Nick Cox

Lucio Baggio

Jean-Loup Bertaux

Anna Fedorova

Gaetan Lacombe

Franck Lefèvre

Mikhail Luginin

Emmanuel Marcq

Franck Montmessin

High-Level Science Products of SPICAM (Mars Express) and SPICAV (Venus Express)

In this contribution we present new high-level science products derived from nadir and stellar/solar occultation observations obtained with the UV/IR spectrometers SPICAM and SPICAV on board the Mars Express and Venus Express spacecrafts. We describe the scope of the new level 2 processing and the data product content. We present examples of scientific results obtained with these data sets.

SPICAM/SPICAV UV/IR data

For the new level 2 processing of UV nadir and stellar occultation measurements all valid data obtained between orbits 0 and 13651 (SPICAM) and between 0 and 3151 (SPICAV) is used. Similarly for SPICAV all valid IR solar occultation measurements are processsed. For SPICAM, solar occultations until orbit 18550 are considered. Observations were reprocessed starting from the most recent level 1b data. We note that for the Venus stellar occultation observations particular care was taken to estimate and subtract the Nitric Oxide (NO) emission from the limb in order to reveal the true occultation signal.

The full details of the L2 processing are described in the Algorithm Theoretical Definition Documents (ATBDs) available through the PSA. The delivered products (Table 1) adhere to the PDS3 standard.

High-level science products

Nadir observations For the UV nadir observations of Mars and Venus the level 2a data products provide the absolute spectral radiances, radiance factors and geometrical parameters. When available, the associated level 2b data provide for Mars the columns of O 3 , the UV-averaged surface albedo, the dust aerosol optical thickness and the geometrical data; and for Venus the mixing ratios of SO 2 and O 3 , cloud top altitude, and the aerosol imaginary refractive index at 250 nm.

Scientific outcome (examples)

Mars: SPICAM-UV nadir The new dataset comprises Martian Years (MY) 27 to 32 (2004-2014) thus covering a time span appropriate for climatological studies of ozone and dust. As an example, Figure 1 shows the assimilated information on the ozone column (top) and dust opacity (bottom) ([1], [START_REF] Willame | Retrieving cloud, dust and ozone abundances in the Martian atmosphere using SPICAM/UV nadir spectra[END_REF]). Venus is darker (UV dust opacity) at lower latitudes, consistent with known features (see [START_REF] Marcq | Discovery of cloud top ozone on Venus[END_REF]).

Stellar occultation on Venus/Mars Stellar occultations provide a radiography of the vertical distribution of CO 2 , O 3 and aerosols. The CO 2 profile gives the vertical temperature profile. These data can then be compared with different global climate/atmosphere models (e.g. [START_REF] Piccialli | Thermal Structure of Venus Nightside Upper Atmosphere Measured by Stellar Occultations with SPICAV/Venus Express[END_REF]).

Summary and Outlook

New high-level science products for the SPICAM and SPICAV UV and IR instruments have been delivered to the ESA PSA and will be available to the community (release is foreseen before the end of 2019). This contribution presents an overview of the delivered data products (level 2a and 2b) and provides some examples of the scientific content.

Figure 1 :

 1 Figure 1: SPICAM retrieval of the ozone column (top) and the dust opacity (bottom) on Mars.

Figure 2 :

 2 Figure 2: SPICAV nadir retrieval results. Example for orbit 0108A07 (from PSA FITS file).

Table 1 :

 1 SPICAM and SPICAV science products delivered to the ESA Planetary Science Archive (PSA). For the level 2b products we include the column (line) and local densities for CO 2 /SO 2 /O 3 (where applicable), as well as the atmospheric temperature profiles. The altitude sampling ∆z is 1-3 km for Mars and 0.2-7 km for Venus.

	Type of	Planet Number of	Product Product
	observation	observations level	type
	UV-Nadir	Mars 2558	L2a	Radiance
		2558	L2b	Atmosphere
		Venus 1508	L2a	Radiance
		1508	L2b	Atmosphere
	UV-Stellar Mars 2093	L2a	Transmittance
	occultation	1920	L2b	Atmosphere
		Venus 901	L2a	Transmittance
		901	L2b	Atmosphere
	IR-Solar	Mars tbd	L2a	Transmittance
	occultation	tbd	L2b	Atmosphere
		Venus 648	L2a	Transmittance
		222	L2b	Atmosphere
	Stellar occultation observations For the UV stel-
	lar occultation measurements the level 2a data prod-
	ucts comprise the wavelength dependent transmission
	spectra as well as the spectral flux of the reference
	star spectrum. Solar occultation observations For the IR solar
	occultation measurements the level 2a data prod-
	ucts comprise the wavelength dependent transmission
	spectra as well as the reference flux of the reference
	solar spectrum. The level 2b products include aerosol

opacities at 4 or 10 wavelengths from 1 to 1.55 µm for SPICAM and 10 wavelenghts from 0.65 to 1.55 µm for SPICAV. In addition for Mars the local density profiles of CO 2 and H 2 O and aerosol size distributions and number density are included. For Venus the size distribution and number density of sulphur acid particles is given.

Acknowledgements

This project is supported by an EXPRO contract with ESA. EM acknowledges support from CNES and PNP.


Jean Caspar

Guillaume Munch-Maccagnoni

Modular efficient deconstruction with typed pointer reversal

Destructors, responsible for releasing memory and other resources in languages such as C++ and Rust, can lead to stack overflows when releasing a recursive structure that is too deep. In certain cases, it is possible to generate an efficient destructor (non-allocating and tail recursive) using a typed variant of pointer reversal. We extend this technique by making it more modular, in order to handle abstract types, separate compilation, and unboxed types.

Introduction

In some programming languages, such as Rust and C++, memory management is performed by the compiler via the insertion of some code at the end of the scope of values, responsible for releasing the memory allocated on the heap. Such code is called a destructor. Since they run predictably and reliably when variables get out of scope, destructors can be used to reason about resources other than memory, such as files, shared data protected by a mutex, network connections, or transactions. The use of destructors as a resource-handling mechanism is called RAII (Resource Acquisition Is Initialization) [START_REF] Koenig | Exception handling for C++[END_REF].

The stack overflow problem

In RAII-based languages, destructors recursively call the destructors of each sub-field; therefore, if the type they operate on is recursive, then the destructor should be recursive as well. In the evolution of this model, as incarnated by the programming languages C++11 and Rust, compiler-generated destructors themselves are potentially recursive (via "smart pointers"), and their naive implementations by contemporary compilers potentially cause stack overflows. It was believed that one cannot implement destructors in general without allocating on the stack or on the heap, or without changing the order and time of destruction, as Herb Sutter explained [START_REF] Sutter | Leak-Freedom in C++[END_REF]. The fact that generated destructors themselves can overflow the stack makes it a problem of compiler correctness.

Typed pointer reversal Given that the programmer can supply an arbitrary nonraising function acting as a destructor on a per-type basis, generated destructors must take programmer-provided destructors into account, and are specific to a given type.

Continuing previous work by Douence and the second author [START_REF] Munch | Efficient deconstruction with typed pointer reversal[END_REF], the subject of this paper is to derive efficient and correct destructors for recursive types, starting from their specifications (that is, their naive implementations) and applying provably-correct transformations. As shown previoulsy, it is indeed possible to obtain implementations which are tail-recursive and which do not allocate, by starting from the naive implementation and applying standard transformations: namely a continuation-passing style (CPS) transformation followed by defunctionalization [START_REF] Reynolds | Definitional Interpreters for Higher-Order Programming Languages[END_REF][START_REF] Wand | Continuation-Based Program Transformation Strategies[END_REF][START_REF] Danvy | Defunctionalization at Work[END_REF], and a transformation to re-use the memory of the (unaliased) value as a way to store the defunctionalized continuations [START_REF] Lafont | The linear abstract machine[END_REF][START_REF] Henry | Lively linear lisp: "look ma, no garbage![END_REF]. A dual transformation, about recycling continuations into data, had also been observed earlier by Sobel and Friedman [START_REF] Sobel | Recycling continuations[END_REF]. In effect, this transformation amounts to a typed generalization of pointer reversal [START_REF] Schorr | An efficient machine-independent procedure for garbage collection in various list structures[END_REF], a graph traversal algorithm developed for garbage collection, notorious for being hard to get right and to reason about.

In particular, the resulting implementation is equivalent to the naive implementation, but it does not overflow the stack. (Another benefit which we can expect is slightly improved performance due to better cache locality, but we do not propose benchmarks in this paper.)

This paper assumes familiarity with neither CPS nor defunctionalization. We directly give the general shape of the resulting implementation below in Section 2, which we use as a starting point.

Limitations to its applicability & goals

The typed pointer reversal method suffers from limitations if we want to use it in the construction of compilers:

• It does not handle abstract types: if a custom destructor supplied for a type is recursive, then it is not clear how to apply the transformation. Therefore, this method cannot be used by the compiler to automatically derive efficient destructors.

• It does not handle separate compilation: if the specification of a destructor is unknown, even if its efficient implementation is provided, then we cannot derive another efficient destructor that calls this destructor, because the method is not compositional.

• It does not handle unboxed types: currently, it assumes that all values are boxed behind a pointer (Lisp-like representation of values), whereas unboxed values (C-like representation of values, as used in Rust and C++) are not handled.

In this paper, we show that is is possible to extend typed pointer reversal by making it modular, in order to handle abstract types and separate compilation (Section 3), and we sketch an application to unboxed types (Section 5). Since pointer reversal is tricky to implement by hand and to prove correct, we ideally want to prove that the new transformations themselves are correct, and maybe relate them to ones that have already been studied; as far as the present work is concerned this is work in progress.

Througout the paper, we propose examples and descriptions in an ML-like syntax. There are two reasons for this choice:

1. the Lisp-like memory representation is simple, so is easy to start with, and there are additional issues with the C-like memory representation of Rust and C++ that we will not touch upon before the later parts of the paper;

2. it is possible to imagine extensions of ML that incorporate RAII-based resourcemanagement [START_REF] Munch-Maccagnoni | Resource polymorphism: A proposal for integrating first-class resources into ML[END_REF]. In this case, the challenges go beyond C++ and Rust, as witnessed by examples from Section 4 involving elaborate forms of polymorphism.

Example: B-trees The B-tree data structure constitutes a good example of the limitations of the approach from the previous work. A B-tree is a tree where each node can have a non-bounded number of children, and holds a non-bounded number of key-value pairs. It can be defined in ML as follows:

type 'a btree = Leaf of (int * 'a) array | Node of (int * 'a) array * 'a btree array This type uses arrays-values whose size is not known during compilation. This is not supported by typed pointer reversal as we described so far, since array is not an algebraic data type. In addition, an efficient destructor for array would be of no help in letting us deal with a recursion occurring via its parameter as in its third occurrence above. In fact, the naive destructor for arrays, which iterates on the elements, is already an efficient destructor (since array is not recursive on its own). Furthermore, we want to be able to derive an efficient destructor for B-trees, even when the implementations of arrays and of its destructor are opaque (for instance opaque to the compiler, when arrays are defined in another compilation unit). This might actually be of a lesser concern in languages that are not so strict about separate compilation, such as C++ and Rust. Lastly, in languages such as C++ and Rust, where we would substitute vectors for arrays, the vectors would be unboxed (as triplets consisting of a size, a capacity, and a pointer to a backing array).

All in all, if we want to apply typed pointer reversal to the implementation of programming languages, we need to determine how to handle abstract types, separate compilation, and unboxed types.

On efficient destructors for algebraic data types

Generic shape for typed pointer reversal

To start from where Douence and the second author [START_REF] Munch | Efficient deconstruction with typed pointer reversal[END_REF] left off, we first describe a generic form of the efficient destructors obtained with their method, for algebraic data types of the following generic shape:

A i = B i,1 (A 1 , . . . , A n ) + • • • + B i,mi (A 1 , . . . , A n ) 1 ≤ i ≤ n B i,j (A 1 , . . . , A n ) = C i,j,1 × • • • × C i,j,li,j 1 ≤ i ≤ n, 1 ≤ j ≤ m i
where C i,j,k is either of the form A i ′ , or some other type which does not contain any A i ′ .

In the latter case, the destructors are assumed given (as drop_C_{i,j,k}), and possibly arbitrary.

The specification of the destructor is to traverse the structure depth-first, from left to right, calling the destructors of the C i,j,k , and freeing the memory of each node it exits-as follows:

(* for all i: *) let rec drop_A_i t = match t with (* for all j: *) | B_{i,j}(c_1, ..., c_l) -> drop_C_{i,j,1} c_1; ...; drop_C_{i,j,l} c_l; free t

Above, recursion takes place when

C i,j,k is of the form A i ′ .
Applying the CPS transformation yields tail-recursive implementation of destructors, with an accumulator (the continuation), so as to avoid consuming stack space. Then, through defunctionalization, the continuation becomes a first-order data structure. We call these implementations of destructors drop_iter_A_i. Then, drop_A_i is equivalent to drop_iter_A_i called with initial continuation.

