Mourad Choulli
email: mourad.choulli@univ-lorraine.fr

Giorgio Metafune
email: giorgio.metafune@unisalento.it

TWO-SIDED GAUSSIAN BOUNDS FOR FUNDAMENTAL SOLUTIONS OF NON-DIVERGENCE FORM PARABOLIC OPERATORS WITH HÖLDER CONTINUOUS COEFFICIENTS

Keywords: 2010 Mathematics Subject Classification. 65M80 Non-divergence form parabolic operator, fundamental solution, parametrix, two-sided Gaussian bounds, Dirichlet-Green function, Neumann-Green function

whether they are published or not. The documents may come

1. Introduction 1.1. Statement of the main result. The tremendous literature on Gaussian bounds for fundamental solutions of second order parabolic operators can be splitted into two classes: divergence or non-divergence operators. In the first class we only quote the deep results obtained by Aronson, following Nash's ideas, and we refer to [5] for a comprehensive treatment. The second class is more classical and can be found in the books [6,[START_REF] Ladyzhenskaja | tzeva, Linear and quasilinear equations of parabolic type[END_REF] where a fundamental solution is constructed, via the parametrix method, assuming Hölder continuity of the coefficients. By construction the fundamental solution satisfies precise upper bounds but, strangely enough, lower bounds are not proved. In this note we show that the parametrix method produces also lower bounds.

Let P = R n x × R t and set Q = {(x, t, ξ, τ ); (x, t), (ξ, τ ) ∈ P, τ < t}.

The space of continuous and bounded functions f : P → R is denoted by C 0 b (P ). Let f ∈ C 0 b (P ). We say that f is Hölder continuous with exponent α, 0 < α ≤ 1, if

[f ] α = sup |f (x, t) -f (x , t )| |(x -x , t -t )| α , (x, t), (x , t ) ∈ P, (x, t) = (x , t ) < ∞, where |(x -x , t -t )| α = |x -x | 2 + |t -t | α/2 .
We define C α (P ) = {f ∈ C 0 b (P ); [f ] α < ∞}. C α (P ) is a Banach space when it is endowed with its natural norm

f α = f ∞ + [f ] α
and we also use the notation

{f } α = sup |f (x, t) -f (x , t)| |x -x | α ; x, x ∈ R n , x = x and t ∈ R .
We consider the second order parabolic operator

(1.1) L = n i,j=1 a ij (x, t)∂ 2 ij + n i=1 b i (x, t)∂ i + q(x, t) -∂ t
with the following assumptions on its coefficients. (a1)

a ij ∈ C α (P ), 1 ≤ i, j ≤ n.
(a2) The matrix a(x, t) = (a ij (x, t)), (x, t) ∈ P , is symmetric, real-valued, and there exist constants κ > 0, M > 0 so that

κ|η| 2 ≤ a(x, t)η, η ≤ M |η| 2 , (x, t) ∈ P, η ∈ R n . (a3) b i , q ∈ C 0 b (P ), 1 ≤ i ≤ n. (a4) There exists a constant N 1 > 0 so that n i,j=1 [a ij ] α ≤ N 1 . (a5) There exists a constant N 2 > 0 so that n i=1 b i ∞ + q ∞ ≤ N 2 . (a6) {b i } α < ∞, 1 ≤ i ≤ n, and {q} α < ∞.
Henceforth we use for convenience the notation D for (n, α, N 1 , N 2 , M, κ).

In this paper the fundamental solution constructed by the parametrix method is denoted by E = E(x, t; ξ, τ ), (x, t, ξ, τ

) ∈ Q. Recall that E is a fundamental solution if E ∈ C 2 (Q), LE = 0 and lim t→τ ˆRn E(x, t; ξ, τ )f (ξ)dξ = f (x), f ∈ C ∞ 0 (R n ). Theorem 1.1. Let c = 1 8M and d = 4 ln e2 3n (M κ -1 ) n/2 Γ(n/2 + 1) κ .
Under assumptions (a1) to (a6), there exist four constants

