Almost Sure Existence of Global Solutions for supercritical Semilinear Wave Equations

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Almost Sure Existence of Global Solutions for supercritical Semilinear Wave Equations

Mickaël Latocca

To cite this version:

Mickaël Latocca. Almost Sure Existence of Global Solutions for supercritical Semilinear Wave Equa-

tions. 2018. �hal-01876990�

SUPERCRITICAL SEMILINEAR WAVE EQUATIONS

MICKAËL LATOCCA

Abstract. We prove that for almost every initial data (u0, u1) ∈ H^s × H^s−1 with s > ^p_p⁻₋³₁ there exists a global weak solution to the supercritical semilinear wave equation

∂_t²u−∆u+|u|^p−1u= 0 where p > 5, in both R³ and T³. This improves in a probabilistic framework the classical result of Strauss [16] who proved global existence of weak solutions associated toH¹×L² initial data. The proof relies on techniques introduced by T. Oh and O.

Pocovnicu in [13] based on the pioneer work of N. Burq and N. Tzvetkov in [5]. We also improve the global well-posedness result in [17] for the subcritical regimep <5 to the endpoints=^p_p⁻−³1.

Acknowledgement

The author warmly thanks Nicolas Burq and Isabelle Gallagher for careful advising during this work and suggesting this problem.

1. Introduction

1.1. Supercritical semilinear wave equations. We consider the Cauchy problem for the energy-supercritical defocusing semilinear equation in dimension 3, that is for p > 5 and s > 0:

(SLW

)

∂

u − ∆u + | u |

u = 0

(u(0), ∂

u(0)) = (u

, u

) ∈ H

(U ) × H

(U ) ,

where u(t) is a real-valued function defined on U = R

or T

. We also consider the associate linear wave equation:

(LW)

∂

z − ∆z = 0

(z(0), ∂

z(0)) = (z

, z

) ∈ H

(U ) × H

(U ) . The formal conserved energy for a solution u to (SLW

) is

E(u(t)) :=

Z

U

| ∂

u(t, x) |

2 + |∇ u(t, x) |

2 + | u(t, x) |

p + 1

!

dx . Moreover (SLW

) is known to be invariant under the dilation symmetry

u(t, x) 7→ u

(t, x) = λ

u(λt, λx) .

A necessary condition for a function u to belong to the energy space, i.e. E(u(t)) < ∞ is that (u(0), ∂

u(0)) ∈ H ˙

(U ) × L

(U ). We observe that

k (u

(0), ∂

u

(0)) k

= λ

k (u(0), ∂

u(0)) k

≫ k (u(0), ∂

u(0)) k

, since p > 5, which explains why for such p, (SLW

) is called energy-supercritical.

We first recall the classical result of existence of weak solutions to (SLW

).

Date: September 19, 2018.

2010 Mathematics Subject Classification. Primary 35L05, 35L15, 35L71.

1

Theorem 1.1 (Strauss, 1970, [16]). Let f be a real smooth function and F be an antiderivative of f . Assume that F(v) & −| v |

and

| F (u) |

| f (u) | → ∞ as | u | → ∞ . Let (u

, u

) ∈ H

( R

) × L

( R

) satisfying

E(u

, u

) :=

Z

R3

| u

|

2 + |∇ u

|

2 + F (u

)

!

dx < ∞ . Then the equation

(1.1)

∂

u − ∆u + f(u) = 0

(u(0), ∂

u(0)) = (u

, u

) ∈ H

× L

,

admits a weak solution, that is a distributional solution u : R → H

( R

) × L

( R

) which is weakly continuous in time and such that

E(u(t), ∂

u(t)) 6 E(u

, u

) for every t ∈ R .

In the following we will seek for global solutions to (SLW

) in the following sense.

Definition 1.2 (Weak solutions for (SLW

)). A function u : R × U → R is said to be a weak solution to (SLW

) with initial data (u

, u

) ∈ H

(U ) × H

(U ) if and only if one can write u = z + v where z is a strong solution to (LW) with initial data (u

, u

) and v is such that (v(0), ∂

v(0)) = (0, 0) and satisfies for every T > 0, v ∈ H

(( − T, T ) × U ) ∩ L

(( − T, T ) × U ), and for every compactly supported ϕ ∈ C

( R × U ):

Z

R

Z

U

∂

v(t)∂

ϕ(t) − ∇ v(t) · ∇ ϕ(t) − (z(t) + v(t)) | z(t) + v(t) |

ϕ(t)

dx dt = 0 . Remark 1.3. Note that this definition differs from the definition of weak solutions in Theorem 1.1.