The type of defunctionalized continuations has the form cont = 1

+ i,j,k K k i,j,1 × • • • × K k i,j,li,j
where the sum is indexed by all tuples (i, j, k) where C i,j,k is of the form A i ′ for some i ′ , and where

K k i,j,k ′ = cont whenever k ′ ≤ k and C i,j,k is of the form A i ′ , and K k i,j,k ′ = C i,j,k otherwise.
After the transformation, we obtain the functions from Listing 1 below, written in ML-like pseudo-code plus explicit freeing of memory. We write K_{i,j,k} the constructor of the variant corresponding to (i, j, k) in the sum cont, and Empty the constructor for 1. In addition, we added the annotation @ on constructors to denote that a memory cell can be re-used, as explained next. The last part of the transformation involves re-using memory to store the continuations, so as to avoid allocating on the heap. We use the notation K@v(a, b, ...) above to mean that K(a, b, ...) can be allocated by reusing the memory cell of the value v. This involves mutating both the fields and the tag of v.

There are two key observations to make here. First, notice that t is no longer used in the rest of the destructor; it also cannot be used later in the program, since the destructor eventually frees its memory. Second, the new continuation always consists of a strict suffix of the previous value, plus a field to store the previous continuation. In particular, the re-used value is large enough, and at most one field needs to be mutated.

As written down, it is manifest that this implementation is tail-recursive and does not allocate.

Possible improvements

The previous section assumes that there is no limit on the number of variants. But there can be more variants for the continuation than there is in the original type. In practice, this can introduce additional constraints for the memory representation if values.

In order to bound the number of variants, the first observation is that we can reduce the amount of variants for the type cont to at most two for each pair (i, j). Indeed, from the second field being destroyed onwards, a second cell is available, which can store an additional integer. The k distinct variants can thus be replaced by two variants plus an auxiliary integer ranging from 2 to k.

We can further extend this solution to recursive types containing arrays (i.e. some of the C i,j,k being arrays possibly containing some A i s). As above, after the first element has been destroyed, we store an integer in the first field, which indicates how many fields have been destroyed so far.

Example: B-trees

Having just explained how to extend the method to arrays, let us go back to the example of B-trees. In Listing 2 we provide a destructor for a simplified version of the B-tree. The keyword cast is used below to coerce the type of values in an unsafe way. We need two variants in cont in order to deal with the array. The first one, K, allows us to drop its first element, and the store the continuation at its place. The second one, K', takes advantage of the fact that there is at least two free memory cells and store in the second one the number of cells already dropped. The variants Leaf and Node contain exactly zero or one element, thus we can further optimize and not adding variants for them inside cont. Listing 2. Efficient B-tree destructor.

An interface for a modular efficient drop

While we have shown that it is possible to handle B-trees by extending the algorithm to arrays, the goal is to obtain a modular interface to handle types that could be defined as an abtract type by a library (e.g. vector in C++/Rust), perhaps in an opaque compilation unit.

An interface in ML-like languages

In order to handle abstract types, we propose a modular interface that any type can implement in order to provide an efficient destructor. It allows composing such destructors automatically for algebraic data types, and can be implemented by hand for more complex data types. This (unsafe) interface is given in Figure 1b in the syntax of ML modules. It can be seen as both as a description of the interface needed by a compiler to generate an efficient destructor, and a refinement of the Drop trait in Rust's terminology (essentially the module type in Figure 1a). The Drop interface allows destructing a type t, and must recursively call the destructor of each of the values it contains. As we have seen so far, an implementation of Drop can be derived automatically for all mutually-recursive types comprised of algebraic types and arrays.

The DropIter interface allows implementing and composing efficient destructors; it is unsafe because the parameter 'a represents a type of continuations, but it is challenging to describe this type in advance. The key idea behind this interface is that each implementation will declare nb_state variants of a cont recursive type (which will never be formally declared), with these variants having a tag belonging to the interval [base; base + nb_state[. The implementation is parameterized by the base value, to be chosen such that the various implementations of DropIter involved in the destruction of a data structure do not interfere with each other.

The values nb_state and base are therefore constant and known at compile-time. The functions drop_iter and invoke play the same roles as in the previous sections, and can be automatically derived for algebraic data types. Their implementation for any base is known at compile-time.

Lastly, the implementation of a DropIter is generally parametrised by the DropIters of its type parameters: the modular efficient drop interface for a parametrised type ('a t) is a functor from DropIter to DropIter.

The type cont represents a type of continuations, which depends on the values of nb_state, base and the other instances of DropIter involved. We have not yet correctly typed the continuation in this interface in any language in a way that respects the required representation of values, we relied on unsafe features instead.

In invoke, implementations should look at the tag of the parameter, and if it is some universal and fixed integer constant, it should return (this means the iteration reached the end). If it belongs to the interval [base; base + nb_state[, it should handle it appropriately (either by doing something directly or by dispatching it to a DropIter implementation instantiated by the current implementation). Lastly, if it belongs to one of the DropIter implementations by which this implementation is parameterized, it must dispatch it accordingly. In correct implementations, other cases do not happen.

Given a DropIter instance, one can build a Drop instance in the following way: Here, Empty is the constant on which all invoke functions must return.

Example with a generic type

Now let us focus on a concrete example. Suppose we have a type parameterized by a type variable, for which we want to implement DropIter, such as the following: The KDropVal variant represents a node whose left child has been dropped. The KDropRight variant represents a node whose left child and the value it holds have been dropped, but not the right child.

This functor can be automatically derived with the typed pointer reversal method (e.g. by the compiler) in the case of mutually algebraic data types and arrays, similarly to the version of section 2.

Limitations due to polymorphism

Nested or non-regular data types are recursive polymorphic data types that can recursively instantiate themselves with different parameters.

Common examples of such data types, as presented by Matthes [START_REF] Matthes | Recursion on nested datatypes in dependent type theory[END_REF], include:

• Perfect binary trees : Tree α = α + Tree(α × α)

• Bushes : Bush α = 1 + α × Bush(Bush α)

• Lambda calculus with de Bruijn indexes and explicit substitutions:

Lam α = α + Lam α × Lam α + Lam(1 + α) + Lam(Lam α)
If we try to apply our algorithm to derive an efficient destructor for such data types, we will be stuck because our algorithm relies on the fact that we encounter a finite number of types in the implementation of a single recursive destructor. But if we want to generate a destructor for perfect binary trees Tree α, we will need to generate the destructor for Tree(α × α), Tree((α × α) × (α × α)), and so on. This would lead to an infinite number of states, at least with this approach.

But, for such types T we can still define a less efficient destructor which is better than the naive one, by using the drop function for β when we try to destruct T β with α ̸ = β. The stack consumption will then increase only when, while destructing a value of type T α, we destruct a value of type T β with β ̸ = α; outside of that, it is constant. Therefore for a value of type T α, the stack usage is proportional to the maximum number of edges T β → T γ where possibly β ̸ = γ that we encounter in a path from the origin to a leaf. This limitation does not apply to languages like C++ or Rust where all polymorphic arguments are monomorphized at compile-time, because we cannot define recursive functions on such a type. Monomorphization (substituting all generic types by a concrete type) is indeed less expressive than polymorphism à la ML.

Deconstructing unboxed types

Refining the interface for unboxed types

Currently, we have only considered values that are boxed, that is, are accessed through an indirection. But some languages such as C++ and Rust allow someone to manipulate unboxed values, that are laid out consecutively in memory. A problem arises when dealing with unboxed values, as we cannot overwrite the unboxed value with the continuation before we have dropped it entirely. It worked with boxed values because we have been exchanging values of the same size.

We propose in the next section a refinement of DropIter which drops a value of type T placed at a specific offset of a type U, rather than accessed by indirection. We deal with transitive subfields in the same way, in terms of offsets of a parent boxed type. Furthermore, a problem arises when dealing with mutually recursive types naively: if we have two mutually recursive types T and U of different unboxed sizes, then it is not clear which should be the size of the continuation.

We can address this problem by using tagged pointers for the continuation, where the tag is used to identify the continuation types before we dereference them. In typical 64-bit architecture, many bits of pointers are non-significant and can be re-used. 1 We call this kind of tagged pointer to a continuation a ContPointer.

Unboxing the efficient destructor

We introduce a refinement of the previous interface, with explicit pointers, in order to drop a type T that is inlined in another type U, at offset o (in bytes):

drop_field T,o,U : U * → ContPointer → 1 invoke_field T,o,U : ContPointer → 1 retrieve : T * → ContPointer
where U* is the type of pointers of U.

We can derive an implementation of this interface called DropField T,o,U for a type T starting with a DropIter interface for T, under a few conditions: we require that all calls to drop_iter for an inlined T take a continuation of the form KRest@u(k, ...) where u is the pointer to U that has been recycled (that is, we do not optimize the last call of a sequence of destructors as we did in subsection 2.2). We call this transformation unboxing.

The function drop_field is essentially drop_iter, but where right at the beginning we retrieve a T* by doing pointer arithmetic with the U* pointer in argument and the offset o. We use this T* pointer as in drop_iter, except that we store the tag in the pointer U* instead of the pointer T*; we still recycle the memory of our T* pointer (since we do not know the state of the rest of the structure, we can only access a portion of it).

For invoke_field, we cast the ContPointer into a U* and also do pointer arithmetic to retrieve a T*. On it, we execute the same code that invoke_iter (the function called invoke formerly) executes.

But inside these two functions we change one thing: we do not free the memory used by the pointer T*. Furthermore, the last recursive call which calls the continuation passed as a parameter (of the form k = KRest(k', ...)) is changed to a call to invoke_iter U KEnd T,o,U (k, ...) where we recycle the U* pointer and store the continuation k somewhere in the current structure, and where KEnd T,o,U is a new state that belongs to U.

And that is where the retrieve function comes in: it must be able to retrieve this continuation after the the destruction of the nested T*. The code to handle a KEnd T,o,U is also inserted into the invoke_iter U function: We need also to change the code calling drop_iter and invoke_iter for T when it is a field of U.

Every call to drop_iter T t KRest@u(k, ...) where t is inlined in some structure U must be replaced with drop_field T,o,U u k where o is the difference in bytes between t and u, and every call to invoke_iter T k must be replaced with invoke_field T,o,U k. When several types are inlined inside one another, the type U and the offset must be taken from the outermost type, that is, from the pointer that currently owns the allocation for the whole structure.

Example: handling vectors

The vector in C++ and Rust is an unboxed type which consists of a pointer to the start of a backing array, a capacity pointer to its end, and a length pointer that points past the last element of the vector.

If the capacity is greater than the length, then we can store the continuation at the end of the backing array. We can decrement the length pointer and drop the next element through drop_iter if there remain elements to destroy, or free the backing array and call the continuation otherwise. If the capacity is equal to the length, then one can first store the continuation by overwriting the capacity (since this information is redundant), while tagging the length pointer to indicate this first step taken. After the first step, the capacity pointer is restored from the length pointer, there is now room at the end, and one proceeds as above.

All this obviously relies on implementation details of vectors. The implementation of the efficient destructor therefore needs to be given by standard library implementors, via the modular efficient drop interface.

Further code and examples, including a detailed implementation for C++ vectors, are available in the public repository https://gitlab.com/JeanCASPAR/artefacts-jfla-2023.

Further work

More work is needed to formally define and prove the correctness of our transformations, especially the unboxing transformation. We would like to find more conceptual explanations of our transformations, which might simplify proofs of correctness. We would also like to clarify the constraints on the memory representation of variants, and distinguish pertinent memory representation details from irrelevant ones.

  type btree = Leaf | Node of btree array type cont = Empty | K of cont | K' of cont * int let rec drop_btree t k = match t with | Leaf -> invoke k | Node x -> free t; drop_array x k and drop_array a k = if Array.length a = 0 then invoke k else let y = a.(0) in drop_btree y (K@a k) and invoke = function | Empty -> () | K k' as k -> let (x : btree array) = cast k in if Array.length x = 1 then invoke k' else let y = x.(1) in drop_btree y (K'@x (k', 2)) | K' (k', i) as k -> let (x : btree array) = cast k in if Array.length x = i then invoke k' else let y = x.(i) in drop_btree y (K'@x (k', i + 1))

  Drop interface. module type Base = sig val base : int end module type DropIter = sig type t val nb_state : int module Make(B : Base) : sig type cont val drop_iter : t -> cont -> unit val invoke : cont -> unit end end (b) DropIter interface.

Figure 1 .

 1 Figure 1. Interfaces for destructors as ML modules.

module

  DropIterToDrop(D: DropIter) : (Drop with type t = D.t) = struct type t = D.t module Impl = D.Make(struct let base = 0 end) let drop x = Impl.drop_iter x Empty end

  type 'a tree = Node of 'a tree * 'a * 'a tree | LeafThe type parameter is required to also have a modular efficient destructor, therefore the implementation of DropIter for 'a tree is parametrised by a DropIter for 'a. In the pseudo-code below, we use ellipses to denote that the tags of variants are chosen among [base; base + 2[. module DropIterTree (D: DropIter) : (DropIter with type t = D.t tree) = struct type t = D.t tree let nb_state = 2 module Make(B : Base) = struct let d_base = B.base + nb_state module DImpl = D.Make(struct let base = d_base end) type cont = Empty | ... | KDropVal of cont * D.t * D.t tree (* with tag offset B.base *) | KDropRight of cont * D.t tree | ... let rec drop_iter t k = match t with | Leaf -> invoke k | Node (l, x, r) -> drop_iter l (KDropVal@t (k, x, r)) and invoke k = match k with | Empty -> () | KDropVal (k',x,r) -> D.drop_iter x (KDropRight@k (k',r)) | KDropRight (k',r) -> free k; drop_iter r k' | K _ when d_base <= tag K && tag K < d_base+D.nb_state -> DImpl.invoke k | _ -> assert false end end Listing 3. Efficient and modular drop implementation for 'a tree.