ℵ i = ℵ i (D), i = 0, 1, 2, 3, ℵ 0 > 0, ℵ 1 ≥ 0, ℵ 2 > 0 and ℵ 3 ≥ 0, such that ℵ 0 e -ℵ1(t-τ ) (t -τ ) -n 2 e -d |x-ξ| 2 t-τ ≤ E(x,t; ξ, τ ) (1.2) ≤ ℵ 2 e ℵ3(t-τ ) (t -τ ) -n 2 e -c |x-ξ| 2 t-τ , for all (x, t, ξ, τ ) ∈ Q. Remark 1.
0 ≤ G D Ω (x, t; ξ, τ ) ≤ ℵ 2 e ℵ3(t-τ ) (t -τ ) -n 2 e -c |x-ξ| 2 t-τ , (x, t, ξ, τ ) ∈ Q,
where the constants in this inequality are the same as in Theorem 1.1.

We say that Ω satisfies the chain condition if there exists a constant > 0 such that for any two points x, y ∈ Ω and for any positive integer m there exists a sequence (x i ) 0≤i≤m of points in Ω such that x 0 = x, x m = y and

|x i+1 -x i | ≤ m |x -y|, i = 0, . . . , m -1.
The sequence (x i ) 0≤i≤m is named a chain connecting x and y.

Since any bounded Lipschitz domain has the chain condition (this fact can be easily deduced from [4, Corollary A.1]), an adaptation of the proof of [2, Theorem 3.1] (see also [3]) and the reproducing property enable us to get the following result.

Corollary 1.2. If the coefficients of L satisfy assumptions (a1) to (a6) then there exist five constants c

0 = c 0 (D) and ℵ i = ℵ i (D) > 0, i = 0, 1, 2, 3, such that ℵ 0 e -ℵ1(t-τ ) (t -τ ) -n 2 e -c0 |x-ξ| 2 t-τ ≤ G N Ω (x,t; ξ, τ ) ≤ ℵ 2 e ℵ3(t-τ ) (t -τ ) -n 2 e -c |x-ξ| 2 t-τ ,
for all (x, t, ξ, τ ) ∈ Q, where c is as in Theorem (1.1).

Preliminaries

In this section the coefficients of L satisfy assumptions (a1) to (a5).

Basic properties of generalized Gaussian kernels.

In the sequel we frequently use

(2.1) ˆR e -ρ 2 dρ = √ π.
The Gaussian heat kernel is defined as follows

(2.2) G(x, t) = 1 (4πt) n 2 e -|x| 2 4t , x ∈ R n , t > 0.
We have, according to Fubini's theorem,

ˆRn G(x, t)dx = ˆR 1 2 √ πt e -y 2 4t dy n , t > 0.
Then the change of variable ρ = y

2 √ t yields (2.3) ˆRn G(x, t)dx = 1, t > 0,
where we used the value of the Gauss integral (2.1).

If a = (a ij ) is n × n symmetric positive definite matrix, we define the generalized Gaussian heat kernel by

(2.4) G a (x, t) = √ det a (4πt) n 2 e -ax,x 4t , x ∈ R n , t > 0.
Let d = diag(d 1 , . . . , d n ) be a diagonal matrix and u an orthogonal matrix, that is u t u = I, so that uau t = d. Then

ax, x = dux, ux , det a = n i=1 d i and ˆRn G a (x, t)dx = ˆRn √ det a (4πt) n 2 e -dux,ux 4t dx, t > 0. Since |det u| = 1, the change of variable y = ux gives ˆRn G a (x, t)dx = ˆRn √ det a (4πt) n 2 e -dx,x 4t dx, t > 0.
Applying again Fubini's theorem, we get ˆRn

G a (x, t)dx = √ det a n j=1 ˆR 1 2 √ πt e -d i ρ 2 4t dρ (2.5) = √ det a n j=1 ˆR 1 2 √ d i πt e -ρ 2 4t dρ = n j=1 ˆR 1 2 √ πt e -ρ 2 4t dρ = 1, t > 0. It is straightforward to check that G a ∈ C ∞ (R n × (0, ∞)) and, since ∂ k ax, x = 2 n j=1 a kj x j = 2(ax) k , x ∈ R n , we have (2.6) ∂ k G a (x, t) = - 1 2t G a (x, t)(ax) k , x ∈ R n , t > 0.
We easily derive from (2.6)

