v 0 (x) , θ(x, 0) = θ 0 (x) .
Throughout the study we will consider two cases: When κ = 0 the process is adiabatic and the resulting model will be the main object of study, as it is the simplest model proposed for studying the phenomenon of formation of shear bands. The case κ > 0 will also be studied in section 3 as a paradigm to assess the effect of thermal diffusion.

2.1. Uniform shearing solutions. The problem (2.1), (2.3), (2.4) admits a class of special solutions describing uniform shear

v s = x, u s = ∂ x v s = 1 θ s = θ s (t) σ s = σ s (t) = µ(θ s (t)) (2.8)
with θ s defined by

   dθ s dt = µ(θ s ) = σ s θ s (0) = θ 0 (2.9)
For the special case of the exponential law (H exp ), θ s and σ s read (2.10)

θ s = 1 α log (αt + c 0 ), c 0 = e αθ 0 , σ s = 1 αt + c 0 .
These are universal solutions, i.e. they hold for all values of the parameters n, α, κ in the model (2.1) including the limiting elliptic initial value problem when n = 0, κ = 0. The graph σ -t is viewed as describing the stress vs. average strain response. As the graph is decreasing, it means that there is always softening response in the course of the deformation.

2.2. Description of the phenomenon of shear bands. The system (2.1) has served as a paradigm for the mathematical study of the phenomenon of shear bands (e.g. [START_REF] Dafermos | Adiabatic shearing of incompressible fluids with temperature dependent viscosity Quart[END_REF][START_REF] Tzavaras | Effect of thermal softening in shearing of strain-rate dependent materials Arch[END_REF][START_REF] Tzavaras | Strain softening in viscoelasticity of the rate type[END_REF][START_REF] Bertsch | The effect of temperature dependent viscosity on shear flow of incompressible fluids[END_REF]) occuring during the high strain-rate deformations of metals. Formation of shear bands is a well known material instability that usually leads to rupture, and has received considerable attention in the mechanics literature (e.g. [START_REF] Clifton | On critical conditions for shear band formation at high strain rates[END_REF][START_REF] Fressengeas | Instability and localization of plastic flow in shear at high strain rates[END_REF][START_REF] Molinari | Analytical characterization of shear localization in thermoviscoplastic materials[END_REF][START_REF] Wright | On stress collapse in adiabatic shear bands[END_REF][START_REF] Wright | The Physics and Mathematics of Shear Bands[END_REF]). The reader is referred to [START_REF] Th | Localization and shear bands in high strain-rate plasticity[END_REF] for a description of the problem intended for a mathematically oriented reader that is briefly also outlined below.

It was recognized by Zener and Hollomon [START_REF] Zener | Effect of strain rate upon plastic flow of steel[END_REF] (see also [START_REF] Shawki | Shear band formation in thermal viscoplastic materials[END_REF]) that the effect of the deformation speed is twofold: (a) To change the deformation conditions from isothermal to nearly adiabatic; under nearly adiabatic conditions the combined effect of thermal softening and strain hardening tends to produce net softening response. (b) Strain rate has an effect per se and induces momentum diffusion that needs to be included in the constitutive modeling. A theory of thermoviscoplasticity is commonly used to model shear bamds and we refer to [START_REF] Shawki | Shear band formation in thermal viscoplastic materials[END_REF][START_REF] Tzavaras | Effect of thermal softening in shearing of strain-rate dependent materials Arch[END_REF][START_REF] Wright | On stress collapse in adiabatic shear bands[END_REF][START_REF] Wright | A model for fully formed shear bands[END_REF][START_REF] Th | Effective equations for localization and shear band formation[END_REF] for various accounts of the modeling aspects.

The basic mechanism for localization is encoded in the behavior of the following simple system of differential equations (2.11)

v t = ∂ x µ(θ)v n x ), θ t = σv x = µ(θ)v n+1
x (which is the adiabatic case κ = 0 of (2.1)). This model is appropriate for a non-Newtonian fluid with temperature dependent viscosity, and is also a simplification of the models proposed in [START_REF] Zener | Effect of strain rate upon plastic flow of steel[END_REF][START_REF] Shawki | Shear band formation in thermal viscoplastic materials[END_REF] to capture the mechanism for shear band formation in high strain-rate plastic deformations of metals. The argument -adapted to the language of a temperature dependent fluid -goes as follows: as the fluid gets sheared the dissipation produces heat and elevates the temperature of the fluid. The viscosity decreases with the elevation of the temperature and the fluid becomes softer and easier to shear. One eventuality is that the shear deformation distributes uniformly across the material in the same manner as it does for the usual Newtonian fluids. This arrangement is described by the uniform shearing solution (2.10). A second mode of response is suggested when comparing the behavior of the fluid in two spots: one hot and one cold. Since the viscosity is weaker in the hot spot and stronger in the cold spots, the amount of shear generated in the hot spot will be larger than the shear generated in the cold spot and in turn the temperature difference will be intensified. It is then conceivable that the deformation localizes into a narrow band where the material is considerably hotter (and weaker) than the surrounding environment. The two possibilities are depicted in Fig. 1. The outcome of the competition of thermal softening and momentum diffusion is captured by the behavior of solutions to (2.11).

A series of works analyze the system (2.11) mostly for power law viscosities (H pl ) µ(θ) = θ -α , α > 0 and hence σ = θ -α v n x . These studies show stable response [START_REF] Dafermos | Adiabatic shearing of incompressible fluids with temperature dependent viscosity Quart[END_REF][START_REF] Tzavaras | Effect of thermal softening in shearing of strain-rate dependent materials Arch[END_REF] when -α + n > 0, and indicate unstable response [START_REF] Tzavaras | Effect of thermal softening in shearing of strain-rate dependent materials Arch[END_REF][START_REF] Bertsch | The effect of temperature dependent viscosity on shear flow of incompressible fluids[END_REF][START_REF] Tzavaras | Nonlinear analysis techniques for shear band formation at high strain rates[END_REF] but without characterizing the behavior in the complementary region -α + n < 0. More precisely, in the unstable regime Tzavaras [START_REF] Tzavaras | Effect of thermal softening in shearing of strain-rate dependent materials Arch[END_REF] shows that development of shear bands can be induced by stress boundary conditions leading to finite-time blow-up, and that their occurence is associated with a collapse of stress diffusion across the band. On the other hand, Bertsch, Peletier and Verduyn-Lunel [START_REF] Bertsch | The effect of temperature dependent viscosity on shear flow of incompressible fluids[END_REF] show that if the problem has global solutions and there is a sufficiently large initial temperature perturbation, then a shear band appears as the asymptotic in time state. Careful and detailed numerical investigations [START_REF] Wright | On stress collapse in adiabatic shear bands[END_REF][START_REF] Estep | Analysis of shear layers in a fluid with temperature-dependent viscosity[END_REF][START_REF] Th | Adaptive finite element computations of shear band formation Math[END_REF] indicate that instability is widespread in the parameter regime -α + n < 0 and is typically followed by the development of shear bands.

The behavior in the "instability domain" is at present poorly understood, especially regarding the precise mechanism of formation of the shear band. The reason is the response in this regime results from the competition between an ill-posed problem and a parabolic regularization, and little mathematical theory is available to study such effects. Nevertheless, an asympotic theory similar to the Chapman-Enskog expansion is used by Katsaounis-Tzavaras [START_REF] Th | Effective equations for localization and shear band formation[END_REF][START_REF] Th | Localization and shear bands in high strain-rate plasticity[END_REF] to calculate an effective equation for the compound effect of thermal softening and strain-rate sensitivity. The effective equation changes type from forward to backward parabolic across the instability threshold, and thus provides an asymptotic criterion for the onset of localization. In the present work, we will develop a mathematical framework for studying the competition and for examining the conjecture of [START_REF] Th | Effective equations for localization and shear band formation[END_REF]. Our analysis is facilitated by the use of the exponential law (H exp ) inducing special scaling properties on the resulting system of partial differential equations.

Hadamard instability.

In the mechanics literature shear localization is typically associated with Hadamard-instability. This is easily put on display for (2.1), by considering the limiting case n = 0 and κ = 0. In the parlance of mechanics, this corresponds to rate-insensitive deformations and absence of heat conduction, and leads to the system of conservation laws (2.12)

∂ t v θ = (µ(θ)) x µ(θ)v x = 0 µ (θ) µ(θ) 0 ∂ x v θ
For µ (θ) < 0 the wave speeds are both imaginary, λ ± = ±ı |µ |µ, and the linearized initial value problem for (2.12) is ill-posed. Two questions then emerge:

(i) To assess the combined effect of the competition between Hadamard instability and momentum diffusion and/or heat conduction (as manifested by small n > 0 and κ > 0 respectively). (ii) While Hadamard instability indicates the catastrophic growth of oscillations around a mean state, it does not by itself explain the formation of coherent structures typically observed in localization. The latter is an outcome of the nonlinear effects and has to be assessed at the level of the nonlinear problem.

2.4. Formulation via relative perturbations. The first objective is to develop quantitative criteria for the stability of the uniform shearing solutions. Since the basic solutions are time dependent an issue arises how to define their stability. Following [START_REF] Molinari | Analytical characterization of shear localization in thermoviscoplastic materials[END_REF][START_REF] Tzavaras | Nonlinear analysis techniques for shear band formation at high strain rates[END_REF], we call the uniform shearing solution asymptotically stable if perturbations of the solution decay faster than the basic solution, and we call it unstable if they grow faster than the uniform shearing solution.

It is expedient to introduce a relative perturbation formulation of the problem: For the exponential law (H exp ), we introduce the state (U, Θ, Σ) describing the relative perturbation and defined through the transformation

(2.13) u(x, t) = U (x, t) θ(x, t) = θ s (t) + Θ(x, t) σ(x, t) = σ s (t)Σ(x, t)
where (θ s , σ s ) is given by (2.10) and satisfies in particular that

(2.14) θs = σ s = e -αθs .
Using (2.1) and (2.14) we derive the system satisfied by the relative perturbation (U, Θ, Σ) in the form (2.15)

U t = σ s (t) Σ xx Θ t -κΘ xx = σ s (t) Σ U -1 Σ = e -αΘ U n
Note that the uniform shearing solution is mapped to the state U 0 = 1, Θ 0 = 0 and Σ 0 = 1, and the latter is an equilibrium for the system (2.15). The problem then becomes to study the stability of the equilibrium (U 0 , Θ 0 , Σ 0 ) = (1, 0, 1) for the non-autonomous system (2.15).