For instance, on the x86_64 architecture, one can re-use at least the lower 3 bits due to alignment, and at least 7 of the upper bits if 5-level paging is used [Int23, Vol. 3A, §4.5].


Phan Van Tin

A COMPACT ATTRACTOR FOR ENERGY CRITICAL AND SUPER-CRITICAL NLS

Keywords: Mathematics Subject Classication. 35Q55 Nonlinear Schrödinger equations, Compact attractor, soliton resolution conjecture, energy super-critical power

We study the asymptotic behavior of large data solutions to nonlinear Schrödinger equations (NLS)

In the case Tao [12] proved that in radial setting, as t → +∞, any solution u of bounded energy can split into a term of form e it∆ u + and remainder term converges in H 1 (R d ) to a compact attractor which is invariant under the NLS ow. In the case p ⩾ 4 d-2 , we work on Sobolev space H s (R d ) for some s > sp := d 2 -2 p , which is more complex than working on H 1 (R d ). Basing on the method in [12], we prove that any radial solution with uniform bounded on H s (R d ) can split as in the work of Tao.

x spatial localisation at xed times.

Second step: L ∞

x spatial localisation on a time interval. 5.3. Third step: L 2

x localisation at xed time 6. Appendix References

Introduction

We consider the following (NLS) equation:

(1.1)

iu t + ∆u = λ|u| p u + V u, (t, x) ∈ R × R d , u(0) = ϕ,
E(u) = sup t∈I ∥u(t)∥ 2 H s ,
For convenience, denote H = H s . We abbreviate ∥f ∥ L p (R d ) by ∥u∥ L p or sometimes by ∥u∥ p . We consider the following conjecture which is called soliton resolution conjecture: any global uniform bounded solution u of dispersive equations can be decomposed as t → +∞ as follows:

u(t) = e it∆ u + + J j=1 R j (t, x) + o(1),
where R j are solitons and o(1) converges to zero as t → +∞. There are two special cases of this conjecture:

(1) As t → +∞, u(t) = J=1 j=1 R j (t, x). In this case, u(t, x) is called multi-soliton. (2) As t → +∞, u(t) = e it∆ u + . In this case, u(t, x) scatters forward in time. Recently, in [START_REF] Duyckaerts | Soliton resolution for the radial critical wave equation in all odd space dimensions[END_REF] and [START_REF] Collot | Soliton resolution for the radial quadratic wave equation in six space dimensions[END_REF], the authors proved the soliton resolution conjecture holds for the radial critical wave equation in all odd space dimensions and in six space dimension. To the best of our knowledge, there is not a proof for the soliton resolution conjecture for general nonlinear Schrödinger equations even in radial setting. In [START_REF] Tao | A (concentration-)compact attractor for high-dimensional non-linear Schrödinger equations[END_REF] and [START_REF] Tao | A global compact attractor for high-dimensional defocusing nonlinear Schrödinger equations with potential[END_REF], the author proved a weaker result than the above soliton resolution conjecture. More precise, the author showed that any radial uniform bounded solution of a nonlinear Schrödinger equation (NLS) can decompose as t → +∞ by u(t) = e it∆ + K + o [START_REF] An | Local well-posedness for the inhomogeneous nonlinear Schrödinger equation in H s (R n ). Nonlinear Anal[END_REF], where K is a compact set which is invariant under ow of NLS. Our goal of this paper is to give a similar result to [START_REF] Tao | A (concentration-)compact attractor for high-dimensional non-linear Schrödinger equations[END_REF][START_REF] Tao | A global compact attractor for high-dimensional defocusing nonlinear Schrödinger equations with potential[END_REF] in the energy critical and supercritical cases i.e p ⩾ 4 d-2 . In energy critical and energy subcritical, there are many works relating to two above special cases of the soliton resolution conjecture for nonlinear Schrödinger equations. Let us give a short introduce for studying NLS in energy supercritical case. In [START_REF] Cazenave | The Cauchy problem for the critical nonlinear Schrödinger equation in H s[END_REF], the authors proved local well posedness of simple power NLS (V = 0 in (1.1)) in high regular Sobolev space H s (R d ) (s ⩾ s p ). In [START_REF] Killip | Energy-supercritical NLS: critical Ḣs -bounds imply scattering[END_REF], the authors showed that for defocusing NLS in high dimensions (d ⩾ 5), any uniform bounded in critical Sobolev space Ḣsp is global and scatters under a suitable condition of s p . Specially, in [START_REF] Merle | On blow up for the energy super critical defocusing nonlinear Schrödinger equations[END_REF], the authors showed that there exists a blow up solutions for NLS in energy supercritical setting even in defocusing case. In focusing case, there is not many studying for asymptotic behavior of solutions for large initial data in energy supercritical setting. Recently, in [START_REF] Duyckaerts | Prole decomposition and scattering for general nonlinear Schrödinger equations[END_REF], we establish prole decomposition and prove some scattering results for general nonlinear Schrödinger equations.

Inspiring by [START_REF] Cazenave | The Cauchy problem for the critical nonlinear Schrödinger equation in H s[END_REF], we study local and global theory for (1.1). Dene It is easy to check that (γ, ρ) is a admissible pair and satisfy the Sobolev embedding Ḣs,ρ → L ρ * , where ρ * such that

1 ρ ′ = p ρ * + 1 ρ .
We have the following properties of ρ, γ:

Proposition 1.1 ([4][Proposition 1.5). ] We have (i) 2 < ρ < 2d d-2 ; (ii) 2/γ = d(1/2 -1/ρ), and hence (γ, ρ) is an admissible pair; (iii) ρ < d/s and hence ρ * > ρ;

(iv) 1/ρ ′ = p/ρ * + 1/ρ; (v) 1/γ ′ ⩾ p/γ + 1/γ.

As in [START_REF] Cazenave | The Cauchy problem for the critical nonlinear Schrödinger equation in H s[END_REF], we have the following result. Theorem 1.2. Let ϕ ∈ H and p, s be as the above. Then there exists a solution of (1.1) u ∈ C(0, T max ; H) ∩ L γ loc (0, T max ; H s,ρ ). Moreover, the following properties hold: i u ∈ L q (0, T ; H s,r ) for every admissible pair (q, r) and every T < T max . ii u is unique in L γ (0, T ; H s,ρ ) for every

T < T max . iii ∥u(t)∥ L 2 = ∥ϕ∥ L 2 and E(u(t)) = E(ϕ) for all t ∈ (0, T max ). iv If T max < ∞ then lim t→ Tmax ∥u(t)∥ H = ∞.
From Theorem 1.2, we see that if solution u has bounded energy then u is global. Inspire by [START_REF] Tao | A (concentration-)compact attractor for high-dimensional non-linear Schrödinger equations[END_REF], we study asymptotic behavior of solutions with uniformly bounded energy. Our main result is the following: Theorem 1.3. Let E > 0. Then there exists a compact subset K E,rad ⊂ H which is invariant under the NLS ow, and such that for every radial global solution u of energy at most E, there exists a unique radiation u + ∈ H such that

(1.3) lim t→+∞ dist H (u(t) -e it∆ u + , K E,rad ) = 0.
Here and in the sequel we write dist H (f, K) := inf{∥f -g∥ : g ∈ K} for the distance between f and K.

Remark 1.4. Consider the compact set K E,rad in the theorem 1.3. Let u 0 ∈ K E,rad and u be the associated solution of (1.1). Since K E,rad is invariant under ow of (1.1), u(t) ∈ K E,rad for all t ∈ R. Thus, {u(t) : t ∈ R} is pre-compact in H s and u is called an almost periodic solution. Assume that each almost periodic solution of (1.1) must be zero, this property is called rigidity property (see e.g [START_REF] Kenig | Global well-posedness, scattering and blow-up for the energy-critical, focusing, nonlinear Schrödinger equation in the radial case[END_REF][Theorem 5.1]). Then Theorem 1.3 implies that each uniformly bounded solution of (1.1) scatters in both directions. However, in the case s = s p , the situation is more complex since local smoothing eect can not hold in this case (see Remark 3.18). Remark 1.5. It seems that we could also obtain a similar result in [START_REF] Tao | A (concentration-)compact attractor for high-dimensional non-linear Schrödinger equations[END_REF][Theorem 1.28] in nonradial setting. However, in this paper, we only focus on solutions in radial setting.

Let u be a solution as in Theorem 1.3. From Lemma 3.20, e -it∆ u(t) is weakly convergent to some u + ∈ H s (R d ) as t → +∞. Dene v(t) = u(t)-e it∆ u + , which be called weak bound component of u. We will show that u + satises the property (1.3).

We recall the following equivalence of precompactness and localisation: Proposition 1.6. (see e.g [START_REF] Tao | A (concentration-)compact attractor for high-dimensional non-linear Schrödinger equations[END_REF][Proposition B.1]) Let K ⊂ H. Then the following are equivalent:

(i) K is precompact in H.

(ii) K is bounded, and for any µ 0 > 0 there exists µ 1 > 0 such that we have frequency localisation estimate

P ⩾1/µ1 f H ≲ µ 0
and the spatial localisation estimate

|x|⩾1/µ1 |f (x)| 2 dx ≲ µ 2 0 ,
for all f ∈ K. (iii) K is bounded, and for any µ 0 > 0 there exists µ 1 > 0 such that we have frequency localisation estimate

P ⩾1/µ1 f H ≲ µ 0 and ∥P ⩽µ1 f ∥ H ≲ µ 0
and the spatial localisation estimate

|x|⩾1/µ1 |f (x)| 2 + |D s f (x)| 2 dx ≲ µ 2 0 ,
for all f ∈ K.

The above proposition is generalized by the below criterion for compact attractor:

Proposition 1.7. (see e.g [START_REF] Tao | A (concentration-)compact attractor for high-dimensional non-linear Schrödinger equations[END_REF][Proposition B.2]) Let U be a collection of trajectories u : R + → H.

Then the following are equivalent: (i) There exists a compact set K ⊂ H such that lim t→+∞ dist H (u(t), K) = 0 for all u ∈ U.

(ii) U is asymptotically bounded in the sense that for all u ∈ U. (iii) U is asymptotically bounded in the sense of (1.4), and for any µ 0 > 0 there exists µ 1 > 0 such that we have the asymptotic frequency localisation estimate

|u(t, x)| 2 + |D s u(x)| 2 dx ≲ µ 2 0 for all u ∈ U.
From Proposition 1.6 and Proposition 1.7, we see that precompactness are formally equivalent to localisation in spatial and frequency. From Proposition 1.7, to prove Theorem 1.3, we only need to prove asymptotic localisation in spatial and frequency of v(t).

This paper is organised as follows. In Section 2, we introduce the useful notations and basic tools in this paper. In Section 3, we prove local well-posedness of solutions of (1.1) in H s and give some basic results in the proof of the main result. In Section 4, we prove asymptotic localisation in frequency of v(t). In Section 5, we prove asymptotic localisation in spatial of v(t) and then prove the main theorem 1.3. In Appendix, we prove some lemmas used in the proof of the main result.

Acknowledgement

The author wishes to thank Prof.Thomas Duyckaerts for his guidance and encouragement. The author is supported by postdoc fellowship of Labex MME-DII: SAIC/2022 No 10078.

Notation and preliminaries

Let L = i∂ t + ∆ be Schrödinger operator.

For convenience, we dene

f 1 (u) = λ|u| p u, f 2 (u) = V u and f (u) = f 1 (u) + f 2 (u).
For each X Banach space, dene ∥(u, v)∥ X = ∥u∥ X + ∥v∥ X . Denote S is the space of Schwartz functions. We shall need some small exponents

1 ≫ η 0 ≫ η 1 ≫ η 2 ≫ η 3 > 0,
where η 0 is small enough depending on E and the other xed parameters, η i is small enough depending on

E, η 0 , η 1 , • • • , η i-1 .
Moreover, in Section 5, we shall need some other constants:

1 > µ 0 ≫ µ 1 ≫ µ 2 ≫ µ 3 ≫ µ 4 > 0.
where µ 0 is small enough depending on E and the other xed parameters, µ i is small enough depending on µ 0 , µ 1 , • • • , µ i-1 .

Lemma 2.1. There exists an admissible pair (q 0 , r 0 ) with q 0 > 2, exponents 2 < Q 0 , Q < 2d d-2s , and an exponent 1 ⩽ R < 2d d+4 such that (2.1)

1 r 0 + p Q 0 = 1 r ′ 0 and
(2.2)

1 2 + p Q = 1 R .
Lemma 2.2. For any u ∈ H, we have

∥f (u)∥ H s,R ≲ ∥u∥ p+1 H + ∥u∥ H .
Proof. By Hölder, we have

D j f 1 (u(t)) L R ≲ ∥u(t)∥ p L Q D j u(t) L 2 ,
for j = 0, s. Let a be such that 1

R = 1 a + 1 2 . Thus, 2 ⩽ a ⩽ d 2 .
By Hölder and Sobolev embedding, for j = 0, s, we have

D j (f 2 (u)) L R = D j (V u) L R ≲ D j (V ) L a ∥u∥ L 2 + ∥V ∥ L a D j u L 2 ≲ ∥u∥ L 2 + D j u L 2 ≲ ∥u∥ H .
This implies the desired result. □ Denote F(f ) = f is Fourier-transformation by space variable. We have the following basic property in Fourier-analysis. Lemma 2.3. Let f, g ∈ S. we have the following identity:

F(f g) = F(f ) * F(g).
Lemma 2.4. For any time interval I ⊂ R, any t 0 ∈ I and any 0 < δ ⩽ 1 2 , we have

∥uv∥ L 2 t,x (I×R d ) ≲ ∥u(t 0 )∥ Ḣ-1/2+δ + |∇| -1/2+δ Lu L q ′ L r ′ (I×R d ) × ∥v(t 0 )∥ Ḣ(d-1)/2-δ + |∇| (d-1)/2-δ Lv L q′ L r′ (I×R d ) ,
for any u, v and any admissible pairs (q, r), (q, r) with q, q > 2.