(2.7) ∂ 2 k G a (x, t) = 1 4t 2 G a (x, t)(ax) k (ax) - 1 2t G a (x, t)a k , x ∈ R n , t > 0. Let a -1 = (a ij ). Inserting the identity n k, =1 a k (ax) k (ax) = a -1 ax, x = ax, x in (2.7) we obtain (2.8) n k, =1 a k ∂ 2 k G a (x, t) = 1 4t 2 ax, x - n 2t G a (x, t), x ∈ R n , t > 0.
On the other hand, it is straightforward to check that (2.9)

∂ t G a (x, t) = 1 4t 2 ax, x - n 2t G a (x, t), x ∈ R n , t > 0.
We define the parabolic operator L a -1 by

L a -1 = n i,j=1 a ij ∂ 2 ij -∂ t .
Comparing (2.8) and (2.9) we see that G a satisfies (2.10)

L a -1 G a (x, t) = 0, x ∈ R n , t > 0.
2.2. The parametrix. Let a -1 (x, t) = (a ij (x, t)), (x, t) ∈ P , where (a ij (x, t)) is the inverse of the matrix (a ij (x, t)), and define

Z(x, t; ξ, τ ) = G a -1 (ξ,τ ) (x -ξ, t -τ ), (x, t, ξ, τ ) ∈ Q, that is (2.11) Z(x, t; ξ, τ ) = det a -1 (ξ, τ ) (4π(t -τ )) n 2 e -a -1 (ξ,τ )(x-ξ),(x-ξ) 4(t-τ ) , (x, t, ξ, τ ) ∈ Q.
This function is usually called the parametrix associated to the parabolic operator L. According to the results of the previous subsection, for any (ξ, τ

) ∈ P , Z(•, •; ξ, τ ) ∈ C ∞ (P τ ) with P τ = {(x, t) ∈ R n ; t > τ }, and
(2.12)

n i,j=1 a ij (ξ, τ )∂ 2 ij Z(•, •; ξ, τ ) -∂ t Z(•, •; ξ, τ ) = 0 in P τ .
Let us define

d i (x, t; ξ, τ ) = - 1 2(t -τ ) n j=1 a ij (ξ, τ )(x j -ξ j ), d ij (x, t; ξ, τ ) = - a ij (ξ, τ ) 2(t -τ ) + d i (x, t; ξ, τ )d j (x, t; ξ, τ ).
From (2.6) and (2.7) we have

∂ i Z = d i Z and ∂ 2 ij Z = d ij Z.
Therefore, taking into account (2.12), we have

(2.13) LZ =   n i,j=1 (a ij (x, t) -a ij (ξ, τ )) d ij + n i=1 d i b i + q   Z = ΨZ, where Ψ = n i,j=1 (a ij (x, t) -a ij (ξ, τ )) d ij + n i=1 d i b i + q.
We need a pointwise estimate for LZ. To this end, we start with the following lemma Lemma 2.1. We have

(2.14) |a -1 (x, t)η| ≤ 1 κ |η|, (x, t) ∈ P, η ∈ R n , (2.15) sup 1≤i,j≤n a ij ∞ ≤ 1 κ .

and

(2.16)

a -1 (x, τ )(x -ξ), x -ξ 4(t -τ ) ≥ 1 4M |x -ξ] 2 t -τ .
Proof. From assumption (a2), we have

a(x, t)η, η ≥ κ|η| 2 , (x, t) ∈ P, η ∈ R n .
In this inequality we get by substituting η by a -1 (x, t)η

|a -1 (x, t)η||η| ≥ a -1 (x, t)η, η ≥ κ|a -1 (x,
d i ∞ ≤ |x -ξ| 2κ(t -τ ) or d i ∞ ≤ 2κ √ t -τ where = |x -ξ| √ t -τ .
It is easy to see that (2.15) and (2.17) entail (2.18)

d ij ∞ ≤ 1 2κ + 2 4κ 2 1 t -τ . Hence (2.19) n i,j=1 (a ij (x, t) -a ij (ξ, τ )) d ij ≤ N 1 1 2κ + 2 4κ 2 (1 + 2 ) α 2 (t -τ ) 1-α 2 .
On the other hand, we get from (2.17)

(2.20) n i=1 b i d i + q ≤ N 2 2k √ t -τ + 1 ≤ N 2 1 + 2κ (t -τ ) 1-α 2 , t -τ ≤ 1.
In light of (2.19) and (2.20) we obtain