In our setting we ask for the solution u to be written in the form u = z + v, where z solves the associate linear problem (LW): the reason why we ask for such a decomposition will appear clearly in the proof of the main result, Theorem 1.4. Note that these weak solutions are a fortiori weak solution as in Theorem 1.1.

1.2. Previous works on probabilistic well-posedness for wave equations. Our purpose is to construct solutions to (SLW

), in the sense of Definition 1.2 using a probabilistic method when initial data are below the energy space. Indeed, up to the knowledge of the author, no global existence result is known for (SLW

), p > 5 and initial data (u

, u

) ∈ H

× H

with s < 1.

Probabilistic methods have been implemented in order to construct solutions to (SLW

) as- sociated to initial data below the energy space. As a result, the local and global well-posedness theory have been widely improved. We briefly recall the existing results in the context of semi- linear wave equations, our list being not exhaustive.

The probabilistic well-posedness theory goes back to J. Bourgain who proved global existence for the two-dimensional nonlinear Schrödinger equation in [2]. Building on Bourgain’s ideas, N. Burq and N. Tzvetkov published a series of two articles [5, 6], introducing a randomization procedure that allows to choose random initial data in Lebesgue and Sobolev spaces. They developed the local and global probabilistic well-posedness theory. They later considered the global well-posedness of (SLW

) for p = 3 in [7] proving that, although for s <

, (SLW

) is ill-posed, there exist unique global solutions for almost every initial data in H

( T

) × H

( T

) as soon as s > 0. The proof relies on a probabilistic improvement of the Strichartz estimates.

In [4], Burq-Thomann-Tzvetkov considered (SLW

) in higher dimensions and proved the almost

sure existence of global infinite energy solutions of (SLW

), for p = 3 in T

, d > 3. Their

argument use compactness techniques just like the ones presented in this article. We mention

that in the context of the Navier-Stokes equation, A.R. Nahmod, N. Pavlović and G. Staffilani proved existence of global weak solutions almost surely in [12].

The work of Lührman-Mendelson in [10, 11] deals with the global well-posedness theory for (SLW

) in the case 3 < p < 5. They prove an almost-sure global well-posedness result associated to initial data

(u

, u

) ∈ H

( R

) × H

( R

) as long as s > p

+ 5p

− 11p − 3 9p

− 6p − 3 which improves the deterministic theory when

(7+ √

73) ≃ 3.88 < p < 5. In [11] they improved their result to

< s < 1 using Oh-Pocovnicu’s ideas from [13].

In [15] O. Pocovnicu proved almost-sure global well-posedness for the energy critical wave equation (SLW

), that is p = 5, in the euclidean space R

of dimension d = 4, 5. The proof relies on the deterministic perturbation theory for critical dispersive equations as well as the probabilistic improvements of the Strichartz estimates coming from the work of Burq-Tzvetkov.

With some more efforts in the domains R

and T

the global well-posedness theory for (SLW

), p = 5 has been treated in the joint work of Oh-Pocovnicu in [13, 14]. In their proof they used a new energy estimate and a new probabilistic Strichartz estimate. Their result shows that almost-sure global well-posedness in known to hold for initial data in H

× H

, s >

.

The global well-posedness theory for (SLW

) and when 3 < p < 5 was then studied in the work of Sun-Xia, in [17]. They proved global existence and uniqueness for s >

interpolating between the results of Oh-Pocovnicu [13] and Burq-Tzvetkov [7].

1.3. Main results and notations.

1.3.1. Statement of the main results. Let (Ω, F , P ) be a probability space, and a randomization map (u

, u

) 7−→ (u

, u

) that will be described in Section 2.1, see (2.2) and (2.4). Let U be an open set. The measure µ

is defined as the pushforward probability measure of P by the above randomization map. We define

M

(U ) :=

(u0,u1)∈H^s(U)×H^s−1(U)

µ

.

We now state our results. The first is an existence result in the supercritical case.