Linearized analysis of the exponential model

In this section we present an analysis of the linearized system. Since the uniform shearing solution is time-dependent the linearized analysis will seek to classify the precise growth in-time of perturbations.

3.1. The linearized problem. We compute the linearized problem, using the ansatz (3.1)

U = 1 + δ ū + O(δ 2 ) Θ = δ θ + O(δ 2 ) Σ = 1 + δσ + O(δ 2 )
where δ a small parameter measuring the size of the perturbation and δ 1. Expansion of the equation (2.15) 3 gives to the leading order of δ

1 + δσ = 1 + (-α θ + nū)δ + O(δ 2 )
while linearizing the other two equations gives

ūt = σ s (t) σxx + O(δ 2 ) θt -κ θxx = σ s (t) (σ + ū) + O(δ 2 ) .
Collecting the equations together and neglecting the terms of O(δ 2 ) we obtain the linearized system:

(3.2)        ūt = σ s (t) nū xx -α θxx θt -κ θxx = σ s (t) (n + 1)ū -α θ σ = nū -α θ
The boundary conditions (2.4) and (2.5) and the transformations (2.13) and (3.1), imply that the induced boundary conditions for the linearized system are

(3.3) ūx (0, t) = ūx (1, t) = 0 , θx (0.t) = θx (1, t) = 0 .
In view of the transformations (2.15) and (3.1), the relation between the original solution (u, σ, θ) and the solution of the linear problem (3.2) is as follows

(3.4) u = v x = 1 + δ ū(x, t) + O(δ 2 ) θ = 1 α log(αt + c 0 ) + δ θ(x, t) + O(δ 2 ) σ = 1 αt + c 0 1 + δσ(x, t) + O(δ 2 )
3.2. A change of time scale. Our analysis of the linearized equation is facilitated by introducing the transformation of variables

(3.5) ū(x, t) = û(x, τ (t)) θ(x, t) = θ(x, τ (t)) σ(x, t) = σ(x, τ (t))
where τ (t) is a rescaling of time, defined by

   dτ dt = σ s (t) = 1 αt + c 0 τ (0) = 0 =⇒ τ (t) = 1 α log c 0 + αt c 0 .
The map τ (t) is invertible and its inverse is given by

t = t(τ ) = c 0 α (e ατ -1) .
One computes that (û(x, τ ), θ(x, τ ), σ(x, τ )) satisfy the linear system

(3.6)              ∂ û ∂τ = nû xx -α θxx ∂ θ ∂τ -κc 0 e ατ θxx = (n + 1)û -α θ σ = nû -α θ with boundary conditions (3.7) ûx (0, t) = ûx (1, t) = 0 , θx (0.t) = θx (1, t) = 0 .
For further reference, we record that the original variables (u, σ, θ) and the solution (û, σ, θ) of the rescaled problem (3.6)-(3.7) are related through the formulas (3.8)

u = v x = 1 + δ û(x, τ (t)) + O(δ 2 ) θ = 1 α log(αt + c 0 ) + δ θ(x, τ (t)) + O(δ 2 ) σ = 1 αt + c 0 1 + δσ(x, τ (t)) + O(δ 2 )
3.3. Linearized analysis with frozen coefficients. We next consider the system (3.9)

       ∂ û ∂τ = nû xx -α θxx ∂ θ ∂τ -k θxx = (n + 1)û -α θ
with boundary conditions (3.7). Note that in (3.6) the thermal diffusion coefficient k = k(τ ) = κc 0 e ατ is time dependent, while in (3.9) this coefficient is frozen in time. We analyze the solutions of (3.9) via an eigenvalue analysis. This provides the following information:

(a) For k = 0 it provides the stability or instability properties of the linearized system (3.6). (b) For k > 0 the eigenvalue analysis will only offer an indication of the effect of thermal diffusion on the linearized analysis. This will be complemented by an energy estimate analysis of the exact system (3.2) in the following section.

The solutions of (3.9)-(3.7) are expressed as a cosine Fourier series

(3.10) û(x, τ ) = ∞ j=0 ûj (τ ) cos jπx θ(x, t) = ∞ j=0 θj (τ ) cos jπx
where the Fourier coefficients (û j (τ ), θj (τ )) satisfy the ordinary differential equations (3.11)

d dt ûj θj = 0 0 0 -k(jπ) 2 ûj θj + -n(jπ) 2 α(jπ) 2 n + 1 -α ûj θj = -n(jπ) 2 α(jπ) 2 n + 1 -α -k(jπ) 2 ûj θj
The eigenvalues of the matrix

A := -n(jπ) 2 α(jπ) 2 n + 1 -α -k(jπ) 2
are the roots of the binomial equation (3.12)

λ 2 + λ α + n(jπ) 2 + k(jπ) 2 + nk(jπ) 4 -α(jπ) 2 = 0 , j = 0, 1, 2, ...
The mode j = 0. For j = 0 the equation becomes λ 2 + αλ = 0 and has two eigenvalues λ 0,0 = 0 and λ -,0 = -α. This corresponds to marginal stability for the linearized system. A closer look shows that the system (3.11) for j = 0 takes the form

dû 0 dτ = 0 d θ0 dτ = (n + 1)û 0 -α θ0
This implies, using (3.10), (3.4), (2.2) and the boundary condition (2.3), that

û0 (τ ) = û0 (0) = 1 0 ū(x, 0)dx = 0.
and clearly θ0 (τ ) decays. Thus, the 0-th mode decays.

The modes j = 1, 2, .... For these modes the eigenvalues are

(3.13) λ ±,j = -1 2 α + n(jπ) 2 + k(jπ) 2 ± 1 2 ∆ j
where the discriminant

∆ j := (n + k)(jπ) 2 + α 2 + 4α(jπ) 2 -4nk(jπ) 4 = (n -k) 2 (jπ) 4 + 2α(n + k) + 4α (jπ) 2 + α 2 > 0 Since λ j,+ + λ j,-= -(n + k)(jπ) 2 -α < 0 , λ j,+ λ j,-= nk(jπ) 4 -α(jπ) 2 we conclude that (3.14) the j-th mode is asymptotically stable iff nk(jπ) 2 > α the j-th mode is unstable iff nk(jπ) 2 < α
In summary: (i) If k = 0 then all modes j = 1, 2, ... are unstable.

(ii) If k > 0 the first few modes might be unstable, that is the modes j for which nk(jπ) 2 < α, but the high frequency modes are stable. (iii) The higher k the fewer the number of unstable modes. For k larger than a certain threshold (here equal to α/(nπ 2 )) all modes are asymptotically stable.

3.4. Asymptotics of the modes and the nature of the instability. We analyze the behavior of the linearized modes as a function of j and discuss the nature of the instability in the various special cases of interest. The analysis reveals the role of the parameters n, k regarding the nature of the instability 3.4.1. Case 1: k = 0, n = 0, Hadamard instability. In this regime

λ ±,j = - α 2 ± 1 2 α 2 + 4α(jπ) 2
one eigenvalue is negative and the other strictly positive and increasing with j. Using the Taylor series development

(3.15) √ 1 + x = 1 + 1 2 x -1 4 x 2 + 3 8 x 3 + ... , for 0 < x 1
we obtain the following asymptotic developments for the eigenvalues

(3.16) λ -,j = - √ αjπ -α 2 -1 8 α 3 2 jπ + O( 1 j 3 ) λ +,j = √ αjπ -α 2 + 1 8 α 3 2 jπ + O( 1 j 3 ) .
Referring to the formulas (3.8), (3.5) and (3.16), we see that the j-th mode of the perturbation grows to the leading order like θj (x, t) = cos(jπx) e λ +,j τ (t) ∼ cos(jπx) exp

√ αjπ 1 α log c 0 + αt c 0 = cos(jπx) c 0 + αt c 0 jπ √ α
Clearly the perturbation grows much faster than the uniform shearing solution θ s (t) given by (2.10) where the growth is logarithmic, and the behavior is characteristic of Hadamard instability, that is the high frequency oscillations grow at a catastrophic rate.

3.4.2. Case 2: k = 0, n > 0, Turing instability. In this case the eigenvalues of the j-th mode are given by the formulas

λ ±,j = - α + n(jπ) 2 2 ± 1 2 (α + n(jπ) 2 ) 1 + 4α(jπ) 2 (α + n(jπ) 2 ) 2
Again we have one negative and one positive egeivalue and thus unstable response in this regime as well. Using once again the Taylor expansion (3.15) we obtan the easymptotic expansions (3.17)

λ -,j = -n(jπ) 2 -α - α(jπ) 2 α + n(jπ) 2 + O( 1 j 2 ) λ +,j = α(jπ) 2 α + n(jπ) 2 + O( 1 j 2 ) .
for the positive and negative eigenvalues respectively.

Lemma 3.1. The positive egenvalue λ +,j satisfies the following properties:

λ +,j is increasing in j (3.18)

λ +,j < α n (3.19) λ +,j → α n as j → ∞ (3.20)
Proof. Set x = (jπ) 2 and recall that

λ + (x) = - α + nx 2 + 1 2 (nx + α) 2 + 4αx
satisfies the identity

(3.21) λ 2 + (x) + λ + (x)(nx + α) -αx = 0 . Differentiating (3.21), we derive ∂λ + ∂x (x) = α -λ + (x)n 2λ + (x) + (nx + α) = α -λ + (x)n (nx + α) 2 + 4αx .
Next, using the elementary bound √ 1 + z < 1 + 1 2 z for z > 0 we obtain

λ + (x) = - α + nx 2 + 1 2 (nx + α) 1 + 4αx (nx + α) 2 < αx nx + α < α n
This shows that ∂λ + ∂x (x) > 0 and provides (3.18) and (3.19). Then (3.20) follows from (3.17).