(2.21) Ψ ∞ ≤ N 1 1 2κ + 2 4κ 2 (1 + 2 ) α 2 (t -τ ) 1-α 2 + N 2 1 + 2κ (t -τ ) 1-α 2 , t -τ ≤ 1. Now (2.16) implies (2.22) |Z(x, t)| ≤ 1 (4κπ(t -τ )) n 2 e -1 4M 2 .
Recall that c = 1 8M and let

(2.23) C = 1 (4κπ) n 2 max λ>0 N 1 1 2κ + λ 2 4κ 2 (1 + λ 2 ) α/2 + N 2 λ κ + 1 e -cλ 2 .
If Φ 1 = LZ = ΨZ, then a combination of (2.21) and (2.22) gives (2.24)

|LZ| = |ΨZ| ≤ C(t -τ ) -n 2 -1+β e -c 2 , t -τ ≤ 1, with β = α 2 .

Two-sided Gaussian bounds

In this section the coefficients of L satisfy (a1) to (a6). Let Φ 1 = LZ,

Φ +1 (x, t, ξ, τ ) = ˆt τ ˆRn Φ 1 (x, t; η, σ)Φ (η, σ, ξ, τ )dηdσ, ≥ 1
and define Φ =

≥1

Φ .

Let E be the fundamental solution, associated to L, constructed by the parametrix method. According to [6, 7], E is given by

(3.1) E(x, t; ξ, τ ) = Z(x, t; ξ, τ ) + ˆt τ ˆRn Z(x, t; η, σ)Φ(η, σ; ξ, η)dηdσ,
for all (x, t, ξ, τ ) ∈ Q.

We refer to [6, Chapter 1] or to [7, Chapter IV] for more details.

Preliminary estimate.

The following lemma will be useful in the sequel.

Lemma 3.1. ([6, Chapter 1, Section 4]) Let λ > 0 and -∞ < γ, δ < 1. Then

ˆt τ ˆRn (t -σ) -n 2 -γ e -λ|x-η| 2 t-σ (σ -τ ) -n 2 -δ e -λ|η-ξ| 2 σ-τ dηdσ = 4π λ n 2 B (1 -γ, 1 -δ) (t -τ ) -n 2 +1-γ-δ e -λ|x-ξ| 2 t-τ
, where B is the Euler beta function.

Let C be the constant given by (2.23) and assume that t -τ ≤ 1. We deduce from (2.24)

(3.2) |Φ 1 | ≤ C(t -τ ) -n 2 -1+β e -c 2 . Let C = 4π c n 2 .
We have by applying Lemma 3.1

|Φ 2 | ≤ CC 2 B(β, β)(t -τ ) -n 2 -1+2β e -c 2 .
By induction in , wo obtain

|Φ | ≤ C -1 C -1 j=1 B(β, jβ)(t -τ ) -n 2 -1+ β e -c 2 , ≥ 2.
If Γ is the Euler gamma function, we recall that

B(β, jβ) = Γ(β)Γ(jβ) Γ((j + 1)β) . Therefore -1 j=1 B(β, jβ) = Γ(β) Γ( β)
and hence

|Φ | ≤ C -1 Λ Γ( β) (t -τ ) -n 2 -1+ β e -c 2 , ≥ 2
,
where Λ = C CΓ(β). Since t -τ ≤ 1, we obtain

(3.3) |Φ | ≤ C -1 Λ Γ( β) (t -τ ) -n 2 -1+β e -c 2 , ≥ 2, If C = C -1 , then (3.3) takes the form (3.4) |Φ | ≤ C Λ Γ( β) (t -τ ) -n 2 -1+β e -c 2 , ≥ 2.
From Stirling's formula for the Γ function (see for instance [8, Chapter V, Section 3]) we have Γ(x + 1) ∼ x x e -x √ 2πx, x → ∞. Therefore, the series

(3.5) S = C + C ≥2 Λ Γ( β) is convergent.
We get from (2.24) and (3.4)

(3.6) |Φ| ≤ S(t -τ ) -n 2 -1+β e -c 2 .
3.2. The upper bound. In light of (2.24) and (3.6), Lemma 3.1 yields

(3.7) ˆt τ ˆRn Z(x, t; η, σ)Φ(η, σ; ξ, τ )dηdσ ≤ SB(1, β) (κc) n 2 (t -τ ) -n 2 +β e -c 2 , for all (x, t, ξ, τ ) ∈ Q and t -τ ≤ 1. Let C = 1 (4κπ) n 2 + SB(1, β) (κc) n 2
.