Theorem 1.4. Let p > 5, s >

and U which stands for R

or T

. Let µ ∈ M

(U ). Then for µ almost every (u

, u

) ∈ H

(U ) × H

(U ) there exists a global weak solution u to (SLW

), in the sense of Definition 1.2.

Remark 1.5. Note that no information is given concerning the uniqueness of the solutions con- structed.

The solutions constructed in Theorem 1.4 enjoy additional properties:

Corollary 1.6. Under the hypothesis of Theorem 1.4, and given µ ∈ M

(U ), there exists a set Σ ⊂ H

(U ) × H

(U ) of full µ-measure which is invariant under the flow of (SLW

).

If s < 1 then for every initial data (u

, u

) ∈ Σ, the solutions u constructed by Theorem1.4 satisfy (u, ∂

u) ∈ C

( R , H

(U ) × H

(U ))

Corollary 1.7. For U = R

, the solutions constructed by Theorem 1.4 enjoy the finite speed of propagation property with speed at most 1.

The next result is an extension of Theorem 1.2 from [17] to the endpoint s =

.

Theorem 1.8. Let p < 5, s :=

and U which stands for R

or T

. Let µ ∈ M

. Then for almost every (u

, u

) ∈ H

(U ) × H

(U ) there exists a unique strong solution to (SLW

).

Again a consequence of the proof of Theorem 1.8 is the following:

Corollary 1.9. Under the hypothesis of Theorem 1.8, and given µ ∈ M

, there exists a set Σ ⊂ H

(U ) × H

(U ) of full µ-measure which is invariant under the flow of (SLW

).

Remark 1.10. One can also prove probabilistic continuous dependence of the flow in the sense of [7], using the proof given in [15], Theorem 1.4 and Remark 1.6 (iii) but this paper does not focus on that matter.

Remark 1.11. Combining these results with the existing results in the case p 6 5, see [7, 13, 14, 17] yields the following classification:

(i) In the energy sub-critical setting i.e p ∈ [3, 5), there exists a unique strong solution to (SLW

) for amlost every initial data in H

(U ) × H

(U ) as soon as s >

. The case p = 3 is treated entirely in [7] in the case of the torus, but similar arguments work in the euclidean setting. The case p ∈ (3, 5) and s >

is treated in [17], and the case p ∈ (3, 5), s =

is Theorem 1.8. In [7], additional results of continuous dependence and flow-invariant set are proven in the case p = 3, s > 0 which remain valid in the case p ∈ (3, 5), s >

.

(ii) In the energy critical setting p = 5, there exists a unique strong solution to (SLW

) with initial data in H

(U ) × H

(U ) as soon as s >

. This result is proven in [13, 14]

both in R

and T

and comes with a continuous dependence result and flow-invariant set construction.

(iii ) In the energy super-critical setting p > 5 there exists a global strong solution as long as s ∈ (

, 1) : this is Theorem 1.4. For s > 1 there still exists a weak solution, in the sense of Definition 1.2. In both cases such solutions can be constructed in a flow-invariant way, this is Corollary 1.6.

1.3.2. General notation. (Ω, F , P ) is called a probability space if Ω is a set and F is a σ-algebra, endowed with a probability measure P . The expectation of a random variable X will be denoted by E [X].

The notation A . B means that there exists a constant C such that A 6 CB . The nota- tion A .

B is used to specify that the constant C depends on x.

For a real number x, ⌊ x ⌋ (resp. ⌈ x ⌉ ) denotes the lower integer part (resp. upper integral part).

We adopt widely used notations for functional spaces: C

denotes the set of k differentiable functions with continuous derivatives up to order k, L

stands for the Lebesgue spaces and W

:= { u, (Id − ∆)

u ∈ L

} (resp. ˙ W

) denotes the usual nonhomogeneous Sobolev spaces (resp. homogeneous spaces). The hilbertian Sobolev spaces are denoted H

:= W

. Besov spaces are denoted B

, see Appendix A for more details. The Fourier transform of a function f is denoted by ˆ f or equivalently F (f ). D denotes the space of test functions, that is C

functions with compact support. S

stands for the space of tempered distributions. If X is a Banach space we often write L

X or L

X for L

((0, T ), X). For x ∈ R

, h x i = (1 + | x |

)

, and h∇i denotes the Fourier multiplier of symbol h ξ i .