As a consequence, we have the following result.

Lemma 2.5. (see e.g [START_REF] Tao | A (concentration-)compact attractor for high-dimensional non-linear Schrödinger equations[END_REF][Corollary 4.4]) For any time interval I ⊂ R, any N, M > 0 dyadic numbers, we have

∥u N u M ∥ L 2 t,x (I×R d ) ≲ ⟨|I|⟩ 1 2 M (d-1)/2-δ N -1 2 +δ ⟨N ⟩ s ⟨M ⟩ s .
Lemma 2.6 (Dispersive estimate). For all 2 ⩽ r ⩽ +∞ and f ∈ L r ′ (R d ), we have

e it∆ f L r ≲ 1 |t| d(1/2-1/r) ∥f ∥ L r ′ . Lemma 2.7. For 1 ⩽ r ⩽ 2, f ∈ D(R d ), we have e it∆ f L r ≲ f (1 + t) M , for all M ∈ N and M > d/2 -d/r.
Proof. Let a be such that

1 r = 1 a + 1 2 .
Let M be a natural number such that M a > d.

We have

e it∆ f L r = F -1 (e -it|ξ| 2 f (ξ))(x) L r x = (1 + |x|) -M (1 + |x|) M F -1 (e -it|ξ| 2 f (ξ))(x) L r x ≲ (1 + |x|) -M L a (1 + |x|) M F -1 (e -it|ξ| 2 f (ξ))(x) L 2
x ≲ e -it|ξ| 2 f (ξ)

H M ξ ≲ f (t + 1) M . □

Local estimate

This section gives some useful tools for studying asymptotic behavior as t → +∞ of solutions of (1.1). First, we introduce some basic tools in studying of fractional derivative.

3.1. Basic tools of fractional derivative. We need the following results on fractional derivative for our analysis:

Lemma 3.1. Let -∞ < s 2 ⩽ s 1 < ∞ and 1 < p 1 ⩽ p 2 < ∞ with s 1 -d p1 = s 2 -d p2 .
Then we have the following embedding:

Ḣs1,p1 ⊂ Ḣs2,p2 , H s1,p1 ⊂ H s2,p2 .
Lemma 3.2 (Product chain rule 1). (see e.g [START_REF] An | Local well-posedness for the inhomogeneous nonlinear Schrödinger equation in H s (R n ). Nonlinear Anal[END_REF] As a consequence of the above lemma, we have the following: Lemma 3.3 (Product chain rule 2). (see e.g [START_REF] An | Local well-posedness for the inhomogeneous nonlinear Schrödinger equation in H s (R n ). Nonlinear Anal[END_REF][Corollary 2.3]) Let s ⩾ 0, q ∈ N (q ⩾ 1),

[Lemma 2.2]) Let s ⩾ 0, 1 < r, r 2 , p 1 < ∞, 1 < r 1 , p 2 ⩽ ∞ such that 1 r = 1 ri + 1 pi (i = 1, 2
1 < r, r i k < ∞, for 1 ⩽ i, k ⩽ q: 1 r = q i=1 1 r i k . Then q i=1 f i Ḣs,r ≲ q k=1   ∥f k ∥ Ḣs,r k k i̸ =k ∥f i ∥ r i k   . Lemma 3.4 (Fractional chain rule). (see e.g [1][Lemma 2.4]) Let G ∈ C 1 (C), s ∈ (0, 1), 1 < r, r 2 < ∞, and 1 < r 1 ⩽ ∞ satisfying 1 r = 1 r1 + 1 r2 , ∥G(u)∥ Ḣs,r ≲ ∥G ′ (u)∥ r1 ∥u∥ Ḣs,r 2 .
Lemma 3.5 (Gagliardo-Nirenberg inequality). (see [START_REF] Brezis | Gagliardo-Nirenberg inequalities and non-inequalities: the full story[END_REF]). Let

s 1 ⩽ s 2 , p 0 > 1, s = θs 1 + (1 -θ)s 2 , 1 p = θ p1 + 1-θ p0 . Then ∥u∥ H s,p ≲ ∥u∥ θ H s 1 ,p 1 ∥u∥ 1-θ H s 2 ,p 0 . Lemma 3.6 (Homogeneous Gagliardo-Nirenberg inequality). Let s 1 ⩽ s 2 , p 0 > 1, s = θs 1 + (1 - θ)s 2 , 1 p = θ p1 + 1-θ p0 . Then ∥u∥ Ḣs,p ≲ ∥u∥ θ Ḣs 1 ,p 1 ∥u∥ 1-θ Ḣs 2 ,p 0 . Lemma 3.7. (see [1][Lemma 3.2]) Let s > 0, 1 < p < ∞ and v = s -⌊s⌋. Then |α|=⌊s⌋ ∥D α f ∥ Ḣv p ≈ ∥f ∥ Ḣs p ,
where ⌊s⌋ is largest integer below s.

Lemma 3.8 (Leibniz rule). Let f ∈ C k be a real variable function and α

= (α 1 , α 2 , • • •, α d ) ∈ N d such that |α| ⩽ k. Then D α f (u)
is written as follows:

(3.1)

D α f (u) = |α| h=1 Λ h α C α,h f (h) (u) h i=1 D βi u, where C α,h ∈ N and Λ h α = {(β 1 , • • •, β h ) : for each i: β i ∈ N d , β 1 + • • • + β h = α, |β i | ⩾ 1}. 3.2.
Local theory in H s . In this subsection, we prove local theory for solutions in H s of (1.1)

Theorem 1.2. Dene G(f (u(t))) = -i t 0 S(t -τ )f (u(τ )) dτ.
We may rewrite (1.1) by

u(t) = S(t)ϕ + G(f (u(t))).
Before proving Theorem 1.2, we need some tools. Lemma 3.9 ([4][Lemma 4.1). ] We have for any (q, r) admissible:

∥G(f (u))∥ L q (0,T ) Ḣs,r (R d ) ≲ T δ ∥u∥ p+1 L γ (0,T ) Ḣs,ρ (R d ) + T γ-2 γ ∥u∥ L γ (0,T ) Ḣs,ρ (R d ) (3.2)
and

∥G(f (u)) -G(f (v))∥ L q (0,T )L r (R d ) ≲ T δ ∥u∥ p L γ (0,T ) Ḣs,ρ (R d ) + ∥v∥ p L γ (0,T ) Ḣs,ρ (R d ) ∥u -v∥ L γ (0,T )L ρ (R d ) + T γ-2 γ ∥u -v∥ L γ (0,T )L ρ (R d ) , (3.3)
where δ = 1 -p+2 γ . Proof. Use the proof of Lemma 3.11 and note that Ḣs,ρ → L ρ * . Remark that in this proof, we only need the condition p even or p > ⌊s⌋.

□ Moreover, we have the following unique result. Proposition 3.10 ([4][Proposition 4.2). ] Let ϕ ∈ H s , and suppose u ∈ L γ (0, T ; H s,ρ ) is a solution of (1.1). Then u ∈ L q (0, T ; H s,r ) ∩ C([0, T ]; H s ) for every admissible pair (q, r). Furthermore, if v ∈ L γ (0, T ; H s,ρ ) is also a solution of (1.1), then u = v. Proof. By Lemma 3.9 and Strichartz, it is easy to prove that u ∈ L q (0, T ;

H s,r ) ∩ C([0, T ]; H s ). Assume u(t) ̸ = v(t) for some t ∈ [0, T ]. Let t 0 = inf{t ∈ [0, T ], u(t) ̸ = v(t)}.
Using Lemma 3.9, we have, for all t ∈ [t 0 , T ]:

∥u -v∥ L γ (t0,t);L ρ = ∥G(f (u)) -G(f (v))∥ L γ (t0,t);L ρ ⩽ C(t -t 0 ) δ ∥u∥ p L γ (t0,t); Ḣs,p + ∥v∥ p L γ (t0,t); Ḣs,p ∥u -v∥ L γ (t0,t);L ρ + ∥u -v∥ L γ (t0,t);L ρ (t -t 0 ) γ-2 γ .
Let t be close to t 0 , we give a contradiction. Thus, u(t) = v(t) for all t ∈ [0, T ], which is the desired result.

□ Proof Theorem 1.2. We use xed point argument to prove existence of solution of (1.1). Let M > 0, we dene

X = X (T, M ) = {u ∈ L γ (0, T ; H s,ρ ) : ∥u∥ L γ (0,T ; Ḣs,ρ ) ⩽ M }.
Endowed with the metric

d(u, v) = ∥u -v∥ L γ (0,T ;L ρ ) ,
X is a complete metric space. We wish to prove that the map

Fu = S(•)ϕ + G(f (u)).
is a contradiction map on X for suitable constants T, M . Let u ∈ X . Using (3.2), we have if

∥S(•)ϕ∥ L γ (0,T ; Ḣs,p ) +CT δ M p+1 +CT γ-2 γ M ⩽ M then Fu ∈ X . From (3.3), if 2CT δ M p +CT γ-2 γ < 1 then F is a strict contradiction on X . By Strichartz, F is a contradiction from X to X if (3.4) C ∥ϕ∥ Ḣs + 2CT δ M p+1 + CT γ-2 γ M ⩽ M.
Combining with Proposition 3.10, there exists a unique maximal solution of (1.1) u ∈ C(0, T max ; H)∩ L γ loc (0, T max ; H s,ρ ). Assume T max < ∞. Using (3.4) and by classical argument (see e.g [START_REF] Cazenave | Semilinear Schrödinger equations[END_REF]), we have

lim t→Tmax ∥u(t)∥ Ḣs = ∞,
which completes the desired result. □ Lemma 3.11. Let p be even or p ⩾ ⌈s⌉ and u, v ∈ S s (I × R d ). We have the following estimates:

∥f 1 (u) -f 1 (v)∥ L q ′ 0 H s,r ′ 0 (I×R d ) ≲ |I| 1/q ′ 0 -1/q0 ∥u -v∥ L q 0 Ḣs,r 0 (I×R d ) ∥(u, v)∥ p L ∞ L Q 0 (I×R d ) + ∥u -v∥ L ∞ L Q 0 (I×R d ) ∥(u, v)∥ p-1 L ∞ L Q 0 (I×R d ) ∥(u, v)∥ L q 0 H s,r 0 (I×R d ) . ∥f 2 (u) -f 2 (v)∥ L q ′ 0 H s,r ′ 0 (I×R d ) ≲ V ∥u -v∥ L q 0 H s,r 0 |I| 1/q ′ 0 -1/q0 .
Proof. It suces to show the rst estimate. The second estimate is proved by using Hölder and

V ∈ D(R d ).
We only consider the case p ⩾ ⌈s⌉; in the case p even the conclusion is easily proved by using Hölder. By Hölder,

∥f 1 (u) -f 1 (v)∥ L q ′ 0 H s,r ′ 0 ≲ ∥f 1 (u) -f 1 (v)∥ L q 0 H s,r ′ 0 |I| 1/q ′ 0 -1/q0 .
Thus, we only need to prove that

∥f 1 (u) -f 1 (v)∥ L q 0 H s,r ′ 0 ≲ ∥u -v∥ L q 0 Ḣs,r 0 (I×R d ) ∥(u, v)∥ p L ∞ L Q 0 (I×R d ) + ∥u -v∥ L ∞ L Q 0 (I×R d ) ∥(u, v)∥ p-1 L ∞ L Q 0 (I×R d ) ∥(u, v)∥ L q 0 H s,r 0 (I×R d ) .
We have

(3.5) f 1 (u) -f 1 (v) = (u -v) 1 0 ∂ z f 1 (v + θ(u -v))dθ + u -v 1 0 ∂ z f 1 (v + θ(u -v))dθ.
By Hölder and using

|∂ z f 1 (u)| + |∂ z f 1 (v)| ≲ |u| p , ∥f 1 (u) -f 1 (v)∥ L q 0 L r ′ 0 ≲ ∥u -v∥ L ∞ L Q 0 sup θ∈[0,1] ∥|v + θ(u -v)| p ∥ L q 0 L a , ( where 1/a + 1/Q 0 = 1/r ′ 0 ) ≲ ∥u -v∥ L ∞ L Q 0 (∥|u| p ∥ L q 0 L a + ∥|v| p ∥ L q 0 L a ) ≲ ∥u -v∥ L ∞ L Q 0 ∥(u, v)∥ p-1 L ∞ L Q 0 ∥(u, v)∥ L q 0 L r 0 .
Thus, it remains to show that

∥f 1 (u) -f 1 (v)∥ L q 0 Ḣs,r ′ 0 ≲ ∥u -v∥ L q 0 Ḣs,r 0 (I×R d ) ∥(u, v)∥ p L ∞ L Q 0 (I×R d ) + ∥u -v∥ L ∞ L Q 0 (I×R d ) ∥(u, v)∥ p-1 L ∞ L Q 0 (I×R d ) ∥(u, v)∥ L q 0 Ḣs,r 0 (I×R d ) .
By Hölder, it suces to show:

(3.6) ∥f 1 (u) -f 1 (v)∥ Ḣs,r ′ 0 ≲ ∥u -v∥ Ḣs,r 0 ∥(u, v)∥ p L Q 0 + ∥u -v∥ L Q 0 ∥(u, v)∥ p-1 L Q 0 ∥(u, v)∥ Ḣs,r 0 .
By using (3.5) (with noting that the contribution of the rst term and the second term are similar) and product rule Lemma 3.2, we have

∥f 1 (u) -f 1 (v)∥ Ḣs,r ′ 0 ≲ ∥u -v∥ L Q 0 1 0 |∇| s ∂ z f 1 (v + θ(u -v))dθ L a + ∥u -v∥ Ḣs,r 0 1 0 ∂ z f 1 (v + θ(u -v))dθ L b (where 1/b + 1/r 0 = 1/r ′ 0 ). Since |∂ z f 1 (u)| ≲ |u| p , we have 1 0 ∂ z f 1 (v + θ(u -v))dθ L b ≲ ∥|u| p + |v| p ∥ L b ≲ ∥(u, v)∥ p L Q 0 ,
hence the second term is acceptable. Thus, to prove (3.6), it suces to show that, for g = ∂ z f 1 , for each u:

(3.7)

∥|∇| s g(u)∥ L a ≲ ∥u∥ p-1 L Q 0 ∥u∥ Ḣs,r 0 .
Since p > ⌈s⌉, the function g belongs to C ⌈s⌉ and satises

(3.8) |g (k) (u)| ≲ |u| p-k ,
for each 0 ⩽ k ⩽ ⌈s⌉ . By Lemma 3.7 and Leibniz rule Lemma 3.8, we only need to prove for each α

∈ N d , 1 ⩽ h ⩽ |α|, each 1 ⩽ i ⩽ h: β i ∈ N d , |β i | ⩾ 1: h i=1 β i = α:
(3.9)

g (h) (u) h i=1 D βi u Ḣv,a ≲ ∥u∥ p-1 L Q 0 ∥u∥ Ḣs,r 0 , (where v = s -⌊s⌋).
We have h ⩽ |α| ⩽ ⌊s⌋ ⩽ ⌈s⌉ < p. Using Lemma 3.3 and (3.8), we have

g (h) (u) h i=1 D βi u Ḣv,a ≲ g (h) (u) Ḣv,m 0 h i=1 D βi u L m i (3.10) + h k=1 |u| p-h L m0 D β k u Ḣv, mk h i=1,i̸ =k D βi u L m i ,
where m i and mi for i = 0, • • • , h are chosen such that

1 m i = θ i r 0 + 1 -θ i Q 0 , ( where
θ i = |β i |/s) 1 m 0 = 1 a - h i=1 1 m i , m0 = Q 0 p -h , 1 mk = 1 a - p -h Q 0 - h i=1,i̸ =k 1 m i = (|β k | + v)/s r 0 + 1 -(|β k | + v)/s Q 0 ,
By the choice of m i and mi , using Gagliardo-Nirenberg inequality Lemma 3.6, we have

D βi u L m i ≲ ∥u∥ θi Ḣs,r 0 ∥u∥ 1-θi L Q 0 , D β k u Ḣv, mk ≲ D |β k |+v u L mk ≲ ∥u∥ θk Ḣs,r 0 ∥u∥ 1-θk L Q 0 , (where θk = (|β k | + v)/s).
Thus, the second term in (3.10) is acceptable. Consider the rst term. If s / ∈ N then ⌈s⌉ ⩾ h + 1 (since h ⩽ |α| = ⌊s⌋), using fractional chain rule Lemma 3.4, we have

g (h) (u) Ḣs,m 0 ≲ g (h+1) (u) L m ∥u∥ Ḣv,n ( where 1/m + 1/n = 1/m 0 ) ≲ |u| p-h-1 L m ∥u∥ Ḣv,n . Choosing m such that (p -h -1)m = Q 0 then 1/n = 1/m 0 -1/m = v/s r0 + ⌊s⌋/s Q0 . Using Gagliardo- Nirenberg inequality Lemma 3.6, we have ∥u∥ Ḣv,n ≲ ∥u∥ v/s Ḣs,r 0 ∥u∥ ⌊s⌋/s L Q 0 .
Thus, the rst term in (3.10) is acceptable. Then, (3.9) holds and the proof is completed. □ Remark 3.12. In the proof of the above lemma, the condition p > ⌈s⌉ is needed. Lemma 3.13 (Perturbation lemma). Let u 0 ∈ H, let I be a compact time interval containing t 0 , let A, μ1 > 0, and v : I → H be strong solution of the following NLS:

iv t + ∆ = f (v) + G where v, G satisfy ∥v∥ L q 0 H s,r 0 ∩C 0 H(I×R d ) + ∥v(t 0 ) -u 0 ∥ H ≲ A and e i(t-t0)∆ (v(t 0 ) -u 0 ) L q 0 H s,r 0 ∩L ∞ L Q 0 (I×R d ) ≲ μ1
and

∥G∥ L q ′ 0 H s,r ′ 0 (I×R d ) ≲ μ1 .
Then, if μ1 is small enough depending on A, |I| there exists a solution u : I → H of (1.1) with initial data u(t 0 ) = u 0 such that

∥u -v∥ L q 0 H s,r 0 ∩L ∞ L Q 0 (I×R d ) ≲ A,|I| μ1 . and ∥u -v∥ C 0 H ≲ A,|I| 1.
Proof. By local theory, u exists on I. Let w = u -v, then w solves the equation

iw t + ∆w = f (w + v) -f (v) -G; w(0) = v(t 0 ) -u 0 . Dene X = ∥w∥ L q 0 H s,r 0 ∩L ∞ L Q 0 (I×R d ) .
We have

X ≲ μ1 + ∥f (v + w) -f (v)∥ L q ′ 0 H s,r ′ 0 .
By Lemma 3.11, we have

X ≲ μ1 + |I| 1/q ′ 0 -1/q0 X((X + A) p + 1).
If |I| ⩽ C(A) (depending only on A) and μ1 is small enough depending on I, A, we conclude that X ≲ μ1 on I. Moreover,

∥u -v∥ C 0 H(I×R d ) ≲ A + ∥f (u) -f (v)∥ L q ′ 0 H s,r ′ 0 ≲ A 1.
Dividing I into |I|/C(A) interval I k , on each I k , we have

∥w∥ L q 0 H s,r 0 ∩L ∞ L Q 0 (I k ×R d ) ≲ μ1
and

∥w∥ C 0 H(I k ×R d ) ≲ A 1.
By inductive argument, the claim follows.

∥u(t)∥ L q ≲ 1 for all 2 ⩽ q ⩽ 2d d -2s and t ∈ [0, +∞).
From Lemma 2.2, we have

(3.13) ∥f (u)∥ H s,R ≲ 1 for all t ∈ [0, +∞).
∥u∥ L q H s,r (I) ≲ ⟨|I|⟩ 1 q
as well as the nonlinearity estimate

(3.15) ∥f (u)∥ L q ′ 0 H s,r ′ 0 (I) ≲ ⟨|I|⟩ 1 q ′ 0 .
Proof. By Theorem 1.2, there exist and uniqueness of solution u are proved. Moreover, u is global and u ∈ L q loc H s,r (R × R d ) for any (q, r) admissible. By subdividing I it suces to prove that if

|I| ≲ 1 then ∥u∥ L q H s,r (I) ≲ 1 and ∥f (u)∥ L q ′ 0 H s,r ′ 0 (I) ≲ 1. Dene X = ∥u∥ L q 0 (I)H s,r 0 (R d ) .
By Strichartz and Hölder inequality, we have

X ≲ ∥u∥ L ∞ H + ∥|u| p u∥ L q ′ 0 (I)H s,r ′ 0 + ∥V u∥ L 1 L 2 ≲ 1 + ∥|u| p u∥ L q ′ 0 (I)H s,r ′ 0 + ∥V ∥ L 1 L ∞ ∥u∥ L ∞ L 2 ≲ 1 + |I| 1 q ′ 0 -1 q 0 ∥u∥ p L ∞ L Q 0 X ≲ 1 + |I| 1 q ′ 0 -1 q 0 X, ( where we use ∥u∥ L ∞ L Q 0 ≲ ∥u∥ L ∞ H ≲ 1) ≲ 1 + 1 2 X,
if we make |I| small enough. This give X = O(1) and then ∥f (u

)∥ L q ′ 0 H s,r ′ 0 = O(1)
. This completes our proof.

□

The following result is useful to prove asymptotic localisation frequency of v: Proposition 3.15. (Local smoothing eect). We have the following result (3.16)

∥P N f (u)∥ L q ′ 0 (I)L r ′ 0 ≲ ⟨N ⟩ -s-η1 ⟨|I|⟩ 1 q ′ 0 .
Proof. It suces to prove Proposition 3.15 for |I| ⩽ 1. Indeed, assume Proposition 3.15 holds for

|I| ⩽ 1. Divide I into |I| subintervals I k such that |I k | ≈ 1.
On each I k , we have

∥P N f (u)∥ L q ′ 0 (I k )L r ′ 0 ≲ ⟨N ⟩ -s-η1 .
Summing the above inequality in k, we obtain (3.16). By Lemma 3.14, we may also assume N ⩾ 1.

For convenience, we shall omit the domain I × R d in all norms in this proof. Consider the case p is not even and p > ⌈s⌉. Our rst task is to show that (3.17)

∥D s P N f 1 (u)∥ L q ′ 0 L r ′ 0 ≲ N -η1 ,
where f 1 (u) = λ|u| p u. For convenience, in the proof of (3.17), we abbreviate f 1 by f . Using Lemma 3.7 and Lemma 3.8, we only need to prove that (3.18)

P N f (h) (u) h i=1 D βi u L q ′ 0 Ḣv,r ′ 0 ≲ N -η1 , where 1 ⩽ h ⩽ ⌊s⌋, β i , α ∈ N d , |β i | ⩾ 1, |α| = ⌊s⌋ and h i=1 β i = α.
We have

P N f (h) (u) h i=1 D βi u L q ′ 0 Ḣv,r ′ 0 ≲ P ≳N f (h) (u) h i D βi u L q ′ 0 Ḣv,r ′ 0 (3.19) + h k=1 f (h) (u)(P ≳N D β k u) h i=1;i̸ =k D βi u L q ′ 0 Ḣv,r ′ 0 , (3.20)
where

P ≳N = P ⩾ N 1+⌊s⌋ . By Lemma 3.3, (3.19
) is controlled by the following

P ≳N f (h) (u) L q 1 Ḣv,r 1 h i D βi u L q 0 i L r 0 i + P ≳N f (h) (u) L q 2 L r 2 h k=1 D β k u L q k k Ḣv,r k k h i=1,i̸ =k D βi u L q k i L r k i ≲ P ≳N u L q 1 1 Ḣv,r 1 1 |u| p-h L q 2 1 L r 2 1 h i D βi u L q 0 i L r 0 i (3.21) + P ≳N u L q 1 2 L r 1 2 |u| p-h L q 2 2 L r 2 2 h k=1 D β k u L q k k Ḣv,r k k h i=1,i̸ =k D βi u L q k i L r k i , (3.22)
where the above parameters satisfy, for each k ∈ [1, h]

1 q ′ 0 = 1 q 1 + h i=1 1 q 0 i , (3.23) 1 r ′ 0 = 1 r 1 + h i=1 1 r 0 i , (3.24)
1 q ′ 0 = 1 q 2 + h i=1 1 q k i , (3.25) 1 r ′ 0 = 1 r 2 + h i=1 1 r k i .
(3.26)

In (3.21), the parameters are chosen such that, for each i ∈ [1, h]

2 q 0 i + d r 0 i = d 2 -(s -|β i |),
(3.27)

2 q 2 1 + d r 2 1 = (p -h) d 2 -s .
(3.28)

Let s 1 be such that 2

q 1 1 + d r 1 1 = d 2 -s 1 .
From (3.23), (3.24), (3.27), (3.28), we see that

s 1 = ⌊s⌋ -2 -p d 2 -s .
It is easy to check that 0 < s 1 and s 1 + v < s (using s > s p ⩾ 1). Thus, by Sobolev embedding Lemma 3.1 and Lemma 3.14, we have

(3.21) ≲ D s1+v P ≳N u L q 1 1 L m 1 1 ∥D s u∥ p-h L (p-h)q 2 1 L m 2 1 h i=1 ∥D s u∥ L q 0 i L m 0 i ≲ N s1+v-s ∥D s u∥ L q 1 1 L m 1 1 ∥D s u∥ p-h L (p-h)q 2 1 L m 2 1 h i=1 ∥D s u∥ L q 0 i L m 0 i ≲ N -(2-p( d 2 -s)) ,
where it is easy to check that m 1 1 , m 2 1 , m 0 1 are such that (q 1 1 , m 1 1 ), ((p -h)q 2 1 , m 2 1 ), (q 0 i , m 0 i ) (for i ∈ [1, h]) are admissible pairs. In (3.22), the parameters are chosen such that,

2 q 2 2 + d r 2 2 = (p -h) d 2 -s , (3.29)
2 q k k + d r k k = d 2 -(s -|β k | -v), (3.30)
2 q k i + d r k i = d 2 -(s -|β i |), for eachi ̸ = k, k ∈ [1, h]. (3.31) Let s 2 be such that 2 q 1 2 + d r 1 2 = d 2 -s 2 . From
(3.22) ≲ D s2 P ≳N u L q 1 2 L m 1 2 ∥D s u∥ p-h L (p-h)q 2 2 L m 2 2 h k=1 ∥D s u∥ L q k k L m k k h i=1,i̸ =k ∥D s u∥ L q k i L m k i ≲ N s2-s ∥D s u∥ L q 1 2 L m 1 2 ∥D s u∥ p-h L (p-h)q 2 2 L m 2 2 h k=1 ∥D s u∥ L q k k L m k k h i=1,i̸ =k ∥D s u∥ L q k i L m k i ≲ N -(2-p( d 2 -s)) ,
where m 1 2 , m 2 2 , m k i are such that (q 1 2 , m 1 2 ), (q 2 2 , m 2 2 ), (q k i , m k i ) (for i ∈ [1, h]) are admissible pairs. Combining the above, we have

(3.19) ≲ N -(2-p( d 2 -s)) .
Similarly, we have

(3.20) ≲ N -(2-p( d 2 -s)) .
This implies that

∥D s P N f (u)∥ L q ′ 0 Ḣv,r ′ 0 ≲ N -(2-p( d 2 -s)) .
Moreover, since s > s p , we have 2 -p d 2 -s > 0.