As an immediate consequence of (2.24) and (3.7), we have

(3.8) |E(x, t; ξ, τ )| ≤ C(t -τ ) -n 2 e -c 2 , (x, t, ξ, τ ) ∈ Q, t -τ ≤ 1.
We recall that E possesses the so-called reproducing property

(3.9) E(x, t; ξ, τ ) = ˆRn E(x, t; η, σ)E(η, σ; ξ, τ ) dη, τ < σ < t.
Applying (3.8), we get

(3.10) |E(x, t, ξ, τ )| ≤ C 2 ˆRn (t -σ) -n 2 e -c |x-η| 2 4(t-σ) (σ -τ ) -n 2 e -c |η-ξ| 2 4(σ-τ ) dη,
for all t -τ ≤ 2, where σ = t+τ 2 . We introduce a variable z so that

c |x -η| 2 4(t -σ) + c |η -ξ| 2 4(σ -τ ) = c |x -ξ| 2 4(t -τ ) + |z| 2 .
Using the identity |x -

η| 2 = |x -ξ| 2 + |ξ -η| 2 + x -ξ, ξ -η , we get |x -η| 2 t -σ + |η -ξ| 2 σ -τ - |x -ξ| 2 t -τ = (σ -τ )|x -ξ| 2 (t -σ)(t -τ ) + (t -τ )|η -ξ| 2 (t -σ)(σ -τ ) + 2 x -ξ, ξ -η (t -σ) 2 . = σ -τ (t -σ)(t -τ ) 1 2 (x -ξ) + t -τ (t -σ)(σ -τ ) 1 2 (ξ -η) 2 .
Therefore, we can for instance take

z = c t -τ t -σ 1 2 η -ξ 2(σ -τ ) 1 2 + c σ -τ t -σ 1 2 ξ -x 2(t -τ ) 1 2
.

Passing to the variable z in (3.10), we deduce

|E(x, t, ξ, τ )| ≤ C C 2 (t -τ ) -n 2 e -c 2 , t -τ ≤ 2.
Next assume that t -τ > 2 and let m be the smallest integer so that t -τ ≤ m. Define

σ 0 = τ, σ 1 = τ + t -τ m , . . . , σ m-1 = τ + (m -1) t -τ m , σ m = t.
Iterating the reproducing property (3.9), we get

E(x, t; ξ, τ ) = ˆRn . . . ˆRm E(x, σ m , η m , σ m-1 )E(η m ,σ m-1 , η m-1 , σ m-2 ) . . . E(η 1 , σ 1 , ξ, σ 0 )dη 1 . . . dη m .
Repeating inductively the case m = 2, we find

|E(x, t, ξ, τ )| ≤ C m-1 C m (t -τ ) -n 2 e -c 2 .
This and the fact that m < t -τ + 1 entail

|E(x, t, ξ, τ )| ≤ C -1 e max 0,ln( C C) e max 0,ln( C C) (t-τ ) (t -τ ) -n 2 e -c 2 .
This is the expected Gaussian upper bound.

A more precise upper bound can be obtained by optimizing the constants appearing in the previous computations. We do it in the special case b i = q = 0, where the iteration procedure based on (3.9) is not needed.

Corollary 3.1. If b i = q = 0, then E(x, t; ξ, τ ) ≤ 1 (4κπ) n 2 (t -τ ) -n 2 e - 2 4M 1 + c 1 (t -τ ) α 2 e c2((t-τ )+ γ ) ,
for all (x, t, ξ, τ ) ∈ Q, where = |x-ξ| √ t-τ and γ = 4α+8 3α+4 < 2.

Proof. First we note that the restriction t -τ ≤ 1 is not needed in (2.21), since it comes from (2.20) only. Then we define C as in (2.23) with c = 4M , N 2 = 0. It is easy to see that C ≤ A -2-α with A > 0 and this leads to (2.24) with this C and c = (1-) 4M . Next we write (3.4) with β instead of β, since we no longer assume that t -τ ≤ 1.