S

:= S

denotes the class of classical symbols of order m ∈ R , that is smooth functions a : R

× R

→ C such that

| ∂

∂

a(x, ξ) | .

h ξ i

, for every α ∈ N

and x, ξ ∈ R

.

In the rest of this article H

(U ) will be used as a shorthand notation for H

(U ) × H

(U ).

1.4. Outline of the paper, heuristic arguments. The purpose of this section is to provide

the reader with heuristic arguments that will help to understand the main ideas behind the

proofs of the two main results. Note that this section is not mathematically needed in order to

follow the rigorous proofs of the result.

1.4.1. Energy estimates. In this paragraph we set U = R

and drop the reference to it. As in [7, 13], the method to construct global solutions to (SLW

) is to seek for solutions u of the form u(t) = z(t) + v(t) with initial data v(0) = ∂

v(0) = 0, and z being the solution to (LW) associated to initial data (u

, u

) ∈ H

. Note that z is globally defined. It is thus sufficient to prove that v globally solves an equation in the energy space H

. A direct computation shows that v formally satisfies

∂

v − ∆v + (z + v)

= 0 .

We now assume that v locally solves this equation in H

and that the local well-posedness result comes with a blow-up criteria that only depends on the size of k (v(t), ∂

v(t)) k

. In this case, proving global existence v reduces to proving that the (not conserved) nonlinear energy

E(t) := k ∂

v(t) k

2 + k∇ v(t) k

2 + k v(t) k

p + 1 is bounded on every time interval.

In order to do so, the standard way is to estimate E(t) using a Grönwall-type estimate and hope for a sublinear estimate that will give non-blowup for E(t). We first begin by writing that

E

(t) =

R3

∂

v(t)

| z(t) + v(t) |

(z(t) + v(t)) − | v(t) |

v(t)

dx .

In the following we will provide a rough argument, ignoring lower order terms and using fractional integration by parts. The terms in powers of z will be estimated using the probabilistic Strichartz estimates from Proposition 2.6 and Proposition 2.7 so they constitute the “good part” when developing the quantity (z + v)

− v

, if we assume p to be an odd integer for instance. These considerations lead to the “worse order approximation” (z + v)

− v

≃ v

z + (better terms).

The other terms are expected to be handled in an easier way so that we write : E

(t) ≃

Z

R³

∂

v v

z dx + (better terms)

| {z }

.E(t)

. From now we will just dismiss the better terms and study the worse term.

Remark 1.12. A crude estimate using the Hölder inequality, putting z ∈ L

, ∂

v ∈ L

and v

∈ L

leads to

E

(t) . E(t

)

−1 p+1

which is sublinear if

+

6 1 i.e. p 6 3. Thus such an argument would provide an energy estimate that does not blow-up for p 6 3 and s > 0. This was the idea behind the energy estimates in [7, 15].

In order to obtain energy estimates for p = 5, Oh-Pocovnicu introduced in [13] an appropriate method that we describe: if one accepts a loss of regularity for the initial data then one can transfer time regularity into space regularity using an integration by part in time, and properties of the wave equation, which state that roughly ∂

z ≃ ∇ z. Since E(0) = 0 we have after integration in time:

E(t) =

0

E

(t

) dt

≃

0

Z

R3

∂

vv

z dx dt

. Then using that ∂

vv

∼ ∂

(v

) an integration by parts yields

E(t) ≃

R³

Z t

0

v(t

)

∂

z(t

) dt

dx

≃

R³

Z _t

0

v(t

)

∇ z(t

) dt

dx .

Pick s ∈ [0, 1] which will be chosen later. We write that ∇ z = ∇

∇

z and integrate by parts in space with the operator ∇

, neglecting the boundary terms:

E(t) ≃

0

Z

R³

∇

(v(t

)

) ∇

z(t

) dx dt

. We expect that ∇

(v

) ≃ v

∇

v so that Hölder’s inequality yields

E(t) ≃

0

Z

R3

∇

v(t

)v(t

)

∇

z(t

) dx dt

. k∇

z k

Z _t

0

E(t

)

−1

p+1

k v(t

) k

W^s,p+12

dt

The term k v(t

) k

W^s,p+12

will be estimated by interpolating the estimates for v(t

) in L

and ˙ H

. The standard tool to do so is the Gagliardo-Nirenberg inequality, see Theorem A.7. We obtain

k∇

v k

L^p+12

. k∇ v k

k v k

. E(t)

−α 2 +_p+1^α

, where s satisfies the homogeneity conditions

(

s 6 α

p+12

−

=

+

which gives α = s

:=

, so that

E(t) . k∇

z k

0

E(t

) dt

and the Grönwall lemma proves the non blow-up of E(t), provided k∇

z k

< + ∞ which is the case for initial data in H

, thanks to probabilistic improvement of the Strichartz estimates that we will prove later.