Our second task is to show the following (3.32)

∥D s P N (V u)∥ L q ′ L r ′ ≲ N -η1 ,
for all (q, r) admissible pair. To prove (3.32), we use the following results.

Lemma 3.16.

∥P ⩾N V ∥ L r ≲ V,k N -k for all k ⩾ 0.
Lemma 3.17. (See [START_REF] Fujiwara | Higher order fractional Leibniz rule[END_REF][Corollary 1.1]) Let p, p 1 , p 2 be such that 1 < p, p 1 , p 2 < ∞ and

1 p = 1 p1 + 1 p2 . The the following holds for all f, g ∈ S ∥D s (f g) -D s f g -D s gf ∥ L p ⩽ C ∥Df ∥ L p 1 D s-1 g L p 2 .
Denote P <N η 0 u = u lo and u hi = u -u lo . We have

∥D s P N (V u)∥ L r ′ = ∥D s P N (V lo u)∥ L r ′ + ∥D s P N (V hi u)∥ L r ′ = D s P N (V lo P >N/10 u) L r ′ + ∥D s P N (V hi u)∥ L r ′
Consider the second term. Using Lemma 3.16, Lemma 3.17 and Hölder inequality, we have

∥D s P N (V hi u)∥ L q ′ L r ′ = ∥D s P N (V hi u)∥ L q ′ L r ′ ⩽ ∥D s (V hi u)∥ L q ′ L r ′ ⩽ ∥D s (V hi )u∥ L q ′ L r ′ + ∥V hi D s u∥ L q ′ L r ′ + ∥DV hi ∥ L k 1 L h 1 D s-1 u L k 2 L h 2 ≲ ∥D s V hi ∥ L q 1 L r 1 ∥u∥ L q L r + ∥V hi ∥ L q 1 L r 1 ∥D s u∥ L q L r + ∥DV hi ∥ L q 1 L r 1 D s-1 u L q L r ≲ V N -η0 .
where (q 1 , r 1 ) is such that

1 q1 = 1 q ′ -1 q and 1 r1 = 1 r ′ -1
r . Thus, this term is acceptable. Consider the rst term. Using Lemma 3.17 and Hölder inequality, we have

D s P N (V lo P >N/10 u) L q ′ L r ′ ⩽ D s (V lo P >N/10 u) L q ′ L r ′ ⩽ D s V lo P >N/10 u L q ′ L r ′ + V lo D s (P >N/10 u) L q ′ L r ′ + ∥DV lo ∥ L q 1 L r 1 D s-1 (P >N/10 u) L q L r ⩽ ∥D s V lo ∥ L q 1 L r 1 P >N/10 u L q L r + V lo D s (P >N/10 u) L q ′ L r ′ + ∥DV lo ∥ L q 1 L r 1 D s-1 (P >N/10 u) L q L r .
The rst term and the third term are acceptable since P >N/10 u L q L r ≲ N -s D s (P >N/10 u) L q L r ≲ N -s and D s-1 (P >N/10 u) L q L r ≲ N -1 D s (P >N/10 u) L q L r ≲ N -1 . Consider now the second term. First, we have

V lo D s (P >N/10 u) L q′ L r′ (I×R d ) ≲ 1,
for all q, r is suciently close to q, r. Moreover, using Lemma 2.5 for 0 < δ < 1 2 , we have

V lo D s (P >N/10 u) L 2 t,x (I×R d ) ⩽ M <N η 0 dyadic, M >N/10 dyadic ∥V M D s u M ∥ L 2 t,x (I×R d ) ≲ M <N η 0 dyadic, M >N/10 dyadic M (d-1)/2-δ M -1/2+δ ⟨M ⟩ s ≲ M <N η 0 dyadic M (d-1)/2-δ N -1/2+δ ≲ N η0((d-1)/2-δ) N -1/2+δ ,
this is an acceptable term if we choose η 0 , δ ≪ 1. By interpolation, this implies that the term V lo D s (P >N/10 u) L q ′ L r ′ is also acceptable. Thus, (3.32) holds. Hence,

∥D s P N f (u)∥ L q ′ 0 (I)L r ′ 0 (R d ) ≲
N -η1 and the proof of Proposition 3.15 is completed.

□ Remark 3.18. In the above proof, the condition s > s p is used since we should have 2 > p d 2 -s . In the critical case s = s p , the situation is more complex and local smoothing eect can not hold.

Lemma 3.19 (Schur's test). If

sup x∈R d y∈R d |K(x, y)| r dy 1/r + sup y∈R d x∈R d |K(x, y)| r dx 1/r < C,
where r satises 1 Lemma 3.20 ([12][Lemma 5.1). ] Let u be a global solution of energy at most E. Then, e -it∆ u(t) is weakly convergent to u + ∈ H as t → +∞. Then there exists a unique decomposition (3.33)

r = 1 -1 p -1 q , for some 1 ⩽ p ⩽ q ⩽ ∞, then the operator T f (x) = y∈R d K(x, y)f (y)dy extends to a continuous operator T : L p (R d ) → L q (R d ) with ∥T ∥ L p →L q ⩽ C.
u(t) = e it∆ u + + v(t),
where u + ∈ H with (3.34)

∥u + ∥ 2 H ⩽ E and (3.35) ∥v(t)∥ 2 H ⩽ 2E,
for all t ⩾ 0. We also have the following identities

(3.36) v(t) = e it∆ (u(0) -u + ) -i t 0 e i(t-t ′ )∆ f (u(t ′ ))dt ′ and (3.37) v(t) = iw-lim T →+∞ T t e i(t-t ′ )∆ f (u(t ′ ))dt ′ .
Remark 3.22. Assume that the solution u in Proposition 3.21 is almost periodic in sense that {u(t) : t ∈ R + } is pre-compact in H. Then u + = 0. Indeed, since e -it∆ u(t) weakly converges to u + in H, e -it∆ u(t) also weakly converges to u + in L 2 . We only need to show that for all ϕ ∈ D(R d ), we have

lim t→∞ u(t), e it∆ ϕ L 2 = 0.
Let ε > 0. By assumption, there exists

C ε > 0 such that ∥u(t, x)∥ 2 L 2 (|x|⩾Cε) ⩽ ε, ∀t ∈ R + .
We have

u(t), e it∆ ϕ L 2 2 ≲ u(t), e it∆ ϕ L 2 (|x|⩾Cε) 2 + u(t), e it∆ ϕ L 2 (|x|⩽Cε) 2 ≲ ∥u(t)∥ 2 L 2 (|x|⩾Cε) ∥ϕ∥ 2 L 2 + ∥u(t)∥ L ∞ t L 2 x |x|⩽Cε |e it∆ ϕ| 2 dx ≲ ε ∥ϕ∥ 2 L 2 + t -d C d ε ∥ϕ∥ 2 L 1
≲ ε, if t large enough. This implies that e -it∆ u(t) also weakly converges to 0 in L 2 , hence, u + = 0. This completes the proof.

Lemma 3.23 (Weakly bound component are approximate solutions, [START_REF] Tao | A (concentration-)compact attractor for high-dimensional non-linear Schrödinger equations[END_REF][Lemma 5.6). ] Let T ∈ R and u, v be as in Proposition 3.21. Then for all t 0 large enough (depending on T, u), we have

S(T )v(t 0 ) = v(T + t 0 ) + o H (1)
where S(t) is the ow of (1.1) and o H (1) goes to zero in H norm as t 0 → +∞. Proof. Fix T , and let t 0 large enough. Dene I = [t 0 , t 0 + T ]. We see that v solves the following NLS:

Lv = f (v) + [f (v + e it∆ u + ) -f (v)].
As t 0 → +∞, we have

e it∆ u + L q 0 H s,r 0 ∩L ∞ L Q 0 (I×R d ) → 0.
Moreover,

e it∆ u + C 0 t H(I×R d ) ≲ 1.
By Lemma 3.14 and triangle inequality, we have

∥v∥ L q 0 H s,r 0 ∩C 0 t H(I×R d ) ≲ T 1. Let X = f (v + e it∆ u + ) -f (v) L q ′ 0 H s,r ′ 0 (I×R d ) .
By Lemma 3.11, we have

X ≲ |T | 1/q ′ 0 -1/q0 e it∆ u + L q 0 H s,r 0 (I×R d ) 1 + e it∆ u + L q 0 Ḣs,r 0 (I×R d ) (u, e it∆ u + ) p L ∞ L Q 0 (I×R d ) + e it∆ u + L ∞ L Q 0 (I×R d ) (u, e it∆ u + ) p-1 L ∞ L Q 0 (I×R d ) (u, e it∆ u + ) L q 0 H s,r 0 (I×R d ) ,
which go to zero as t 0 → +∞. Thus, by perturbation Lemma 3.13, the claim follows.

Frequency localisation

In this section, we prove the following asymptotic localisation frequency of weakly bound component v of u: Proposition 4.1. Let u be a global solution of energy at most E and let v be the weakly bound component of u. Then, we have

lim sup t→+∞ sup N ⩽1 N -η3 ∥P ⩽N v(t)∥ H ≲ 1 and (4.2) lim sup t→+∞ sup N ⩾1 N η3 ∥P ⩾N v(t)∥ H ≲ 1.
Proof. See [START_REF] Tao | A (concentration-)compact attractor for high-dimensional non-linear Schrödinger equations[END_REF][Proposition 6.1]. We sketch the proof here. The main idea is to use the double Duhamel formula (3.36) and (3.37)

. Fix ε > 0. Let u ε ∈ D(R d ) be such that u(0) -u + = u ε + O H (ε 2 ).
Hence, from (3.36), we have

(4.3) v(t) = e it∆ u ε -i t 0 e i(t-t ′ )∆ f (u(t ′ ))dt ′ + O H (ε 2 ).
To Prove (4.1), it suces to prove that (4.4)

∥P ⩽N v(t)∥ H ≲ N η3 + ε,
for t large enough (can depend on E, u, ε but is independent on N ) and N ⩽ 1.

Using (4.3) and (3.37), we have

∥P ⩽N v(t)∥ 2 H = ⟨P ⩽N v(t), P ⩽N v(t)⟩ H ⩽ iw-lim T →+∞ T t e i(t-t ′ )∆ P ⩽N f (u(t ′ ))dt ′ , P ⩽N e it∆ u ε -i t 0 e i(t-t ′′ )∆ P ⩽N f (u(t ′′ ))dt ′′ H + O(ε 2 ) ⩽ T t e i(t-t ′ )∆ P ⩽N f (u(t ′ )), P ⩽N e it∆ u ε H dt ′ + T t t 0 Y N (t ′ , t ′′ )dt ′′ dt ′ + O(ε 2 ),
for T large enough and Y N is dened by

Y N (t ′ , t ′′ ) = e i(t-t ′ )∆ P ⩽N f (u(t ′ )), e i(t-t ′′ )∆ P ⩽N f (u(t ′′ )) H .
For the rst integral, we have

P ⩽N e it ′ ∆ u ε H s,R ′ ≲ uε 1 |t ′ | d(1/R-1/2) . Since R < 2d
d+4 , the exponent of ⟨t ′ ⟩ is larger than 2, hence, the rst integral is bounded by O(ε 2 ) if t, T (t < T ) large enough (not depend on N ). The second integral is bounded by

∞ t t 0 |Y N (t ′ , t ′′ )| dt ′ dt ′′ .
Using Lemma 2.2, we have

|Y N (t ′ , t ′′ )| = P ⩽N f (u(t ′ ))), e i(t ′ -t ′′ )∆ P ⩽N f (u(t ′′ )) H ≲ ∥P ⩽N f (u(t ′ ))∥ H s,R e i(t ′ -t ′′ )∆ P ⩽N f (u(t ′ )) H s,R ′ ≲ ∥P ⩽N f (u(t ′ ))∥ L R ∥P ⩽N f (u(t ′′ ))∥ L R 1 |t ′ -t ′′ | d(1/R-1/2) ≲ 1 |t ′ -t ′′ | d(1/R-1/2) .
Moreover, using Lemma 2.2, Bernstein's inequality, we have

|Y N (t ′ , t ′′ )| ≲ ∥P ⩽N f (u(t ′ ))∥ H s ∥P ⩽N f (u(t ′′ ))∥ H s ≲ ∥P ⩽N f (u(t ′ ))∥ L 2 ∥P ⩽N f (u(t ′′ ))∥ L 2 ≲ N d(2/R-1) ∥f (u(t ′ ))∥ L R ∥f (u(t ′′ ))∥ L R ≲ N d(2/R-1) .
Combining all the above, we have

∞ t t 0 |Y N (t ′ , t ′′ )| dt ′ dt ′′ ≲ ∞ t t 0 min 1 t ′ -t ′′ , N 2 d(1/R-1/2) dt ′′ dt ′ . Since R < 2d d+4 , the exponent d(1/R -1/2) > 2,
hence, the claim (4.4) follows.