Entering this estimate in the constants C, Λ defining S (see (3.5)), using [1, Theorem 2, Section 15, Chapter V] and Stirling's formula again, we deduce that

≥2 Λ (t -τ ) β Γ( β) ≤ c 1 (t -τ ) 2β e c2((t-τ )+Λ 1 β )
and S ≤ c 1 e c2((t-τ )+ -(2+ 4 α

) ) . Then we use this estimate in (3.7) with c = (1-) 4M to get ˆt τ ˆRn Z(x, t; η, σ)Φ(η, σ; ξ, η)dηdσ

≤ c 1 (t -τ ) -n 2 +β e -(1-) 4M 2 +c2 -(2+ 4 α ) +c2(t-τ ) .
Optimizing over and using (3.1), the corollary follows.

3.3. The lower bound. From the previous analysis, we easily get

Z(x, t; ξ, τ ) ≥ 1 (4πM ) n 2 (t -τ ) -n 2 e -1 κ ρ 2 . Hence, (3.11) Z(x, t; ξ, τ ) ≥ e -1 (4πM ) n 2 (t -τ ) -n 2 , |x -ξ| 2 ≤ κ(t -τ ).
A combination of (3.7) and (3.11) yields

E(x, t; ξ, τ ) ≥ e -1 (4πM ) n 2 (t -τ ) -n 2 - SB(1, β) (κc) n 2 (t -τ ) -n 2 +β ,
for all |x -ξ| 2 ≤ κ(t -τ ) and t -τ ≤ 1. Fix δ ≤ 1 sufficiently small in such a way that

e -1 (4πM ) n 2 - SB(1, β) (κc) n 2 δ β ≥ e -1 2(4πM ) n 2 . Then, with µ = e -1 2(4πM ) n 2 , (3.12) E(x, t; ξ, τ ) ≥ µ(t -τ ) -n 2 , |x -ξ| 2 ≤ κ(t -τ ), t -τ ≤ δ.
Let x and ξ be given so that 2|x -ξ| > κ(t -τ ) and let m ≥ 2 be the smallest integer so that (3.13) 4|x -ξ| 2 m ≤ κ(t -τ ).

Define the sequence (x k ) 0≤k≤m where we used

x k = x + k m (ξ -x), 0 ≤ k ≤ m.
|x i+1 -x i | = 1 √ m |x -ξ| √ m ≤ 1

  -τ ) √ m andσ k = τ + k m (t -τ ), 0 ≤ k ≤ m.Using (3.12), the positivity of E and the reproducing property, we getE(x, t;ξ, τ ) ≥ µ m ˆB(x1,r) . . . ˆB(xm-1,r) (σ 1 -σ 0 ) -n 2 . . . (σ m -σ m-1 ) -n 2 dη 1 . . . dη m-1 ,

  By inspecting the proof of Theorem 1.1 we see that, in the Gaussian upper bound, we can substitute c by c = 4M , 0 < < 1, and ℵ i by ℵ i , i = 2, 3, with an explicit dependence of ℵ 2 and ℵ 3 on .

	1.2. Consequences. Let Ω be a C 1,1 -bounded domain of R n . We denote the
	parabolic Dirichlet-Green (resp. Neumann-Green) function on Ω by G D Ω (resp. G N Ω ). It is well known that, according to the maximum principle, 0 ≤ G D Ω ≤ E.
	Therefore as a consequence of Theorem 1.1, we have
	Corollary 1.1. Let the coefficients of L satisfy assumptions (a1) to (a6). Then
	the Dirichlet-Green function G D Ω satisfies

1.

  -η i | ≤ |η i+1 -x i+1 | + |x i+1 -x i | + |x i -η i | < 2r + |x i+1 -x i | ≤ 4r = κ(t -τ ) √ m = κ(σ i+1 -σ i ).If C 0 = min µ, κ -n 2 e -| ln ν| and d = 4| ln ν| κ , then the last inequality and (3.12) yieldE(x, t; ξ, τ ) ≥ C 0 (t -τ ) -n 2 e -d |x-ξ| 2 t-τ , t -τ ≤ δ.We now proceed similarly to the case of the upper bound to remove the condition t -τ ≤ δ. If m is the smallest integer so that t -τ ≤ mδ, we get E(x, t; ξ, τ ) ≥ C -1 CC 0