The following remarks are in order: Observe first that, by virtue of (3.8), (3.5) and (3.16), the j-th mode of the perturbation grows like

(3.22) θj (x, t) = cos(jπx) e λ +,j τ (t) = cos(jπx) exp α(jπ) 2 α + n(jπ) 2 + O( 1 j 2 ) 1 α log c 0 + αt c 0 ∼ cos(jπx) c 0 + αt c 0 1 n
Clearly the perturbation grows much faster than the uniform shearing solution θ s (t), but the rate of growth is bounded in j. On the other hand, the growth becomes very fast as the rate sensitivity n → 0. The behavior in this regime is characterized by Turing instability. Recall, that Turing instability [START_REF] Turing | The chemical basis of morphogenesis Phil[END_REF] corresponds to the following scenario that occurs in some instances of dynamical systems. There are two dynamical systems ẋ = Ax , ẋ = Bx where the spectrum of both matrices A and B lies in the left-hand plane, but the Trotter product of the two systems ẋ = (A + B)x exhibits unstable response.

Indeed, the dynamical system (3.11) with k = 0 may be visualized as the Trotter product of the two systems

(3.23) d dt ûj θj = -1 2 n(jπ) 2 α(jπ) 2 0 -1 2 α ûj θj and (3.24) d dt ûj θj = -1 2 n(jπ) 2 0 n + 1 -1 2 α ûj θj
Each of (3.23) and (3.24) has spectrum in the left half space but the eigenvalues of (3.11) have unstable response.

3.4.3. Case 3: k > 0, n > 0, thermal diffusion stabilizes the high modes. In this regime the eigenvalues are given by (3.13) and they are real and have the properties

λ -,j < 0 ∀j (3.25) λ +,j < 0 ⇐⇒ k(jπ) 2 > α n (3.26)
Therefore, the high modes are stable, but the low modes might be unstable if k is sufficiently small. The "most unstable" mode is the first mode.

One can again carry out the large j asymptotics of the eigenvalues. The result, in the case that n > k, reads

(3.27) λ -,j = -n(jπ) 2 -α - α(k + 1) n -k + α (jπ) 2 + O( 1 j 2 ) , λ +,j = -k(jπ) 2 + α(k + 1) n -k + α (jπ) 2 + O( 1 j 2 ) .
A similar formula holds when n < k.

3.5.

Effect of thermal diffusion on the linearized problem. Consider now the linearized system (3.2) with boundary condition (3.3). Since we are interested in perturbations of the uniform shearing solution we will consider initial data that satisfy 1 0 u 0 dx = 0 and thus focus on solutions with (3.28)

1 0 u(x, t)dx = 0
The objective is now to analyze the effect of the time-dependent diffusion on the dynamics of the system. The linearized analysis of the system with frozen coefficients suggests to expect that diffusion stabilizes the system and its effect is intensified with the passage of time. Indeed, here we provide a rigorous proof of this statement via energy estimates.

Lemma 3.2.

There exist A > 0 and T > 0 such that

(3.29) 1 0 A 2 ū2 + 1 2 θ2 (x, t) dx ≤ 1 0 A 2 ū2 + 1 2 θ2 (x, T ) dx for t ≥ T and (3.30) 1 0 A 2 ū2 + 1 2 θ2 (x, t)dx → 0 as t → ∞.
Proof. We multiply (3.2) 1 by ū and (3.2) 2 by θ and obtain the identities ū2 dx + n 2 σ s (t)

d dt 1 0 1 2 ū2 dx + nσ s (t) 1 0 ū2 x dx = ασ s (t) 1 0 θx ūx dx d dt 1 0 1 2 θ2 dx + κ 1 0 θ2 x dx + ασ s (t) 1 0 θ2 dx = (n + 1)σ s (t)
1 0 ū2 x dx ≤ α 2 2n σ s (t) 1 0 θ2 x dx d dt 1 0 1 2 θ2 dx + κ 1 0 θ2 x dx + α 2 σ s (t) 1 0 θ2 dx ≤ (n+1) 2 2α σ s (t) 1 0 ū2 dx .
In view of (3.28), we may use the Poincare inequality (3.32)

1 0 ū2 dx ≤ C p 1 0 ū2 x dx .
Combining the above, we note that for A a positive constant we have

d dt 1 0 A 2 ū2 + 1 2 θ2 dx + An 2Cp σ s (t) 1 0 ū2 dx + α 2 σ s (t) 1 0 θ2 dx + κ 1 0 θ2 x dx ≤ Aα 2 2n σ s (t) 1 0 θ2 x dx + (n+1) 2 2α σ s (t) 1 0 ū2 dx .
Next we select A such that An 2Cp ≥ (n+1) 2 α , and deduce that for A thus selected there is a choice c = c(A) > 0 such that d dt

1 0 A 2 ū2 + 1 2 θ2 dx + cσ s (t) 1 0 A 2 ū2 + 1 2 θ2 dx + κ 1 0 θ2 x dx ≤ Aα 2 2n σ s (t) 1 0 θ2 x dx .
Select now T such that Aα 2 2n σ s (t) < κ for t ≥ T (recall from (1.2) that σ s (t) → 0 when t → ∞). On the interval t ≥ T , we have the differential inequality

d dt 1 0 A 2 ū2 + 1 2 θ2 dx + cσ s (t) 1 0 A 2 ū2 + 1 2 θ2 dx ≤ 0 .
The latter implies (3.29) and yields, upon using Gronwall's inequality and (2.10),

1 0 A 2 ū2 + 1 2 θ2 (x, t) dx ≤ 1 0 A 2 ū2 + 1 2 θ2 (x, T ) dx exp -c t T σ s (z)dz → 0 , as t → ∞ ,

and concludes the proof

The next lemma provides control of the intermediate times.

Lemma 3.3. There exist B > 0 and a constant C B > 0 such that

(3.33) 1 0 1 2 ū2 + B 2 θ2 (x, t) dx ≤ 1 0 1 2 ū2 + B 2 θ2 (x, 0) dx exp{C B t} for t ≥ 0.
Proof. Using again (3.31) and (3.32) we have for any B > 0

d dt 1 0 1 2 ū2 + B 2 θ2 dx + n 2Cp σ s (t) 1 0 ū2 dx + Bα 2 σ s (t) 1 0 θ2 dx + Bκ 1 0 θ2 x dx ≤ α 2 2n σ s (t) 1 0 θ2 x dx + B (n+1) 2 2α σ s (t) 1 0 ū2 dx
Select now B large so that Bκ > α 2 2n σ s (t) for all t ≥ 0, and then we obtain d dt

1 0 1 2 ū2 + B 2 θ2 dx ≤ C B 1 0 1 2 ū2 + B 2 θ2 dx
from where (3.33) follows via Gronwall's inequality.

We conclude now the analysis of stability. Suppose that for some δ > 0 the initial data satisfy

1 0 1 2 ū2 + B 2 θ2 (x, t) dx ≤ δe C B t for t ≥ 0 1 0 A 2 ū2 + 1 2 θ2 (x, t) dx ≤ Aδe C B T for t ≥ T
Together they imply that the state (0, 0) is stable in L 2 , and in fact on account of (3.30) it is asymptotically stable. We state the result in the following: Proposition 3.4. For the initial-boundary value problem (3.2), (3.3) with κ > 0, if the initial perturbation (ū 0 , θ0 ) satisfies 1 0 ū0 dx = 0 then the equilibrium (0, 0) is asymptotically stable in L 2 . The proof of the above lemmas and the analysis of the linearized problem with frozen coefficients indicate that the perturbation of the base solution may drift far from the origin until eventually it is recaptured as the effect of the diffusion intensifies with time.

This phenomenon of metastability also appears in numerical runs for the nonlinear problem; an instance of these runs appears in Figure 2. We solve numerically the rescaled system (4.2) for Figure 2. Metastability for the nonlinear problem : v(x, t) (left), θ(x, t) (right). κ = 0.5, α = 0.5, n = 0.05. The initial conditions are perturbations of the uniform shearing solutions and they are : v(x, 0) = x and θ(x, 0) = 1 + δ(x), where δ(x) is a small Gaussian perturbation centered at the middle of the interval. The numerical method is based on adaptivity in space and time. An adaptive finite element method is used for the spatial discretisation while a Runge-Kutta method with variable time step is used for the discretisation in time. Further details concerning the numerical scheme and the adaptivity criteria can be found in [START_REF] Th | Adaptive finite element computations of shear band formation Math[END_REF]. In Figure 2 the evolution in time is presented for the original variables v and θ in (4.1). We notice that, after a transient period of localization, thermal diffusion dominates the process, inhomogeneities of the state variables dissipate, and eventually the solution approaches a uniform shearing solution (2.8).

Localization -Part I

We consider next the problem (2.1) on the whole real line and put aside the boundary conditions. The objective is to construct special solutions that capture a process of localization. This is done over three sections. The present one is preliminary in nature and discusses various properties that make the construction feasible and the setting of the problem. The main step is performed in section 5 where a special class of self-similar solutions are constructed. The localized solutions are then presented in section 6.

4.1.

Relative perturbations and a change of time scale. We consider (2.1) on R × (0, ∞) and recall the notation u = v x . The uniform shearing solutions (2.8) are defined by solving (2.9). Introduce the transformation (4.1)

u(x, t) = Û (x, τ (t)) θ(x, t) = θ s (t) + Θ(x, τ (t)) σ(x, t) = σ s (t) Σ(x, τ (t))
which involves a passage to relative perturbations and a change of time scale from the original to a rescaled time τ (t). The scaling τ (t) is defined by

τ (t) = 1 α log c 0 + αt c 0 = θ s (t) -θ s (0) , it satisfies    dτ dt = dθ s dt = σ s (t) = e -αθs τ (0) = 0
where θ s (t) is as in (2.10), c 0 = e αθ 0 . The scaling τ (t) satisfies τ (t) → ∞ as t → ∞, and τ (t) is invertible with inverse map given by t = t(τ ) = c 0 α (e ατ -1) .

The functions ( Û (x, τ ), Θ(x, τ ), Σ(x, τ )) satisfy the system of partial differential equations (4.2)

Ûτ = Σxx Θτ -κc 0 e ατ Θxx = Σ Û -1 Σ = e -α Θ Û n
We emphasize that • For κ > 0 the system is non-autonomous, which is natural in view of the nature of the transformation (4.1). • Notably, for κ = 0 the system (4.2) becomes autonomous.

4.2.