In our context it will not be possible to construct such a strong solution v, even locally in time because of the lack of local Cauchy theory for energy supercritical wave equations. However the strategy used in [4] applies: this is the strategy one uses to construct Leray solutions in the context of the Navier-Stokes equations, which consists in first finding approximate solutions u

= z

+ v

that are global in time. Then the previous energy estimate provides uniform bounds for v

that allow strong compactness arguments in order to pass to the limit.

1.4.2. Yudovich-Wolibner argument. The Yudovich-Wolibner argument was first presented in the work of Wolibner in the context of the 2-dimensional Euler equations, see [20]. A similar argument was provided by Yudovich in the same context, see [21]. We will recall this argument, in its simplest version. Note that this kind of argument has been widely used since, in particular in the study of (SLW

), p = 3, s = 0, see [7].

Let us explain it in the context of the Schrödinger equation on R

: (NLS)

i∂

u + ∆u + u | u |

= 0 u(0) = u

∈ H

( R

) .

Let u

, u

∈ C

([0, T ), H

) be two solutions to (NLS). The following aims at proving uniqueness of solutions, that is u

= u

. In order to do so consider E(t) := k u

(t) − u

(t) k

. Then a computation, using that u

, u

solve (NLS) yields

E

(t) .

R²

| u

(t) − u

(t) | ( | u

(t) |

+ | u

(t) |

) dx .

If the embedding H

( R

) ֒ → L

( R

) were true, then we would have E

(t) . E(t) and would

deduce that E(t) = 0 for t ∈ [0, T ]. Unfortunately there is no such continuous embedding, and

we only have the Trudinger type estimate [1]

k u k

. √ p k u k

for all p > 2 . Using the previous inequality and the Hölder inequality gives

E

(t) . pE(t)

(1.2)

for all p > 1. After integration by separation of variables this implies E(t) 6 (Ct)

for a constant C > 0. Thus for a fixed t <

and letting p → ∞ , we get E(t) = 0. This argument can be iterated on time intervals

,

for n > 0 so that E(t) = 0 for all t > 0.

Remark 1.13. Another way to conclude is to optimize in p in (1.2) so that E

(t) . − E(t) log(E(t)) and gives the same result.

As mentioned before our setting will be a little more complicated: we will need some bootstrap argument to conclude rather than this simple integration techniques. The method will be used to prove existence of global solutions in a limiting case rather than proving uniqueness, the framework being very similar.

1.4.3. Organization of the paper. Section 2 explains the the randomization procedure and recalls the probabilistic improvement for the Strichartz estimates. We then provide a generalization of these estimates in the context of Besov spaces, see Proposition 2.7.

Section 3 is devoted to the proof of Theorem 1.4 in the case of the euclidean space R

. More precisely, sub-section 3.1 proves existence of global solutions u

to approximate equations, sub- section 3.2 provides uniform bounds in n for the nonlinear energies that will allow to use a compactness argument in sub-section 3.3.

Section 4 provides the proof of Theorem 1.8 and its corollary.

For reader’s convenience some useful facts concerning Sobolev and Besov spaces are gathered in Appendix A.

2. Probabilistic estimates

2.1. The probabilistic setting. We first recall some standard notation in Littlewood-Paley analysis.

Let C be the annulus { ξ ∈ R

, 3/4 6 | ξ | 6 8/3 } , then there exists radial functions χ, ϕ taking values in [0, 1] belonging to D (B(0, 4/3)) and D ( C ) satisfying

χ(ξ) +

j>0

ϕ(2

ξ) = 1 for all ξ ∈ R

and

j∈Z

ϕ(2

ξ) = 1 for all ξ ∈ R

\ { 0 } and such that

| j − j

| > 2 = ⇒ supp ϕ(2

· ) ∩ supp ϕ(2

· ) = ∅ j > 1 = ⇒ supp χ ∩ supp ϕ(2

· ) = ∅ . We now define the nonhomogeneous Littlewood-Paley projectors:

∆

u :=







0 for j 6 − 2 χ(D)u for j = − 1 ϕ(2

D)u for j > 0,

where χ(D) (resp. ϕ(2

D)) denotes the Fourier multiplier of symbol χ (resp. ϕ(2

· )). As a homogeneous Littlewood decomposition will be needed, we set ˙ ∆

:= ϕ(2

D) for all j ∈ Z . We set :

P

=

j^′6j−1

∆

and ˙ P

= χ(2

D).

In the case of the torus T

we construct a similar decomposition, with a bump function

ϕ ∈ D (B(0, 2)) such that ϕ = 1 on B(0, 1). Let (e

)

be the hilbertian sequence of L

( T

)

defined by x 7→ e

(x) = e

. For a function u =

c

e

, and for j > 0, we set P

u =

ϕ(2

| n | )c

e

and ∆

u := P

u − P

u, with the convention that P

= 0

An account of useful facts in Littlewood-Paley theory is given in Appendix A.

The randomization that is widely used in the context of the torus T

or more generally a compact manifold is presented in [5]. Consider (X

)

a sequence of random variables defined on a probability space (Ω, A , P ) which satisfy the following definition.

Definition 2.1. Let (Ω, F , P ) be a probability space and

X

n∈Z³;i=0,1

be a sequence of complex random variables defined on Ω, satisfying the symmetry property X

= X

and such that the random variables

X

, Re

X

, Im

X

n∈I;i=0,1

where

I =

Z

× { 0 }

∪ ( Z × Z

× { 0 } ) ∪

Z

× Z

, are independent; and that there exists a constant c > 0 such that

(2.1) E

e

6 e

for all γ ∈ R when n = 0, for all γ ∈ R

when n 6 = 0.

Remark 2.2. Definition 2.1 immediately implies that the X

, i = 0, 1 are real-valued and that the X

are of mean zero. Note that complex-valued Gaussian random variables, standard Bernoulli or random variables with compact support distributions satisfy the hypothesis of Definition 2.1.

Every u ∈ L

( T

) can be written in the hilbertian basis (e

)

as u =

u

e

with u

the Fourier coefficients of u. We introduce the Fourier randomization associated to a couple (u

, u

) with the randomization map:

(2.2) Θ

:











Ω −→ { map from T

to C } ω 7−→



X

n∈Z³

X

(ω)u

e

,

n∈Z³

X

(ω)u

e





.

There is a similar procedure in the euclidean setting. In this case the standard randomization setup is called the Wiener randomization and randomizes frequencies annuli in a way that we explain. Note that a rough version of that randomization was introduced by Wiener in [19], and that the smooth version used here has been developed by Benyi-Oh-Pocovnicu in [8]. Let ψ ∈ D (] − 1, 1[

) with the symmetry property ψ( − ξ) = ψ(ξ) for all ξ ∈ R

and satisfying the unit partition condition

ψ( · − n) = 1, so that for every u ∈ S

there holds

u =

n∈Z³

ψ(D − n)u . One readily sees that

(2.3) ξ 7−→

n∈Z³

| ψ(ξ − n) |

is bounded.

The randomization of a couple (u

, u

) is then defined with the randomization map (2.4) Θ

:











Ω −→ { map from R

to C }

ω 7−→



X

n∈Z³

X

(ω)ψ(D − n)u

,

n∈Z³

X

(ω)ψ(D − n)u





.

In both cases (randomization in T

or R

) we set µ

(A) := P

Θ

0,u1)

(A)

for every mea- surable set A, and define

M

(U ) :=

(u0,u1)∈H^s(U)×H^s−1(U)

µ

, the set of all measures that we will work with.

Remark 2.3. In the following, when there is no possible confusion we will denote by (u

, u

) the couple of random variables defined by the randomization maps (2.2) and (2.4), rather than (u

, u

) or any other notation.

These randomizations have been studied in [5], in which Burq-Tzvetkov proved the flolowing theorem. For a proof see Lemma B.1 in [5] and also Lemma 2.2 in [8].