								2	κ(t -τ ) √ m	= 2r,
	and
	|η i+1 Whence
		E(x, t; ξ, τ ) ≥ κ -n 2 ν m (t -τ ) -n 2 ,
	with	ν =	eM	n 2 2 2 3n Γ(n/2 + 1) κ n	< 1.
	Noting that
				m <	4|x -ξ| 2 κ(t -τ )	+ 1,
	we obtain
							m	(t -τ ) -n 2 e -d 2	,
	from which we deduce
	E(x, t; ξ, τ ) ≥ C -1 e	min 0,ln( CC0) e	min 0,	ln( CC 0 ) δ	(t-τ )	(t -τ ) -n 2 e -d 2	.

E(x, t; ξ, τ

) ≥ κ -n 2 e -| ln ν|m (t -τ ) -n 2 ≥ κ -n 2 e -| ln ν| (t -τ ) -n 2 e -4| ln ν| κ |x-ξ| 2 t-τ , t -τ ≤ δ.

MC is supported by the grant ANR-17-CE40-0029 of the French National Research Agency ANR (project MultiOnde).


Ruben Betsema

Silvia V Nedea

Arjan J H Frijns
email: a.j.h.frijns@tue.nl

A Marie-Curie-ITN within H2020 ISTEGIM 2019 -285315 MEASUREMENT OF THE INTERFACIAL TEMPERATURE JUMP DURING STEADY-STATE EVAPORATION OF A DROPLET

Keywords: Evaporating droplets, interfacial temperature profile, heat and mass transfer at liquid-vapor interface

Introduction

Evaporation is an important phenomena that occurs in a wide range of natural and industrial processes. Although this phenomena has been a subject of research for many years, it is still not fully understood. Experimental results of the last few decades seem to contradict with each other, and with the theory which describes this process, e.g. the kinetic theory of gasses (KTG) and non-equilibrium thermodynamics (NET) [START_REF] Badam | Experimental and theoretical investigations on interfacial temperature jumps during evaporation[END_REF][START_REF] Fang | Temperature measured close to the interface of an evaporating liquid[END_REF][START_REF] Gatapova | The temperature and pressure jumps at the vapor-liquid interface: Application to a two-phase cooling system[END_REF][START_REF] Kazemi | Experimental and Numerical Study of the Evaporation of Water at Low Pressures[END_REF]. Temperature jumps of about 3.2-8.1 o C at the interface of a steady state evaporating water droplet at a pressure of about 245 P a were measured [START_REF] Badam | Experimental and theoretical investigations on interfacial temperature jumps during evaporation[END_REF][START_REF] Fang | Temperature measured close to the interface of an evaporating liquid[END_REF]. In order to determine whether this temperature jump exists and what influences this temperature jump, an experimental setup has been developed.

Experimental setup

The experimental setup (Figure 1) consists of an evaporating spherical droplet (diameter of 8 mm) placed on temperature controlled, stainless steel tube in a cylindrical stainless steel vacuum chamber (diameter of 150mm and length of 275mm). One end of the cylinder is closed by a stainless steel plate, while the other end is closed by glass window for optical access. Purified, degassed, temperature controlled water of 4 o C (controlled by a build-in heat exchanger) was supplied to an evaporating geometry by a mass flow controlled syringe (VWR gas tight syringe 549-0536). The evaporating mass flow is prescribed by the syringe and controlled by a CCD camera (Teledyne Dalsa Genie), which measures the height of the droplet with a precision of 35µm per pixel. The appropriate mass flow was determined by adjusting the flow rate until the droplet height remains steady state. The pressure in the chamber was controlled by a rotary vane pump (vacuumbrand PC8 / RC6) which can reach a vacuum of 0.4 Pa . The pressure is measured by a pressure transducer at the side of the cylinder (Keller PAA 33x) with an accuracy of 10 Pa. The temperature profile at the interface of this droplet was measured by a K-type thermocouple made of two 25.4µm wires with an

Results and Discussion

Before and after the start of the experiments, the thermocouple was calibrated in a bath of boiling and ice water and the offset and a proportionality constant have been determined. The measured uncertainty was about 0.4 o C and did not change over time.