Effective equation for the adiabatic system. Next, we consider the adiabatic case κ = 0 where (4.2) takes the form

(4.3) Ûτ = Σxx Θτ = Σ Û -1 Σ = e -α Θ Û n .
To understand the long time response of (4.3) we perform a formal asymptotic analysis following ideas from [START_REF] Th | Effective equations for localization and shear band formation[END_REF]. Fix an observational time scale T and introduce the change of variables

Û (x, τ ) = ŨT x √ T , τ T Σ(x, τ ) = ΣT x √ T , τ T Θ(x, τ ) = ΘT x √ T , τ T
One easily checks that ( ŨT (y, s), ΣT (y, s), ΘT (y, s)) satisfies the system (4.4)

Ũs = Σyy 1 T Θs = Σ Ũ -1 Σ = e -α Θ Ũ n
We are interested in calculating the equation describing the effective response of (4.4) for T sufficiently large. This asymptotic analysis involves a parabolic equation (4.4) 1 , which can be thought as a moment equation, and equations (4.4) 2 and (4.4) 3 , which may be thought as describing a relaxation process towards the equilibrium curve described by Σ Ũ -

1 = 0 Σ = e -α Θ Ũ n ⇐⇒ Σ = 1 Ũ , Θ = n + 1 α log Ũ .
Note that the equilibria are a one-dimensional curve parametrized by Ũ .

To calculate the effective equation we perform a procedure analogous to the Chapman-Enskog expansion of kinetic theory. We refer to [START_REF] Th | Localization and shear bands in high strain-rate plasticity[END_REF]Sec 5] for the details of the asymptotic analysis. The prodcedure produces the following effective equation (4.5)

u s = ∂ yy 1 u + 1 T n+1 α ∂ yy 1 u 2 ∂ yy 1 u ,
which captures the effective response up to order O( 1 T 2 ). The leading term is backward parabolic and the first order correction is fourth order. One checks that the effect of the fourt order term is to stabilize the equation, in the sense that the linearized equation around the equilibrium u 0 = 1 is stable (see [START_REF] Th | Localization and shear bands in high strain-rate plasticity[END_REF] for the details).

It was further shown in [START_REF] Th | Localization and shear bands in high strain-rate plasticity[END_REF] that (4.5) admits special solutions that exhibit localization, which is not surprising as the leading response in (4.5) is backward parabolic. Localized solutions of a similar nature will be constructed here directly for the system (4.3). This is a more "unexpected" behavior, because (4.3) is at least pro-forma a "hyperbolic-parabolic" system, and the explanation for this behavior lies precisely in the asymptotic relation between (4.3) and (4.5).

4.3.

Invariance properties and a related ansatz. It is easy to check that for κ = 0 the system (4.2) is invariant under the one-parameter scaling transformation

(4.6) Ûa (x, τ ) = a Û (ax, τ ) Σa (x, τ ) = 1 a Σ(ax, τ ) Θa (x, τ ) = n + 1 α log a + Θ(ax, τ )
for any a > 0. It is also easy to check that this is the only scaling transformation under which the problem is invariant. This transformation has the distinctive feature that the scaling of space is independent of the scaling of time. Finally, the system (4.2) is not scaling invariant when the thermal diffusion κ is positive. The scaling invariance (4.6) motivates a change of variables and a related ansatz of solutions. Introduce a variable in time scaling r(τ ), the variable ξ = x r(τ ) , and the change of variables

(4.7) Û (x, τ ) = 1 r(τ ) Ū x r(τ ) , τ Σ(x, τ ) = r(τ ) Σ x r(τ ) , τ Θ(x, τ ) = - n + 1 α log r(τ ) + Θ x r(τ ) , τ
We denote by ṙ = d dτ r(τ ) and by = d dξ . Introducing (4.7) to (4.2), we obtain that the function ( Ū (ξ, τ ), Σ(ξ, τ ), Θ(ξ, τ )) satisfies the system of partial differential equations

(4.8) Ūτ - ṙ r ξ Ū ξ = Σξξ Θτ - ṙ r n+1 α + ξ Θξ -κc 0 e ατ r 2 (τ ) Θξξ = Σ Ū -1 Σ = e -α Θ Ū n
The resulting system (4.8) depends on the choice of the scaling function r(τ ). We will be interested in solutions that are steady states in the new variables. Of course such solutions are dynamically evolving in the original variables. There are two case that produce simplified results of interest:

(a) Consider the case κ > 0, and select r(τ ) = e 1 2 ατ . Then the system (4.8) is consistent with the ansatz of steady state solutions ( Ū (ξ), Σ(ξ), Θ(ξ)) and the latter are constructed by solving the system of ordinary differential equations (4.9)

- α 2 ξ Ū = Σ - α 2 n+1 α + ξ Θ -κc 0 Θ = Σ Ū -1 Σ = e -α Θ Ū n
Since r(τ ) → ∞ as τ increases, solutions of (4.9) will produce defocusing solutions for (4.8). A study of (4.9) is expected to give information about the convergence to the uniform shearing state in the non adiabatic setting. This is not the focus of the present work and we will not deal further with the system (4.9). (b) Instead we consider the case κ = 0. We may now select any exponential function for r(τ ) but are most interested in r(τ ) = e -λτ with λ > 0 as this produces focusing solutions. With this selection, ( Ū (ξ), Σ(ξ), Θ(ξ)) is a steady solution for (4.8) with κ = 0 provided it satisfies (

Σ = λ ξ Ū λ n+1 α + ξ Θ = Σ Ū -1 Σ = e -α Θ Ū n 4.10)
where λ > 0 is a parameter.

Remark 4.1. Note that:

• The form of the transformation (4.7) is motivated by the scaling invariance (4.6).

• The scaling connects the two variables and produces interesting systems even in the case κ positive when the system (4.2) is not scale invariant.

The auxiliary problem

In this section we consider the system of ordinary differential equations (5.1)

Σ = ξU ν n+1 α + ξΘ = Σ U -1 Σ = e -αΘ U n ,
on the domain 0 < ξ < ∞ with ν > 0, subject to the initial condition

(5.2) Σ(0) = Σ 0 > 0.
This is an auxiliary problem that will be studied in detail in this section. We emphasize that it does not coincide with the integrated form of (4.10), however it will be used together with additional properties in order to construct solutions of (4.10) and localizing solutions for (4.3) in the following section.

In this section, we are interested in constructing solutions (U (ξ), Σ(ξ), Θ(ξ)) for the singular initial value problem (5.1), (5.2) for any ν > 0. The following elementary remarks are in order: (a) When ν = 0, (5.1) simplifies to

(5.3) Σ = ξU Σ(0) = Σ 0    Σ U = 1 Θ = n + 1 α log U
The solution of (5.3) is computed explicitly by (5.4)

Σ(ξ) = ξ 2 + Σ 2 0 U (ξ) = 1 Σ(ξ) = 1 ξ 2 + Σ 2 0 Θ(ξ) = - n + 1 α log ξ 2 + Σ 2 0 .
(b) When ν > 0, the system (5.1) is invariant under the family of scaling transformations (5.5)

U b (ξ) = bU (bξ) Σ b (ξ) = 1 b Σ(bξ) Θ b (ξ) = n + 1 α log b + Θ(bξ)
for b > 0. There is a special solution of (5.1) that is self-similar with respect to the scaling invariance (

Σ(ξ) = ξ , Û (ξ) = 1 ξ , Θ(ξ) = -n+1 α log ξ . 5.5) and reads (5.6)
This solution does not however satisfy the initial condition (5.2).

We next construct a function (U (ξ), Σ(ξ), Θ(ξ)) which satisfies (5.1), (5.2) for all values of ν > 0 and Σ 0 > 0. We will show that the constructed solution will have the inner behavior of (5.4) and the outer behavior of (5.6).

Theorem 5.1. Let Σ 0 > 0 be fixed. Let us note c ν = 1 + ν(n + 1)/α. Given ν > 0 there exists a solution (Σ ν , U ν , Θ ν ) of (5.1), (5.2) defined for ξ ∈ (0, ∞) and satisfying the properties:

(i) It achieves the initial data (5.7) lim ξ→0   Σ(ξ) U (ξ) Θ(ξ)   =   Σ 0 U 0 Θ 0   ,
where Σ 0 as in (5.2) and U 0 , Θ 0 are defined via

(5.8) U 0 Σ 0 = c ν Θ 0 = n+1 α log U 0 -1 α log c ν , (ii)
As ξ → ∞ it has the limiting behavior (5.9)

U (ξ) = 1 ξ + o 1 ξ Σ(ξ) = ξ + o(ξ) Θ(ξ) = -n+1 α log ξ + o(1) (iii) The solution has the regularity U , Θ ∈ C 1 [0, ∞), Σ ∈ C 2 ([0, ∞)). It satisfies (5.10) dΣ dξ (0) = dU dξ (0) = dΘ dξ (0) = 0 ,
and has the Taylor expansion as ξ → 0, (

U (ξ) = U 0 + o(ξ) Σ(ξ) = Σ 0 + 1 2 U 0 ξ 2 + o(ξ 2 ) Θ(ξ) = Θ 0 + o(ξ) , 5.11)
where as usual lim ξ→0 1 ξ o(ξ) = 0 (but the rate may degenerate as ν tends to 0).

Proof. Step 1. Motivated by the special solution (5.6) we introduce a change of variables in two steps:

U (ξ) = 1 ξ ū(ξ) = 1 ξ u(log ξ) Σ(ξ) = ξσ(ξ) = ξσ(log ξ) (5.12) Θ(ξ) = -n+1 α log ξ + θ(ξ) = -n+1 α log ξ + θ(log ξ)
Indeed, we easily check that (ū, σ, θ)(ξ) satisfies the system of equations (5.13)

ξ σ = ū - σ νξ θ = σ ū -1 σ = e -α θ ūn .
In the second step, we introduce in (5.13) the change of independent variable η = log ξ and derive that (u, σ, θ)(η) satisfy (5.14)

dσ dη = u -σ ν dθ dη = σ u -1 σ = e -αθ u n .
The system (5.14) is far simpler than the original (5.1) in that it is autonomous and non singular.