We now prove (4.2). As before, we have

∥P ⩾N v(t)∥ 2 H ≲ ε 2 + T t t 0 Z N (t ′ , t ′′ )dt ′′ dt ′ ,
for T large enough, where

Z N (t ′ , t ′′ ) = e i(t-t ′ )∆ P ⩾N f (u(t ′ )), e i(t-t ′′ )∆ P ⩾N f (u(t ′′ )) H .
Thus, it suces to show that

T t t 0 Z N (t ′ , t ′′ )dt ′′ dt ′ ≲ N -η3 .
Consider the region where t ′ ⩾ t + N η2 . We have

|Z N (t ′ , t ′′ )| ≲ |t ′ -t ′′ | -d(1/R-1/2) . Since d(1/R -1/2) > 2, the contribution of this term is ≲ t ′ >t+N η 2 t ′′ <t |t ′ -t ′′ | -d(1/R-1/2) dt ′′ dt ′ = O(N -η3 ),
where we use N -η2 ≪ N -η3 . Similar argument for the region t ′′ ⩽ t -N η2 . It remains to prove that

t+N η 2 t t max(0,t-N η 2 ) Z N (t ′ , t ′′ )dt ′′ dt ′ ≲ N -η3 .
By Hölder inequality, the left hand side is bounded by

∥P ⩾N f (u(t ′ ))∥ L q ′ 0 t ′ H s,r ′ 0 x ((t,t+N η 2 )×R d ) × t max(0,t-N η 2 ) e i(t ′ -t ′′ )∆ P ⩾N f (u(t ′′ ))dt ′′ L q 0 t ′ H s,r 0 x ((t,t+N η 2 )×R d )
.

By Strichartz, the left hand side is bounded by

∥P ⩾N f (u)∥ 2 L q ′ 0 H s,r ′ 0 ((max(t-N η 2 ,0),t+N η 2 )×R d ) ,
which by Proposition 3.15 and dyadic decomposition can be bounded by

O(N 2η2/q ′ 0 N -2η1 ),
this is acceptable. The proof is completed. □

Spatial localisation

In this section, we prove localisation in spatial of v and then the main result Theorem 1.3. First, we have the following localisation result: Theorem 5.1. Let E > 0, 0 < µ 0 < 1. There exist J, µ 4 (depending only on E, µ 0 ) with the following property: For each u is a forward-global solution of energy at most E, then there exists functions x 1 , x For each t, let x 1 (t), x 2 (t), • • • , x J(t) (t) be a set of points such that |x j (t) -x k (t)| > µ -1 3 for each j ̸ = k and (5.4)

|v med (t, x j (t))| ⩾ µ 1/η1 3 for all 1 ⩽ j ⩽ J(t).
By the rapid decay of the convolution kernel of P µ2<•<µ -1 2 and Hölder, we have

|v med (t, x j (t))| 2 ≲ µ -d 2 |x-xj (t)|⩽1/2µ -1 3 |v med (t, x)| 2 dx + O(µ 3/η1 3
) and thus

|x-xj (t)|⩽1/2µ -1 3 |v med (t, x)| 2 dx ≳ µ 2/η1 3 µ d 2 ≳ µ 2/η1+d 3
for all j. Thus, by uniformly bounded of H-norm of v, there exists J depending on E, µ 3 such that J(t) ⩽ J for all t. If J(t) < J then for each J(t) < j ⩽ J, we dene x j (t) = x 1 (t). By the above construction, for all t, there exist x 1 (t), x 2 (t), • • • , x J (t) such that (5.5)

|v med (t, x)| < µ 1/η1 3 whenever inf 1⩽j⩽J |x -x j (t)| ⩾ µ -1 3 .
5.2. Second step: L ∞ x spatial localisation on a time interval. Fix t 0 a sucient late time and

I = [t 0 -µ -1 1 , t 0 + µ -1 1 ]. Let D : R d → R + be the distance function D(x) = inf 1⩽j⩽J |x -x j (t 0 )|; thus |v med (t 0 , x)| ⩽ µ 1/µ1 3 whenever D ⩾ µ -1
3 . Let χ : R d → R + be a smooth cuto function which equals to one when D(x) ⩽ 2µ -1 3 , vanishes when D(x) ⩾ 3µ -1

3 and obeys the bounds

∇ k χ = O k (µ k 3 ) for k ⩾ 0.
We can let χ(x) = χ 1 (µ 3 x) and chose suitable function χ 1 .

We have the following result:

Lemma 5.2. We have lim sup t→+∞ e i(t-t0)∆ ((1 -χ)u(t 0 ))

L q 0 H s,r 0 ∩L ∞ L Q 0 (I×R d ) ≲ µ η3 2 .
Proof. Let ũ+ ∈ D(R d ) be such that

ũ+ = u + + O H (µ η3
2 ), this gives

(1 -χ)u(t 0 ) = e it0∆ ũ+ + (1 -χ)v med (t 0 ) + O H (µ η3 2 ).
The contribution of the rst term on the right hand side is acceptable if we choose t 0 large enough.

For the second term on the right hand side, we have

(1 -χ)v med (t, x) = (1 -χ)P <100µ -1 2 (1 D⩾µ -1 3 v med (t 0 )) + (1 -χ)P <100µ -1 2 (1 D<µ -1 3 v med (t 0 ))
Using Lemma 6.3, the second term in the right hand side is acceptable. It remains to consider the rst term. We only need to prove that

e i(t-t0)∆ (1 -χ)P 100µ -1 2 (1 D⩾µ -1 3 v med (t 0 )) L q 0 H s,r 0 ∩L ∞ L Q 0 (I×R d ) ≲ µ η3 2
for t 0 large enough. Using Lemma 6.2, the left hand side is bounded by

O µ1 (µ -1/η0 2 ) 1 D⩾µ -1 3 v med (t 0 ) L r 0 ∩L Q 0 .
From (5.5),

1 D⩾µ -1 3 v med (t 0 ) L ∞ ⩽ µ 1/η1 3 .
Interpolating this with bounded L 2 norm of v, we obtain the desired result.

□ the contribution of

P ⩽µ -1 2 (1 D⩽2µ -2 3 f (u(t ′ )χ 2 )) is acceptable.
It remains to consider the contribution of the term P ⩾µ -1 2 f (u(t ′ )χ 2 ). We have

1 D>µ -3 3 I ′ e i(t0-t ′ )∆ P ⩾µ -1 2 f (u(t ′ )χ 2 )dt ′ L 2 ≲ P ⩾µ -1 2 f (uχ 2 ) L q ′ 0 L r ′ 0 (I×R d ) ≲ µ s 2 ∥f (uχ 2 )∥ L q ′ 0 Ḣs,r ′ 0 (I×R d ) ≲ µ η3 2 .
this term is acceptable. This completes the proof. 3 , equals 0 when D ⩽ µ -3

3 and χ 3 has bounded derivatives. Then for t 0 large enough, we have

χ 3 v(t 0 ) = O L 2 (µ c 1 ),
for some c > 0.

Proof. From corollary 5.4, we have

χ 3 I ′ e i(t0-t ′ )∆ f (u(t ′ ))dt ′ = O L 2 (µ η3/2 2
).

for all I ′ ⊂ I. Thus, by Duhamel's formula, we have

χ 3 v(t 0 ) = χ 3 e -iµ -1 1 ∆ v(t 0 + 1/µ 1 ) + O L 2 (µ η3/ 2 2
) and

χ 3 v(t 0 ) = χ 3 e +iµ -1 1 ∆ v(t 0 -1/µ 1 ) + O L 2 (µ η3/2 2 )
This implies that

∥χ 3 v(t 0 )∥ 2 L 2 = χ 3 e -iµ -1 1 ∆ v(t 0 + 1/µ 1 ), χ 3 e +iµ -1 1 ∆ v(t 0 -1/µ 1 ) L 2 + O(µ η3/2 2
).

Thus, it suces to show that

e -iµ -1 1 ∆ v(t 0 + 1/µ 1 ), χ 2 3 e +iµ -1 1 ∆ v(t 0 -1/µ 1 ) L 2 ≲ µ c 1 .
Let ϕ be a Schwartz function be such that u(0) -u + = ϕ + O L 2 (µ 2 ). Thus, by (3.36), we have

v(t 0 -1/µ 1 ) = e i(t0-1/µ1)∆ ϕ -i t0-1/µ1 0 e i(t0-µ1-t ′ )∆ f (u(t ′ ))dt ′ + O L 2 (µ 2 )
and hence

χ 2 3 e iµ -1 1 ∆ v(t 0 -1/µ 1 ) = e it0∆ ϕ -(1 -χ 2 3 )e it0∆ ϕ -iχ 2 3 t0-1/µ1 0 e i(t0-t ′ )∆ f (u(t ′ ))dt ′ + O L 2 (µ 2 ).
From dispersive estimate we see that (1 -χ

3 )e it0∆ ϕ = O L 2 (µ 2 ) for t 0 large enough. Using (3.37), it suces to show that (5.8)

+∞ t0+1/µ1 e i(t0-t ′ )∆ f (u(t ′ )), e it0∆ ϕ L 2 dt ′ ≲ µ 2c 1 ,

and

(5.9)

+∞ t0+1/µ1 t0-1/µ1 0 e i(t0-t ′ )∆ f (u(t ′ )), χ 2 3 e i(t0-t ′′ )∆ f (u(t ′′ )) L 2 dt ′′ dt ′ ≲ µ 2c 1 .
To prove (5.8), using (2.2) and Hölder, we have

e i(t0-t ′ )∆ f (u(t ′ )), e it0∆ ϕ L 2 = f (u(t ′ )), e it ′ ∆ ϕ L 2 ≲ e it ′ ∆ ϕ L R ′ ≲ ϕ |t ′ | -d(1/R-1/2)
. Since d(1/R -1/2) > 2, we obtain (5.8) for t 0 large enough. Now, we prove (5.9). Writing

χ 2 3 = 1 -(1 -χ 2
3 ), we have (5.10)

e i(t0-t ′ )∆ f, χ 2 3 e i(t0-t ′′ )∆ g L 2 = e i(t ′′ -t ′ )∆ f, g L 2 - R d R d f (x)g(z)K t ′ ,t ′′ (x, z)dxdz, Consider the case M 3 ⩽ 1. Since |M 3 x 2 | ⩾ 10 and M 3 |y -x 2 | ⩽ 2 and |y -x 2 | + |y| ⩾ |x 2 |, we have |y| ≳ |x 2 |. Thus, Q 0 ≲ |x 2 | M M3|y-x2|⩽2 (1+|y| 2 ) -M/2 |y| -M dy ≲ M3|y-x2|⩽2 (1+|y| 2 ) -M/2 dy ≲ 1.
This completes the proof of Lemma 5.6.

Thus, K t ′ ,t ′′ (x, z) ≲ |t ′ -t ′′ | -d/2
. This implies that (5.10) is estimated by |t ′ -t ′′ | -d/2 ∥f ∥ L 1 ∥g∥ L 1 . Moreover, (5.10) is also estimated by ∥f ∥ L 2 ∥g∥ L 2 . Hence, by bilinear interpolation, (5.10) is estimated by |t ′ -t ′′ | -d(1/R-1/2) ∥f ∥ L R ∥g∥ L R . By the choice of R, d(1/R -1/2) > 2, hence (5.9) is proved. This completes the proof of Proposition 5.5. □ Theorem 5.1 is followed from Proposition 5.5. As a consequence of Theorem 5.1, we have the following spatial localisation result: Proposition 5.7. Let E > 0 and 0 < µ 0 < 1. Then there exists µ 5 = µ 5 (E, µ 0 ) depending on E, µ 0 (and on the other xed parameters) with the following properties: for any u radial global solution of energy at most E, we have (5.12) lim sup

t→+∞ |x|⩾1/µ5 |v(t, x)| 2 dx ≲ µ 2 0 ,
where v is the weak bound component of u.