Before the start of the experiment, water was placed in the degasification channel and degassed for about one hour. Also the vacuum chamber was degassed for one hour at a pressure <10 Pa. Meanwhile the cooling device and pump were turned on to cool the evaporating liquid. After this hour, the thermocouple above the interface (at a fixed position) and the pressure was set at 780 Pa. Due to the occurrence of bubbles inside the syringe and the channel between the syringe and the droplet geometry, it was not possible to do an experiment lower than 780 Pa. The measurements started when both measured temperature and measured pressure in the vacuum chamber were at steady state.

The vapor and liquid temperatures were measured between 4mm above the interface and 1.5mm underneath it. The measured temperature profile is shown in Figure 2. Due to evaporation the liquid temperature near the interface is lower than the bulk liquid temperature T bl . The measured temperature profile has been analyzed by the same model that has been used for the experiment conducted by Fang et al. [START_REF] Fang | Temperature measured close to the interface of an evaporating liquid[END_REF]. The liquid temperature T l is described by:

(

With

For the vapor temperature T v holds:

(

With .

Where r i is the interface location, Q i is the mass flux at the interface, j i the mass flux at the interface, h the enthalpy, c p the specific heat, κ the thermal conductivity and T bv and T bl the bulk temperature of the vapor and liquid. The subscripts v and l indicate the liquid and vapor phase respectively.

The calculated temperature profile seems to approximate the measurement profile well (Figure 2). However, the calculated evaporative mass flux of seems to be lower than would be expected from the experiments by Fang et al. [START_REF] Fang | Temperature measured close to the interface of an evaporating liquid[END_REF] , who measured a mass flux of for an evaporating spherical conditions. The difference between this value and the mass flow calculated model can partly be explained by the lower pressure of the experiment by Fang et al. (493 Pa) compared to our experiment that was performed at 780 Pa, and the fact that the liquid temperature entering the evaporation chamber was 15 °C in the experiments by Fang et al, while we controlled it to be 4 • C.

Proceedings of the International Symposium on

Thermal Effects in Gas flows In Microscale October 24-25, 2019 -Ettlingen, Germany

4

A Marie-Curie-ITN within H2020

Conclusions and Future Research

An experimental setup has been developed to measure the temperature profile at the liquid-vapor interface of an evaporating droplet. At 780 Pa the measured temperature profile shows good agreement with other experiments and models from literature . However the evaporative mass flow is lower than expected from the experiments by Fang et al. [START_REF] Fang | Temperature measured close to the interface of an evaporating liquid[END_REF].

In order to investigate the evaporation phenomena further and to determine the relation between temperature jump and external pressure, the setup will be further modified in order to go to lower pressures (order 200-600 Pa) without bubble formation.

2 A

 2 Proceedings of the International Symposium onThermal Effects in Gas flows In Microscale October 24-25, 2019 -Ettlingen, Germany Marie-Curie-ITN within H2020 accuracy of about 0.4 o C. The thermocouple is moved towards the interface by a micrometer spindle (Freinmeyer) mounted on top of the vacuum cylinder this spindle can move the thermocouple toward the interface with an accuracy of 2µm.

Figure 1 .

 1 Figure 1. (Left) Schematic overview of the experimental set-up and (right) front view of the experimental setup with the glass window through which the camera can measure the height of the droplet.

3 AFigure 2 .

 32 Figure 2. Theoretical (solid line) and measured (circles) temperature profile at the liquid-vapor interface of an evaporating droplet. The vapor pressure is set at 780 Pa. The position of the liquid-vapor interface is indicated by the vertical dashed line.

Acknowledgements

This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 643095.


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  Pathway to National Guidelines for Laboratory Diagnostics of Chronic Kidney Disease -Examples from Diverse European Countries. Vanja Radišić Biljak, Kristin Moberg Aakre, Dogan Yucel, Anne-Sophie Bargnoux, Jean-Paul Cristol, Laurence Piéroni To cite this version: Vanja Radišić Biljak, Kristin Moberg Aakre, Dogan Yucel, Anne-Sophie Bargnoux, Jean-Paul Cristol, et al.. A Pathway to National Guidelines for Laboratory Diagnostics of Chronic Kidney Disease -Examples from Diverse European Countries.. EJIFCC [electronic resource] / IFFC, 2017, pp.289-301. hal-01825561