Step 2. We next consider (5.14) and perform a further simplification, via the change of dependent variables Note that u = a b . We now compute

da dη = - 1 σ 2 dσ dη = - 1 σ 2 σ 1 n e α n θ -σ = a - a 3 b
and

db dη = -1 + 1 n b 1 σ dσ dη -α n b dθ dη = -n+1 n b 1 σ 1 σb -σ -α νn b 1 b -1 = α νn 1 + ν n+1 α b -1 -n+1 α νa 2
In summary, we conclude that (a, b)(η) satisfies the autonomous system of ordinary differential equations (5.16)

d dη a b = a 1 -a 2 b α νn c ν b -1 -n+1 α νa 2 =: F (a, b)
where (5.17)

c ν := 1 + ν n+1 > 1 .
There are two advantages in (5.16) relative to (5.14): (i) (5.16) is expressed via only two independent variables (a, b).

(ii) The equilibrium at infinity in (5.14) is pulled at the axis for (5.16) via the transformation (5.15).

Step 3. We perform a phase space analysis of (5.16) and establish the existence of a heteroclinic connection. We show: Proposition 5.2. Consider the autonomous planar system of differential equations (5.16) with c ν defined by (5.17). For any ν > 0 the system has the following properties:

• F has two equilibria P = (0, 1/c ν ), Q = (1, 1) in the first quadrant. Both P and Q are hyperbolic points, P is a repelling node while Q is a saddle point. • There exists a heteroclinic orbit connecting P to Q. At the equilibrium P we have ∇F 0, 1 cν = 1 0 0 α nν c ν and we have eigenvalues λ 1 = 1 with eigenvector r 1 = (1, 0) T , and λ 2 = α nν c ν > 0 with eigenvector r 2 = (0, 1) T . In view of (5.17), we have λ 2 > 1. The local behavior of solutions of (5.16) around the equilibrium P is that of a repelling node. Standard phase plane theory for second order systems (e.g. [START_REF] Ph | Ordinary differential equations[END_REF]) states that there is a small neighborhood of the equilibrium P such that for any given solution (a(η), b(η)) of (5.16) there are constants κ 1 , κ 2 such that (5.19) a(η) b(η) = 0 3. We will show that R is negatively invariant.

1 cν + κ 1 1 0 e λ 1 η + κ 2 0 1 e λ 2 η
= 0 • l 1 is the horizontal line b -1 = 0 • l 2 is the parabola b -a 2 = 0 The region R is expressed as R = (a, b) : a 2 ≤ b ≤ 1 , 0 ≤ a ≤ 1 see Figure
Indeed, the flow of (5.16) is parallel to l 0 as indicated in (b). On the horizontal line l 1 ∩R the system (5.16) gives

da dη = a(1 -a 2 ) > 0 , db dη = n+1 n (1 -a 2 ) > 0 .
On the curve l 2 ∩ R the system (5.16) gives da dη = 0 , db dη = α νn (a 2 -1) < 0 .

Therefore R is negatively invariant. (f) For completeness we also show that the vector r -associated to the eigenvalue λ -at Q points towards the interior of the region R. Recall that Q is the upper right corner of the region R and that the tangent vector to l 1 is τ 1 = (1, 0) T the tangent vector to l 2 at Q = (1, 1) is τ 2 = (1, 2) T the eigenvector r -= (1, 2 + λ -) T . We will show that (5.21) 0

< 2 + λ -< 2 , ∀ν > 0 .
This then implies that the vector -(r -) when placed at the corner Q points strictly towards the interior of R.

To show (5.21) recall that λ -< 0 and use (5.20) and (5.17 where

I := ( α nν c ν -2) 2 + 8 α nν = ( α nν c ν ) 2 + 4 + 4 α nν -4 n+1 n < ( α nν c ν ) 2 + 4 + 4 α nν < ( α nν c ν + 2)
2 This shows λ -+ 2 > 0 and concludes (5.21). (g) It is easy to check that there can be no closed limit cycle in R because in that region

da dη = a b (b -a 2 ) > 0 .
The Poincaré-Bendixson theorem implies that there exists a heteroclinic orbit joining (0, 1 cν ) and (1, 1) which emanates along the direction of the stable manifold at (1, 1) and progresses backwards in time towards the node at (0, 1 cν ).

Step 4. We turn now to defining the solution of (5.1), (5.2). Let (A(η), B(η)) stand for a parametrization of the heteroclinic orbit. In view of (5.19) and the fact that λ 2 > λ 1 = 1, there exists a constant κ 1 such that

(5.22) e -η A(η) B(η) - 0 1 cν → κ 1 1 0 as η → -∞ ,
Due to the fact that the vertical axis is a trajectory of its own and thus distinct from the trajectory we are considering, and due to the form of the phase portrait we can assert that κ 1 > 0.

For any η 0 ∈ R the function (A(η + η 0 ), B(η + η 0 )) offers an alternative parametrization of the same heteroclinic orbit. Applying (5.22) to this parametrization yields

(5.23) e -η A(η + η 0 ) B(η + η 0 ) - 0 1 cν → κ 1 e η 0 1 0 as η → -∞ .
Choose η 0 such that κ 1 η 0 = 1 Σ 0 and let (a(η), b(η)) denote the corresponding parametrization of the heteroclinic :

a(η) b(η) := A(η + η 0 ) B(η + η 0 )
The selected parametrization satisfies as η → -∞

(5.24)

a(η) = 1 Σ 0 e η + o(1)e η b(η) = 1 c ν + o(1)e η σ(η) = 1 a(η) = Σ 0 e -η 1 + o(1) = Σ 0 e -η (1 + o(1))
where as usual o(1) → 0 as η → -∞.

Recalling the changes of variables (5.12) and (5.15) we define

U (ξ) = 1 ξ u(log ξ) = 1 ξ a b (log ξ) Σ(ξ) = ξσ(log ξ) = ξ 1 a (log ξ) (5.25) Θ(ξ) = -n+1 α log ξ + θ(log ξ) = -n+1 α log ξ + n+1 α log a (log ξ) -n α log b (log ξ) .
The function (5.25) clearly satisfies the system (5.1) and, since (a(η), b(η)) → (1, 1) as η → ∞, it satisfies

U (ξ) = 1 ξ (1 + o(1)) Σ(ξ) = ξ(1 + o(1)) Θ(ξ) = -n+1 α log ξ + o(1)
as ξ goes to infinity, which shows (5.9).

Step 5. In the last step, we study the behavior near the (apparent) singular point at ξ = 0 and prove that the solution is in fact regular there and satisfies (5.7), (5.10) and (5.11).

Clearly (5.24) and (5.25) 2 imply that Σ(ξ) → Σ 0 as ξ → 0. Let U 0 and Θ 0 be defined by (5.8). Since u(η) = a b (η), (5.24) gives (5.26)

u(η) = a b (η) = c ν Σ 0 e η + o (
(ξ) = -n+1 α log ξ + n+1 α log ξ 1 Σ 0 (1 + o(1)) -n α log 1 c ν + o(1) → n+1 α log 1 Σ 0 -n α log 1 c ν = Θ 0 as ξ → 0.
and together with (5.26) and (5.25) provide (5.7).

Observe now that

dU dξ (ξ) = 1 ξ 2 du dη -u (log ξ) .
Using (5.16), (5.24) and (5.26), we obtain Turning to Θ, observe that by (5.24) and (5.25)

du dη -u = 1 b da dη - a b 2 db dη - a b = 1 b a(1 - a 2 b ) - a b - a b 2 α νn (c ν b -1) -n+1 n a 2 = 1 n u 3 b -u α νn c ν b b - 1 c ν = 1 n U 3 0 cν e 3η (1 + o( 1
3 θ(η) = n+1 α log a(η) -n α log b(η) = n+1 α log 1 Σ 0 e η (1 + o(1)) -n α log 1 c ν (1 + o(1)e η ) = n+1 α log 1 Σ 0 -n α log 1 c ν + n+1 α log e η + n+1 α log(1 + o(1)) -n α log 1 + o(1)e η = Θ 0 + n+1 α η + n+1 α log(1 + o(1)) -n α log(1 + o(1)) , as η → -∞ , while Θ(ξ) = -n+1 α log ξ + θ(log ξ) = Θ 0 + o(1)
as ξ → 0. In addition,

Θ (ξ) = -n+1 α 1 ξ + 1 ξ dθ dη (log ξ) combined with the formula dθ dη = n+1 α 1 a da dη -n α 1 b db dη = n+1 α 1 - a 2 b -1 ν c ν - 1 b + n+1 α a 2 b = n+1 α -1 ν c ν b b -1 cν gives Θ (ξ) = - 1 ξ c ν b b -1 cν (log ξ) = - 1 ξ c ν b log ξ o(1)ξ = o(1) → 0 as ξ → 0 .
This shows Θ ∈ C 1 [0, ∞) has the Taylor expansion (5.11) 3 at the origin.

Localization -Part II

We are ready to state and prove the main result of the paper. This concerns the adiabatic case κ = 0 of system (2.1), taken now on the whole real line (x, t) ∈ R × R + and expressed in terms of the variable u = v x in the form: (6.1)

u t = σ xx θ t = σu σ = e -αθ u n .
Our analysis is based on the transformations of section 4 and the self-similar solution constructed in section 5. Theorem 6.1. Let θ 0 > 0 and Σ 0 > 0 be given. For any λ > 0, the system (6.1) admits a special class of solutions describing localization of the form:

u(x, t) = α c 0 t + 1 λ α U λ √ λx α c 0 t + 1 λ α (6.2) σ(x, t) = σ s (t) α c 0 t + 1 - λ α Σ λ √ λx α c 0 t + 1 λ α (6.3) θ(x, t) = 1 + λ n+1 α θ s (t) -λ n+1 α θ 0 + Θ λ √ λx α c 0 t + 1 λ α (6.4)
where σ s , θ s are the uniform shear solutions in (2.10), c 0 = e αθ 0 , while (U λ , Σ λ , Θ λ )(ξ) is an even function defined on R, which solves the initial value problem (5.1), (5.2) with ν = λ > 0 for ξ ∈ [0, ∞).

Remark 6.2.