Proof. Apply Theorem 5.1, for all t suciently large, there exist

x 1 (t), • • • , x J (t) such that R d 1 inf 1⩽j⩽J |x-xj (t)|⩾1/µ4 |v(t, x)| 2 dx ≲ µ 2 0 .
Since u is radial, u + and v are also radial. Thus, we may average the above estimate over rotation and obtain that

R d S d-1 1 inf 1⩽j⩽J ||x|ω-xj (t)|⩾1/µ4 dω |v(t, x)| 2 dx ≲ µ 2 0 .
In particular, we have

|x|⩾1/µ5   S d-1   1 - J j=1 1 ||x|ω-xj (t)|<1/µ4   dω   |v(t, x)| 2 dx ≲ µ 2 0 .
But if µ 5 is suciently small depending on µ 4 , by using |x| ⩾ 1/µ 5 , we have

S d-1 1 ||x|ω-xj (t)|<1/µ4 dω = S d-1 1 |ω-xj (t)/|x||< 1/µ 4 |x| dω ⩽ S d-1 1 |ω-x0|<µ5/µ4 dω (where x 0 = x j (t)/|x|) = Vol({||ω -x 0 | < µ 5 /µ 4 |} ∩ S d-1 ) ⩽ (µ 5 /µ 4 ) d-1 .
Thus, if µ 5 is small enough depending on both J and µ 4 , we have

S d-1   1 - J j=1 1 ||x|ω-xj (t)|<1/µ4   dω ⩾ 1 2 .
for all |x| ⩾ 1/µ 5 and the desired result is proved.

□

Combining the above analysis, we give the proof of the main result:

Proof of Theorem Let K E,rad be the closure of the set of all limit points lim tn→+∞ v(t n ), where t n ranges over sequences of times which converge to innity. Then, K E,rad is close and hence is a compact subset of K. Let us prove that K E,rad is also a attractor. Indeed, let t n → +∞. Since, dist H (v(t n ), K) → 0 as n → +∞, there exists k n ∈ K such that ∥v(t n ) -k n ∥ K → 0. By compactness of K, for each subsequence of (t n ) there exists φ ∈ K and a sub-sequence of (t n ) which for convenience still denotes by (t n ) such that ∥v(t n ) -φ∥ H → 0. Thus, φ ∈ K E,rad and hence, lim t→+∞ dist H (v(t), K E,rad ) = 0. By local theory and Lemma 3.23, K E,rad is invariant under S(t) for t small and hence for all t large also.

Finally, we show the uniqueness of u + . If ũ+ satises the same properties of u + then {e it∆ (u + -ũ+ ), t ⩾ 0} is pre-compact in H. Moreover, for 2 < p ⩽ 2d d-2s , e it∆ (u + -ũ+ ) L p → 0, as t → +∞.

Combining the above e it∆ (u + -ũ+ ) H → 0 as t → +∞. Thus, u + = ũ+ , which completes the proof. □

Appendix

In this section, we prove some useful results which is used in the proof of the main result. We rst study the graph of D in R d+1 i.e the set of elements of form (x, D(x)), x ∈ R d . For each j, the graph of |x -x j | is {(x, |x -x j |), x ∈ R d } ⊂ R d+1 , which is a co-dimension one surface having a critical point at x j . The graph of |x -x k | can be given by move the graph of |x -x j | by vector x j -x k . For each x j ̸ = x k , the intersection of the graph (x, |x -x j |) and (x, |x -x k |) is a co-dimension two surface: {(x, |x -x j |) : |x -x j | = |x -x k |}. Thus, the graph of D is a co-dimension one surface which is singular at each x j and at each co-dimension two surface {(x, |x -x j |) : |x -x j | = |x -x k |}. We can easily take a regular surface H close to the graph of D, which is graph of a smooth function D. We see that D ∈ C ∞ (R d ) and D satises the desired properties if we take H is close enough to the graph of D.

□

Let t 0 , t, t ′ , χ, µ 2 , µ 3 , v med be as in Subsection 5.2. We have the following results: Lemma 6.2. The operator ∇ k e i(t-t0)∆ (1 -χ)P ≲µ -1 2 have an operator norm of O µ1 (µ -c(k)

2

) on L r for 2 < r < 2d d-2s , c(k) depending on k and ∀k ∈ N. Similar result holds for D j e i(t-t0)∆ (1-χ)P ≲µ -1 2 for j = 0, s.

Proof. By density, we need only work on Schwartz functional space. We consider the case k = 0, the general case is obtained by bounded derivatives of χ and interpolation. Let ψ be smooth function such that ψ(ξ) = 1 if |ξ| ⩽ 1 and ψ(ξ) = 0 if |ξ| ⩾ 2.

For each t ∈ R and N dyadic, we have e it∆ P ≲N f = F -1 e -it|ξ| 2 ψ(ξ/N ) f (ξ) = F -1 (e -it|ξ| 2 ψ(ξ/N )) * f.

Thus, by Young inequality, for M > d an integer, the operator norm of e it∆ P ≲N f on L r is bounded by ≲ e -i(tN 2 )|ξ| 2 ψ(ξ)

F -1 (e -it|ξ| 2 ψ(ξ/N ))(x) L 1 = N d F -1 (e -itN
H M ξ ≲ (1 + |ξ|) M ψ(ξ) H M ξ (1 + |tN 2 |) M ≲ (1 + |tN 2 |) M .
Thus, the operator norm of e i(t-t0)∆ P ≲µ -1 2 on L r is bounded by

(1 + |t -t 0 | 2 µ -2 2 ) M = O µ1 (µ -2M 2
).

It remains to prove that e i(t-t0)∆ (1 -χ)P ≲µ -1 2 has bounded operator on L r by O µ1 (µ -c 2 ) for some constant c > 0. The above expression is of form e it∆ (1 -χ)P ≲µ

where d ⩾ 5 , 2 p

 52 λ = ±1, p ⩾ 4 d-2 , p is even or p > ⌈s⌉ and V ∈ C ∞ 0 (R d ). Let s p = d 2 -. Fix s such that d2 > s > s p , the energy is dened by(1.2)

□ 3 . 3 .

 33 Fixed time estimate. From (1.2) and our convenience to suppress dependence on E we have(3.11) ∥u(t)∥ H ≲ 1 for all t ∈ [0, +∞) and hence by Sobolev embedding we have(3.12)

3. 4 .

 4 Local-in-time estimates. Next we prove a local-in-time Strichartz estimate. Lemma 3.14. (Local Strichartz control). For any time interval I ⊂ [0, +∞) and any admissible pair of exponents (q, r) we have(3.14)

  (3.25), (3.26), (3.29), (3.30), (3.31), we see that s 2 = s -2 -p d 2 -s . Moreover, by Sobolev embedding Lemma 3.1 and Lemma 3.14, we have

Proposition 3 .

 3 21([12][Proposition 5.2). ] Let u be a global solution of energy at most E > 0.

□

□ 5 . 3 .

 53 Third step: L 2 x localisation at xed time. We now prove the following result: Proposition 5.5. Let χ 3 : R d → R + be a smooth cuto function which equals 1 when D ⩾ 2µ-3

Lemma 6 . 1 .

 61 Let x 1 , x 2 , • • • , x J ∈ R d and D(x) = inf 1⩽j⩽J |x -x j |. There exists a function D ∈ C 2 (R d ) such that D -D L ∞ ⩽ 1 and ∇ D, ∇ 2 D ≲ 1. Proof. We see that D ∈ C 0 (R d )∩C ∞ (R d )\A for some zero measure set A. Indeed, A = ∪ i {x i }∪ j̸ =k {x : |x -x j | = |x -x k |}. Moreover, D is small inside ∪ J i=1 B(x i , 1) and |∇D|, |∇ 2 D| ≲ 1 outside ∪ J i=1 B(x i , 1) ∪ j̸ =k {x : |x -x j | = |x -x k |}.

2 |ξ| 2 ψ 1 x=F 1 x= 1 ξ≲ ( 1 +

 21111 (ξ))(N x) L -1 (e -i(tN 2 )|ξ| 2 ψ(ξ))(x) L F(e -i(tN 2 )|•| 2 ψ(•))(ξ) L |ξ|) M F(e -i(tN 2 )|ξ| 2 ψ(ξ)) L 2 (1 + |ξ|) -M L 2

-1 2 =

 2 e i t∆ (1 -χ)Ψ(µ 2 D) for |t| ⩽ µ -1 1 and Ψ is a cut o function which vanishes in {ξ : |ξ| ⩾ 2}. Recall that e it∆ f (x) = 1 t d/2 e -i|x-y| 2 tf (y)dy (up to multiply by a constant).Thus,e it∆ (χf )(x) = 1 t d/2 e -i|x-y| 2 t χ(y)f (y)dy,which means that the Schwartz kernel of the operatorT 1 : f → e it∆ (χf ) is K 1 (x, y) = 1 t d/2 e-i|x-y| 2 t χ(y).

  and for any µ 0 > 0 there exists µ 1 > 0 such that we have the asymptotic frequency localisation estimate

	(1.4)	sup	lim sup
		u∈U
		lim sup t→+∞	∥P 1/µ1 u(t)∥ H ≲ µ 0
		and the spatial localisation estimate
		lim sup t→+∞ |x|⩾1/µ1	|u(t, x)| 2 dx ≲ µ 2 0

t→+∞ ∥u(t)∥ H < ∞

  ). Then, ∥f g∥ Ḣs,r ≲ ∥f ∥ r1 ∥g∥ Ḣs,p 1 + ∥f ∥ Ḣs,r 2 ∥g∥ p2 .

  [START_REF] Brezis | Gagliardo-Nirenberg inequalities and non-inequalities: the full story[END_REF]

  , • • • , x J : R + → R, such that First step: L ∞ x spatial localisation at xed times. Dene v med = P µ2<•<1/µ2 v. We exploit the frequency localisation of v med . From Proposition 4.1, we have(5.2)

	5.1. ∥v(t) -v med (t)∥ H ≲ µ η3 2
	and thus,
	(5.3)	u(t) = e it∆ u + + v med (t) + O H (µ η3 2 ).
	(5.1)	lim sup t→+∞ inf 1⩽j⩽J |x-xj (t)|⩾1/µ4	|v(t, x)| 2 dx ≲ µ 2 0 .

  for all radial global bounded solution u of energy at most E.

1.3. By Proposition 1.7, Proposition 4.1 and Theorem 5.1, there exists a compact set K ⊂ H such that lim t→+∞ dist H (v(t), K) = 0

f (u(t ′ )χ 2 )). Using Lemma 6.4, and ∥f (u(t ′ ))∥ L R ≲ 1, we obtain

We have the following spatial decay: Lemma 5.3. We have

Proof. Let ũ be the solution of (1.1) with initial data ũ(t 0 ) = χu(t 0 ). From Lemma 3.13 and Lemma 5.2, we see that

and (5.6) ∥u -ũ∥ L q 0 H s,r 0 ≲ µ1 µ η3 2 .

Let W = 1 + µ 3 D be such that

⩽ 1 (see Lemma 6.1 for the choice of such D). We have ∇W, ∇ 2 W = O(µ 3 ) and W χ = χ + µ 3 Dχ + µ 3 ( D -D)χ = O(1). Thus, (5.7)

We have

By Strichartz and e it∆ f L q 0 L r 0 ≲ ∥f ∥ L 2 , we have

for any I ′ ⊂ I, t ′ ∈ I ′ . Denote the left hand side by X(I ′ ), by Hölder, we have

and hence (for I ′ small enough), we have

By iterating this using (5.7), we have

Thus,

and hence,

3 by the choice of D. Thus,

Combining with (5.6), we obtain the desired result.

□

As a consequence, we have local in time of Strichartz Duhamel:

Corollary 5.4. For t 0 large enough and any I ′ ⊂ I, we have

Proof. Let χ 2 : R d → R + be a smooth cuto such that

3 and χ 2 has bounded derivatives. We have

By Lemma 5.3, Hölder, we have

and hence this term is acceptable. Consider the term

for each test functions f, g and K t ′ ,t ′′ is dened by

for some constant c.

We prove that (5.11)

Without loss of generality, we assume supp(φ) ⊂ B(x 0 , 2µ -3

3 ) for some

We have:

)dy, (where

.

Thus, for some M ∈ N large enough chosen later,

))dy.

To prove (5.11), we only need to prove the following lemma Lemma 5.6. The following estimate holds for M ∈ N large enough:

Proof. For convenience, dene

By induction, we see that

Moreover,

Combing to uniform bounded of y k φ(y), using c k0 +c

, we only need to check the case h = 0 and h = M . Choosing M = 2d. The rst case is easy to estimate Q 0 . Consider the second case.

Similarly, using the inverse Fourier transform:

and thus the Schwartz kernel of the operator

Thus, the Schwartz kernel of f → e it∆ (χΨ(µD)f ) is

Then write K = K i + K e , where

We have

It remains to control K e . We have

After integration by part, we get:

By Schur's test 3.19, we get the desired result.

□ Lemma 6.3. We have

Proof. Since the convolution kernel of P <100µ -1 2 is a Schwartz function and the support of 1 -χ belongs to {|x| : D(x) > 2µ -1

3 }, we have kernel of (1 -χ)P <100µ -1

for any l > 0. Thus, from Young inequality and H s -norm bounded of v med we obtain the desired result.

□ Lemma 6.4. The operator 1 D>µ -3

3 ) for each t ′ ∈ I.

Proof Lemma 6.4. Let f ∈ L R . We have, for φ is a smooth cuto function which equals 1 in |ξ| ⩽ 1 and equals 0 in |ξ| ⩾ 2:

Thus, by Young inequality and using ∥f ∥ H s = (1 + |x|) s F -1 f (x) L 2 , we have

2 /2 F -1 e -i(t0-t ′ )µ -2

F -1 e -i(t0-t ′ )µ -2

(1 + |x|) M (1 + |x|) -M F -1 e -i(t0-t ′ )µ -2

O((µ 3 3 µ 2 ) l ) e -i(t0-t ′ )µ -2

for M > d and any l > 0

3 ), (for l large enough) This completes the proof. □