(i) Because α c 0 t + 1 λ α goes to infinity in time, the level curves of ξ(x, t) = √ λx α c 0 t + 1 λ α accumulate along the {x = 0} axis in the (x, t) plane. The functions (6.2), (6.3), (6.4) thus describe a localizing solution where the flow tends to concentrate around x = 0 as the time proceeds. (ii) The solution (6.2), (6.3), (6.4) is defined on R × R + and emanates from initial data The initial data depend on two parameters: The parameter λ which can be thought as a length scale in initial data, and Σ 0 in (5.2) which describes the amplitude of the initial perturbation. (iii) This parametric family of solutions is valid for any λ > 0. If λ is large then the length scale of the initial perturbation is short while the growth in time and the rate of localization for the solution become fast.

Proof. We introduce the changes of variables (4.1) and the stationnary variant of (4.7), which once combined together reads (6.6) u(x, t) = 1 r(τ (t)) Ū

x r(τ (t)) σ(x, t) = σ s (t)r(τ (t)) Σ x r(τ (t)) θ(x, t) = θ s (t) -n + 1 α log r(τ (t)) + Θ x r(τ (t))

.

We select r(τ ) = e -λτ with λ > 0. According to the analysis in section 4.3 the function ( Ū , Σ, Θ)(ξ) is sought as a solution of (4.10) defined on the interval (-∞, ∞).

The desired ( Ū , Σ, Θ)(ξ) is constructed as follows. First, observe the relation between the system (4.10) and the system (5.1). Let Σ 0 be fixed, and let (U λ , Σ λ , Θ λ )(ξ) be the solution of (5.1) with ν = λ and (5.2), defined for ξ ∈ [0, ∞). This solution is well defined by Theorem 5.1. Using a direct computation, the smoothness properties in (iii) of Theorem 5.1, and (5.5) we conclude that we combine (6.6) with (6.7) and (2.10) to arrive at (6.2), (6.3), (6.4).

In Figure 4 the profiles of u(x, t) and θ(x, t) in (6.2) and (6.4), respectively, are drawn at various instances of time from t = 0 to t = 200. To achieve these profiles, we construct numerically the heteroclinic orbit for (5.16) whose existence is justified in Proposition 5.2. The construction is simple, if one exploits the special properties of system (5.16), and proceeds as follows: Initial conditions (a 0 , b 0 ) are selected close to the equilibrium Q = (1, 1) inside the region R and in the direction of the stable manifold of Q for the linearized problem. Then (5.16) is solved backward in time, using an ode solver, and since P = (0, 1/c ν ) is an unstable node (for the forward problem) the resulting orbit is a good approximation of the heteroclinic. For the integration of (5.16) we use an explicit Runge-Kutta predictor-corrector method of order (4,5) with adaptive time stepping. Then Σ 0 is selected by choosing an appropriate parametrisation η 0 following the construction in (5.23).

For the numerical runs the constitutive parameters are n = 0.1, α = 0.5. The parameters related to the solution are θ 0 = 10, λ = 0.1 while for the selected reparametrization of the heteroclinic orbit the value of Σ 0 = 1.88. Once the heteroclinic has been constructed, the profiles U λ , Θ λ are computed via the changes of variables (5.15) and (5.12). Finally, u(x, t) and θ(x, t) are computed by (6.2), (6.4) respectively with θ 0 = 10, and are presented in Fig 4.

The solution presented in Fig 4 evidently exhibits localisation as time increases: a coherent structure forms where the deformation localizes in a narrow band. In this self-similar solution there is no trace of the oscillations, captured by the higher order modes (3.22), that are everywhere present in solutions to the linearized problem (3.9) with k = 0. The nonlinearity supresses the oscillations and a seamless coherent structure emerges.

Figure 1 .

 1 Figure 1. Uniform shear versus shear band.

1 0

 1 u 0 = 0 and are controlled in L 2 by

1 - a 2 b = 0 c ν b - 1 1 -3a 2 b a 3 b 2 - 2 Figure 3 .

 101223 Figure 3. Phase diagram of System (5.16)

1 + 8 α nν 2 ,

 12 + higher order terms as η → -∞. (d) At the point Q = (1, 1) equation (5.18) gives ∇F The eigenvalues λ ± , computed by (5.20) λ ± = ( α nν c ν -2) ± α nν c ν -2 2 are λ -< 0 with eigenvector r -= (1, 2+λ -) T and λ + > 0 with eigenvector r + = (1, 2+λ + ) T . Q is a saddle point. (e) Consider the region R bounded by the curves • l 0 is the b-axis defined by the equation a

  ) to rewrite 2(λ -+ 2) = ( α nν c ν + 2) -√ I

  1)e η = U 0 e η + o(1)e η . Now (5.25), (5.24), (5.26), (5.8) imply Θ

  )) -α νn U 0 o(1)e 2η as η → -∞ and thus(5.27) dU dξ (ξ) ≤ o(1) as ξ → 0. Since U ∈ C[0, 1] ∩ C 1 (0, 1) the mean value theorem implies U (ξ) -U 0 ξ = |U (ξ * )| for some ξ * ∈ (0, ξ),and (5.27) implies U (0) exists and U (0) = 0. We conclude that U ∈ C 1 [0, ∞) has the Taylor expansion (5.11) 1 . In turn, Σ ∈ C 2 [0, ∞), Σ (0) = 0, Σ (0) = 1 2 U 0 and Σ achieves the Taylor expansion (5.11) 2 .

(6. 5 )

 5 u(x, 0) = U λ √ λx σ(x, 0) = σ s (0)Σ λ √ λx θ(x, 0) = θ s (0) + Θ λ √ λx .

  (a) The function ( Ū , Σ, Θ) defined by(6.7) Ū (ξ) = U λ ( √ λξ) , Σ(ξ) = Σ λ ( √ λξ) , Θ(ξ) = Θ λ ( √ λξ) satisfies (4.10) on R + and the condition Σ(0) = Σ 0 . (b) If we extend ( Ū , Σ, Θ) on (-∞, 0) by setting ( Ū , Σ, Θ)(-ξ) = ( Ū , Σ, Θ)(ξ) for ξ < 0, the resulting function Ū , Θ ∈ C 1 (R), Σ ∈ C 2 (R) is a solution of (4.10) defined on the whole real line that is even.Noting that r(τ (t)) = e -λτ (t) = α c 0 t + 1λ α

  ,t), α =0.5, n=0.1, λ =0.1 ,t), α =0.5, n=0.1, λ =0.1

Figure 4 .

 4 Figure 4. The solution u(x, t), (6.2) and θ(x, t), (6.4).

Research partially supported by the EU FP7-REGPOT project "Archimedes Center for Modeling, Analysis and Computation" and the "Aristeia" program of the Greek Secretariat of Research. Part of this work was completed at the Institute of Applied and Computational Mathematics, FORTH, Greece.


Alessandro Ferreira Leite
email: alessandro.ferreira-leite@u-psud.fr

Claude Tadonki
email: claude.tadonki@mines-paristech.fr

Christine Eisenbeis
email: christine.eisenbeis@inria.fr

Tainá Raiol
email: tainaraiol@unb.br

Maria Emilia

M T Walter

Alba Cristina
email: albamm@cic.unb.br

Magalhães Alves De Melo

Excalibur: An Autonomic Cloud Architecture for Executing Parallel Applications

Keywords: C.2.4 [Cloud computing]: Software architecture Cloud computing architecture, parallel execution, autonomic computing

IaaS providers often allow the users to specify many requirements for their applications. However, users without advanced technical knowledge usually do not provide a good specification of the cloud environment, leading to low performance and/or high monetary cost. In this context, the users face the challenges of how to scale cloud-unaware applications without re-engineering them. Therefore, in this paper, we propose and evaluate a cloud architecture, namely Excalibur, to execute applications in the cloud. In our architecture, the users provide the applications and the architecture sets up the whole environment and adjusts it at runtime accordingly. We executed a genomics workflow in our architecture, which was deployed in Amazon EC2. The experiments show that the proposed architecture dynamically scales this cloud-unaware application up to 10 instances, reducing the execution time by 73% and the cost by 84% when compared to the execution in the configuration specified by the user.

Introduction

Nowadays, the cloud infrastructure may be used for high performance computing (HPC) purposes due to characteristics such as elastic resources, pay-as-you-go model, and full access to the underlying infrastructure [START_REF] Abdelbaky | Enabling High-Performance Computing as a Service[END_REF]. These characteristics can be used to decrease the cost of ownership, to increase the capacity of dedicated infrastructure when it runs out of resources, and to respond effectively to changes in the demand. However, doing high-performance computing in the cloud faces some challenges such as differences in HPC cloud infrastructures and the lack of cloud-aware applications.

The cloud infrastructure requires a new level of robustness and flexibility from the applications, as hardware failures and performance variations become part of its normal operation. In addition, cloud resources' are optimized to reduce the cost to the cloud provider often without performance guarantees at low cost to the users. Furthermore, cloud providers offer different instance types (e.g. Virtual Machine (VM)) and services that have costs and performance defined according to their purpose usage. In this scenario, cloud users face many challenges. First, re-engineering the current applications to fit the cloud model requires expertise in both domains: cloud and highperformance computing, as well as a considerable time to accomplish it. Second, selecting the resources that fits their applications' needs requires data about the application characteristics and about the resources purpose usage. Therefore, deploying and executing an application in the cloud is still a complex task [START_REF] Zhao | Opportunities and Challenges in Running Scientific Workflows on the Cloud[END_REF][START_REF] Juve | Comparing FutureGrid, Amazon EC2, and Open Science Grid for Scientific Workflows[END_REF].

Although some efforts have been made to reduce the cloud's complexity, most of them target software developers [START_REF] Nordberg | BioPig: a Hadoop-based analytic toolkit for largescale sequence data[END_REF][START_REF] Schumacher | SeqPig: simple and scalable scripting for large sequencing data sets in Hadoop[END_REF] and are not straightforward for unexperienced users [START_REF] Juve | Comparing FutureGrid, Amazon EC2, and Open Science Grid for Scientific Workflows[END_REF]. Therefore, in this paper, we propose and evalu-ate an architecture to execute applications in the cloud with three main objectives: (a) provide a platform for high performance computing in the cloud for users without cloud skills; (b) dynamically scale the applications without user intervention; and (c) meet the users requirements such high performance at reduced cost.

The remainder of this paper is organized as follows. Section 2 presents our cloud architecture. In Section 3, experimental results are discussed. Section 4 presents related work and discusses cloud architectures to perform highperformance computing. Finally, Section 5 presents the conclusion and future work.

Design of the Proposed Architecture

Our architecture aims to simplify the use of the cloud and to run applications on it without requiring re-design of the applications.

We propose an architecture composed of micro-services. A micro-service is a lightweight and independent service that performs single functions and collaborates with other services using a well-defined interface to achieve some objectives. Micro-services make our architecture flexible and scalable since services can be changed dynamically according to users' objectives. In other words, if a service does not achieve a desirable performance in a given cloud provider, it can be deployed in another cloud provider without requiring service restart.

In this paper, an application represents a user's demand/work, being seen as a single unit by the user. An application is composed of one or more tasks which represent the smallest work unit to be executed by the system. A partition is a set of independent tasks. The tasks that form an application can be connected by precedence relations, forming a workflow. A workflow is defined to be a set of activities, and these activities can be tasks, as said above, or even another workflows. The terms application and job are used interchangeably in this paper.

The proposed architecture has three layers: Physical, Application, and User layer (Figure 1). In the Physical layer, there are services responsible for managing the resources (e.g. Virtual Machine (VM) and/or storage). A resource may be registered by the cloud providers or by the users through the Service Registry. By default, the resources provided by the public clouds are registered with the following data: resource type (e.g. physical or virtual machine, storage), resource URL, costs, and resource purpose (e.g. if the resource is optimized for CPU, memory or I/O). The Resource Management service is responsible to validate these data and keep them updated. For instance, a resource registered at time t i may not be available at time t j , t j > t i , either because it failed or because its maximum allowed usage was reached. With the Monitoring and Deployment service, we deploy the users' jobs and monitor them. Monitoring is an important activity for many reasons. First, it collects data about the resources' usage.

Second, it can be used to detect failures -sometimes the providers terminate the services when they are using/stressing the CPU, RAM memory, or both. And finally, to support the auto scaling mechanism.

We provide a uniform view of the the cloud providers' APIs implementing a Communication API. This is necessary because each provider may offer different interfaces to access the resources.

On top of the Physical layer, the Application layer provides micro-services to schedule jobs (Provisioning), to control the data flows (Data Event Workflow), to provide data streaming service (Data Streaming Processing), and to control the jobs execution. The architecture uses the MapReduce [START_REF] Dean | MapReduce: simplified data processing on large clusters[END_REF] strategy to distribute and execute the jobs. This does not mean that the users' applications must be composed of MapReduce jobs, but only that they are distributed following this strategy.

The Coordination service manages the resources, which can be distributed across different providers, and provides a uniform view of the system such as the available resources and the system's workload.

The Provisioning service uses high-level specifications to create a workflow. This workflow contains tasks which will set up the cloud environment and an execution plan which will be used to execute the application. In fact, the Provisioning service communicates with the Coordination service to obtain data about the resources and to allocate them for a job. After that, it submits the workflow for the Workflow Management service.

An execution plan consists of the application, the data sources, the resources to execute it, a state (initializing, waiting data, ready, executing, and finished), and a characteristic that can be known or unknown by the system. A characteristic represents the application's behavior such as CPU, memory, or I/O-bound.

The Workflow Management service coordinates the execution of the workflow and creates the data flows in the Workflow Data Event service.

The Workflow Data Event service is responsible for collecting and moving data for the execution plans. A data flow has a source and a sink and it can supply data for multiple execution plans. This avoids multiple accesses to the Distributed File System (DFS) to fetch the same data.

The User layer has two micro-services: Job Submission and Job Stats Processing services. A user submits a job using the Job Submission service. A job has the following data: the tasks which compose it, the constraints, the data definition (input and output), and the data about the cloud providers (name and access key).

The users can monitor or get the results of their jobs through the Job Stats Processing.

Scaling cloud-unaware application without technical skills requires an architecture that abstracts the whole environment, taking into account the users' objectives. In the next subsections, we explain how the proposed architecture achieves these goals.

Scaling cloud-unaware applications with budget restrictions and resource constraints

The applications considered in this paper are workflows but some parts of the workflow can be composed of a set of independent tasks that can be run in parallel. These independent tasks are the target of our scaling technique. They are split into P partitions, assigned to different resources. One important problem here is to determine the size and the number of partitions. Over-partitioning can lead to a great number of short duration tasks that may cause a considerable overhead to the system and can result in inefficient resources usage. To avoid this, a partition is estimated by [START_REF] Angiuoli | CloVR: A virtual machine for automated and portable sequence analysis from the desktop using cloud computing[END_REF]:

P = N q * R T (1)
where N q is the workload size; T is the estimated CPU time for executing N q in the partition; and R is a parameter for the maximum execution time for partition P . A partition can be adjusted according to the node characteristics. For instance, if the resource usage by a partition P i is below a threshold, P i can be increased. Partitions exist due to the concept of splittable and static files. It is the user who defines which data are splittable and how to split the data when the system does not know. Splittable data are converted to JavaScript Object Notation (JSON) records and persisted onto the distributed database, so a partition represents a set of JSON records. On the other hand, static data are kept in the local file system.

Minimizing data movement to reduce cost and execution time

Data movement can increase the total execution time of the application (makespan) and sometimes it can be higher than the computation time due to the differences in networks' bandwidth. In that case, we can invert the direction of the logical flow, moving the application as close as possible to the data location. Actually, we distribute the data using a Distributed File System and the MapReduce strategy.

Although MapReduce is an elegant solution, it has the overhead to create the map and the reduce tasks every time a query must be executed. We minimize this overhead using a data structure that keeps data in memory. This increases the memory usage and requires data consistency policies to keep the data updated, however it does not increase the monetary cost. We implemented a data policy that works as follows. For each record read by a node, it is kept in memory and its key is sent to the Coordination service (Figure 1). The Coordination stores the key/value pairs, where they were read. When a node updates a record, it removes the record from its memory and notifies the Coordination service. Then, the Coordination notifies asynchronously all the other nodes that have the key to remove it from memory.

Minimizing job makespan through workload adjustment

In an environment with incomplete information and unpredictable usage pattern as in the cloud, load imbalance can impact the total execution time and the monetary cost. For instance, assigning a CPU-bound task to a memory optimized node is not a good choice. To tackle this problem, we propose a workload adjustment technique that works as follows. For execution plans in the ready state and with an unknown application's characteristics, the scheduler selects similar execution plans and submits them for each available resource (i.e. CPU, memory or I/O optimized) and waits. As soon as the first execution finishes, the scheduler checks if there are similar execution plans in the ready state and submits them.

When there are no more ready execution plans, the scheduler assigns one in the executing state. Note that, in this case, the cost can increase, since we have more than one node executing the same task. In fact, we minimize this, finishing the slow node according to the difference between the elapsed time and the time to charge the node usage.

Making the cloud transparent for the users

As our architecture aims to make the cloud transparent for the users, it automates the setup process. However for some users, this is not sufficient since some jobs still require programming skills. For instance, consider the following scenarios: (i) a biologist who wants to search DNA units that have some properties in a genomics database, and to compare these DNA units with another sequence that he/she has built; (ii) a social media analyst who wants to filter tweets using some keywords.

Normally, these works require a program to read, parse, and filter the data. However, in our solution the users only have to know the data structure and to use a domain specific language (DSL) to perform their work. Listings 1 and 2 show how those works are defined, where b, P1, P2, T and w are users' parameters. s e l e c t t w e e t from t w e e t s where t e x t c o n t a i n s (w) ✆ Listing 2. Specification of a Twitter analysis application.

In this case, a data structure (e.g. a file) is seen as a table whose fields can be filtered. Although we have similar approaches in the literature such as BioPig [START_REF] Nordberg | BioPig: a Hadoop-based analytic toolkit for largescale sequence data[END_REF] and Se-qPig [START_REF] Schumacher | SeqPig: simple and scalable scripting for large sequencing data sets in Hadoop[END_REF], they still require programming skills to register the drivers and to load/store the data. In other words, to use them, the users have to know the system's internals.

In order to illustrate our architecture, consider the bioinformatics scenario described above. In this case, the biologist submits a XML or a YAML file with the application, the requirements, and the data definition (the genomics database and the built sequence) using a console application (a client of the Job Submission service) at the User layer. The Job Submission sends the job description to the Provisioning service at the Application layer and waits for the job's ID. When the Provisioning service receives the application, it executes the following steps. First it creates a workflow with five activities: (i) select the cheapest Virtual Machine to setup the environment; (ii) get non splittable files (e.g. a reference genome) to store them in the local file system; (iii) get the splittable files (the genomics database) and persist them into the DFS; (iv) create a Virtual Machine Image (VMI) of the configured environment; and (v) finish the VM used to configure the environment. Second, it selects the resources returned by the Coordination service that match the users' requirements or the applications' characteristics. Third, it creates an execution plan for the application; selects a resource to execute it; and starts the execution. Finally, it returns the job's ID.

In this scenario, a partition has a set of genomics sequences read from the DFS by the Workflow Data Event assigned to an execution plan. During the application's execution, the Provisioning service monitors the applications through the Monitoring service and if the partition's execution time reaches the expected time it creates more VMs to redistribute the workload. After all tasks have finished, the user receives the output through the Job submission service.

Experimental Results

We deployed an instance of our architecture on Amazon EC2. Our goal was to evaluate the architecture when instanced by a user without cloud skills.

We executed a genomics workflow that aims to identify non-coding RNA (ncRNA) in the fungi Schizosaccharomyces pombe (S. pombe). This workflow, called Infernal-Segemehl, consists of four phases (Figure 2): (i) first the tool Infernal [START_REF] Nawrocki | Infernal 1.0: inference of RNA alignments[END_REF] maps the S. pombe sequences onto a nucleic acid sequence database (e.g. Rfam [START_REF] Griffiths-Jones | Rfam: annotating non-coding RNAs in complete genomes[END_REF]); (ii) then, the sequences with no hit or with a hit with a low score are processed by the segemehl tool [START_REF] Hoffmann | Fast Mapping of Short Sequences with Mismatches, Insertions and Deletions Using Index Structures[END_REF] (iii) SAMTools [START_REF] Li | The Sequence Alignment/Map format and SAMtools[END_REF] is used to sort the alignments and convert them to the SAM/BAM format. (iv) finally, the RNAFold tool [START_REF] Hofacker | Fast folding and comparison of RNA secondary structures[END_REF] is used to calculate the minimum free energy of the RNA molecules obtained in step (iii). We used the Rfam version 11.1 (with 2278 ncRNA families) and S. pombe sequences extracted from the EMBL-EBI (1 million reads). Rfam is a database of non-coding RNA families with a seed alignment for each family and a covariance model profile built on this seed to identify additional members of a family [START_REF] Griffiths-Jones | Rfam: annotating non-coding RNAs in complete genomes[END_REF].

Although, in its higher level, this workflow executes only four tools, it is data oriented. In other words, each step processes a huge amount of data and, in all tools, each pairwise sequence comparison is independent. So, the data can be split and processed by parallel tasks.

Infernal

Segemehl RNAfold SAMtools

Figure 2. The Infernal-Segemehl workflow. The Amazon EC2 micro instance (t1.micro) was used to setup the environment (e.g. install the applications, copy the static files to the local file system) and to create a Virtual Machine Image (VMI). We chose it because it is cheaper and also eligible for the free quota.

In addition to the cloud's executions, we also executed the workflow in a local PC (Table 1) to have an idea of the cloud overhead.

Case study 1: execution without auto scaling

This experiment aims to simulate the users' preferences, where an instance is selected either upon their knowledge about the applications' requirements or the amount of computational resources offered by an instance. We executed the workflow in the first four instances listed in Table 1. The t1.micro instance was used exclusively to setup the environment and it was not used to run the application.

Figure 3 shows the costs and execution time for the four instances. The time was measured from the moment the application was submitted until the time all the results are produced (wallclock time). Therefore, it includes the cloud overhead (data movement to/from the cloud, VM instantiation, among others). The instance hs1.8xlarge, which was selected based on the application requirements (≥ 88GB of RAM), outperformed all other instances. Although it was possible for the user to execute his/her application without any technical cloud skills, the amount paid (USD 78.00) was high. This happened because the user specified that the application would need more than 88GB of RAM and in fact, the application used only 3GB of RAM.

Considering this scenario, the cloud is not an attractive alternative for the users due to its execution times; those were 22% and 31% higher than the local execution (PC Table 1). Even in the best configuration (hs1.8xlarge), the execution time was only 60% lower with a high monetary cost. These differences are owing to the multitenant model employed by the clouds.

Case study 2: execution with auto scaling

This experiment aims to evaluate if the architecture can scale a cloud-unaware application.

Based upon the previous experiment (Figure 3), the system discarded the I/O optimized instance (hs1.8xlarge) due to its high cost(Table 1) and also because the application did not really require the amount of memory defined by the user. In a normal scenario, this instance is selected only if the monitoring service confirms that the application is I/O intensive.

To scale, the system creates P partitions (P 1 , P 2 , • • • , P n ) using the Equation 1 (Section 2.1) with R equals to 1 hour and T equals to 9 hours. These values represent respectively the expected execution time for one partition P i and for the whole workflow. They were defined because Amazon charges the resource by the hour and because, in the previous experiment, the best execution time took approximately 9 hours to finish (Figure 3). This means that this experiment aims to at least decrease the cost. In this case, were created 9 partitions.

As the beginning, the system had not sufficient data to decide if the workflow was memory or CPU-bound, so it submitted two similar partitions -for two instance types (m1.xlarge and c1.xlarge) -to realize which was the most appropriate for the partition.

Figure 4 shows the execution time for each partition in the selected instance types. As soon as execution in the partition assigned to the c1.xlarge instance finished, the system created one VM for each partition in the ready state and executed them. Although there were only 7 partitions in the ready state and 1 in execution (execution state) the architecture duplicates the partition in execution, since its execution time in the m1.xlarge instance was unknown. After one hour, more three instances were created to redistribute the tasks as shown in Figure 5.

Due to the cloud infrastructure, which provided in nearly real time the requested resources, and the auto scaling mechanism, which selected the resources based on the partitions' characteristics, we decreased both the cost (5 times) and the makespan (10, 830 seconds) using 10 c1.xlarge instances (80 vCPUs) and one m1.xlarge (4 vCPUs).

Our strategy differs from the scaling services 1 offered by the cloud providers, since the users do not have to select an instance type nor to split the work manually. 1 Amazon CloudWatch (aws.amazon.com/cloudwatch/).

Related Work

In the last years, many works have described the challenges and opportunities of running high-performance computing in the cloud [START_REF] Abdelbaky | Enabling High-Performance Computing as a Service[END_REF][START_REF] Juve | Comparing FutureGrid, Amazon EC2, and Open Science Grid for Scientific Workflows[END_REF]. Many of the benefits identified by these works, such as easy access to the resources, elasticity, stability and resource provisioning in nearly real time, as well as the technical skills required to administrate the cloud, confirms the requirements of reducing the complexity for the users, and are consistent with our work in scaling the workflow Infernal-Segemehl using the Amazon EC2.

Recently, many works have focused on developed new architecture to execute users' applications in the cloud considering both cost and performance. For instance, the Cloud Virtual Service (CloVR) [START_REF] Angiuoli | CloVR: A virtual machine for automated and portable sequence analysis from the desktop using cloud computing[END_REF] is a desktop application for automated sequences analyses using cloud computing resources. With CloVR, the users execute a VM on the their computer, configure the applications, insert the data in a special directory, and CloVR deploys an instance of this VM on the cloud to scale and to execute the applications. CloVR scales the application by splitting the workload in P partitions using Equation 1, and the Cunningham BLAST runtime to estimate the CPU time for each BLAST query.

In [START_REF] Karlsson | Enabling Large-Scale Bioinformatics Data Analysis with Cloud Computing[END_REF], biological applications are run on the Microsoft Windows Azure showing the required skills and challenges to accomplish the work. Iordache and colleagues [START_REF] Iordache | Resilin: Elastic MapReduce over Multiple Clouds[END_REF] developed Resilin, an architecture to scale MapReduce jobs in the cloud. The solution has different services to provision the resources, to handle jobs flow execution, to process the users requests, and to scale according to the load of the system. Doing bioinformatics data analysis with Hadoop requires knowledge about the Hadoop internal and considerable effort to implement the data flow. In [START_REF] Nordberg | BioPig: a Hadoop-based analytic toolkit for largescale sequence data[END_REF], a tool for bioinformatics data analysis called BioPig is presented. In this case, the users select and register a driver -bioformatics algorithms -provided by the tool and write their analysis' jobs using the Apache Pig (pig.apache.org) data flow language. SeqPig [START_REF] Schumacher | SeqPig: simple and scalable scripting for large sequencing data sets in Hadoop[END_REF] is another tool that has the same objective of BioPig. The differences between them are the drivers provided by each tool. These tools reduce the needs to know Hadoop internal to realize bioinformatics data analysis.

The closest works to ours are [START_REF] Angiuoli | CloVR: A virtual machine for automated and portable sequence analysis from the desktop using cloud computing[END_REF], [START_REF] Nordberg | BioPig: a Hadoop-based analytic toolkit for largescale sequence data[END_REF], and [START_REF] Schumacher | SeqPig: simple and scalable scripting for large sequencing data sets in Hadoop[END_REF]. Our work differs from these approaches in the following ways. First, the users do not need to configure a VM in their computers to execute the applications in the cloud. Second, our architecture tries to match the workload to the appropriate instance type. Third, the data flow is defined using an abstract language freeing the users to write any code. The language is the same as used by BioPig and SeqPig but with the difference that the users write the data flow only considering the data structure. For instance, to filter the sequences using BioPig or SeqPig the users have to register the loaders, the drivers, and write a script to execute the analysis, which is more appropriate for software developers.

Conclusion and Future Work

In this paper, we proposed and evaluated a cloud architecture based on micro-services to execute application in the cloud.

With a user-oriented perspective, we could execute a genomics workflow without requiring programming skills or cloud knowledge from the users. We executed two experiments using Amazon EC2 to evaluate the architecture when instantiated with and without auto scaling. In the first case, the user was responsible to define an instance type to execute the workflow without auto scaling. In the second case, an instance type was selected based on the applications' characteristics and the work was split to reduce the execution time. Using 11 VMs we decreased both the cost and the execution time when compared to an execution without the auto scaling.

As future work, we intend to instantiate the architecture running other applications in a hybrid cloud. Also, we will consider a dynamic scenario, where both the number of tasks are unknown and the resources usage are restricted. Finally, we intend to incorporate QoS and budget requirements.

Figure 1 .

 1 Figure 1. The proposed architecture and its micro-services.

  e x e c u t e T w i t h ( s e l e c t r e a d s from genomicd a t a b a s e where P1 = X and P2 = Y) -s e q = b ✆ Listing 1. Specification of a genomics analysis application.

  hs1.8xlarge m1.xlarge PC Instance type Execution time in seconds (b) Execution time

Figure 3 .

 3 Figure 3. Cost and execution time of the Infernal-Segemehl workflow (Figure 2) in the cloud allocating the resources based on users' preferences.

  time in seconds (b) Execution time for one partition when executed in the c1.xlarge and m1.xlarge instances. One partition was defined to finish in 1 hour with the deadline of 9 hours for the workflow.

Figure 4 .Figure 5 .

 45 Figure 4. Cost to execute the workflow (Figure 2) with auto scaling enabled.

Table 1 .

 1 Resources used during the experiments.

	Instance type	CPU	RAM	Cost ($/hour)
	PC	Intel Core 2 Quad CPU 2.40GHz	4 GB	Not applicable
	hs1.8xlarge	Intel Xeon 2.0 GHz 16 cores	171 GB	4.60
	m1.xlarge	Intel Xeon 2.0 GHz 4 cores	15 GB	0.48
	c1.xlarge	Intel Xeon 2.0 GHz 8 cores	7 GB	0.58
	t1.micro	Intel Xeon 2.0 GHz 1 core	613 MB	0.02

Acknowledgments

The authors would like to thank CAPES/Brazil and CNPq/Brazil though the STIC-AmSud project BioCloud, and INRIA/France for their financial support.