Theorem 2.4 (Non-smoothing effect of the randomization setup). Let µ = µ

∈ M

(U ), U = R

or T

. We have the following:

(i) The measure µ is supported by H

(U ).

(ii) For s

> s if (u

, u

) ∈ H /

(U ) then µ( H

(U )) = 0.

(iii) In the case of the torus, if all the Fourier coefficients of u

and u

are different from zero and the support of the distributions of the

X

, Re(X

), Im(X

)

n∈I;i=0,1

are R then the support of µ is exactly H

(U ).

This theorem proves that there is no gain in regularity by randomization. However a gain in integrability is known, see Theorem 2.5, and is responsible for the improvement in Strichartz inequalities and thus the local wellposedness and global well-posedness theory in dispersive equations.

2.2. Probabilistic semigroup estimates. The starting point of every probabilistic improve- ment of the Strichartz estimates is the following well-known theorem in the theory of random Fourier series. For a proof see Lemma 3.1 in [5] and Lemma 2.1 in [13].

Theorem 2.5 (Kolmogorov-Paley-Zygmund). Let (X

)

be a sequence of independent and identically distributed real-valued random variables such that there exists a constant c > 0 such that for every γ ∈ R , the inequality (2.1) holds. Let (a

)

∈ ℓ

( Z ) be a complex valued sequence with the symmetry property a

= a

for every integer n (resp. a real-valued sequence (a

)

∈ ℓ

( N )). For q ∈ [1, ∞ ) one has:

(2.5)

X

n∈Z

a

X

L^q

. √ q k (a

)

k

.

We turn to the probabilistic improvement of Strichartz estimates and introduce semi-groups associated to the linear wave equation. Let s > 0 and (u

, u

) ∈ H

(U ). We set:

z(t) := S(t)(u

, u

) := cos(t |∇| )u

+ sin(t |∇| )

|∇| u

,

which is a solution to (LW). One of the key features of the linear wave equation is that “two time derivatives equal two space derivatives” so that one expects that “one time derivative equals one space derivative” which can be turned more rigorously writing ∂

z(t) = h∇i z(t) where ˜

˜

z(t) := ˜ S(t)(u

, u

) := − |∇| sin(t |∇| )

h∇i u

+ cos(t |∇| ) h∇i u

. For our purposes we will need a smooth version of both z(t) and ˜ z(t), namely

(2.6) z

(t) := S

(t)(u

, u

) := P

S(t)(u

, u

) and ˜ z

(t) := ˜ S

(t)(u

, u

) := P

S(t)(u ˜

, u

)

for every integer n > 1. We recall the probabilistic Strichartz estimates, proven in [15, 14]

for (2.7) and [13, 14] for (2.8).

Proposition 2.6 (Probabilistic Strichartz estimates for the wave operator). Let U standing for either T

or R

. Let (u

, u

) ∈ H

(U ) and still write (u

, u

) its randomization (Fourier randomization procedure or Wiener randomization procedure). Let z

stand for either z, z, z ˜

or z ˜

. For any q

∈ [1, ∞ ) and q

∈ [2, ∞ ]:

(2.7) P ( k z

k

> λ) .

exp







− cλ

max

T

, T

k (u

, u

) k







with ε = 0 if q

< ∞ and ε > 0 arbitrarily small otherwise.

For any q

∈ [2, ∞ ] and arbitrarily small ε > 0:

(2.8) P ( k z

k

> λ) .

(1 + T ) exp − cλ

max { 1, T

}k (u

, u

) k

!

. For our purposes we will need a counterpart of Proposition 2.6 in the context of Besov spaces:

Proposition 2.7 (Besov norm probabilistic Strichartz estimates). Let (u

, u

) ∈ H

(U ) where U stands for either T

or R

and still write (u

, u

) its randomization (Fourier randomization procedure or Wiener randomization procedure). Let z

stand for either z, z, z ˜

or z ˜

. For any q

∈ [1, ∞ ), q

∈ [2, ∞ ] and r ∈ [1, ∞ ]:

(2.9) P

k z

k

2,r)

> λ

.

exp





− c λ

max { T

, T

}k (u

, u

) k





with ε = 0 if q

< ∞ and r > 2; ε > 0 otherwise.

For any q

∈ [2, ∞ ] and r ∈ [1, ∞ ]:

(2.10) P

k z

k

> λ

.

exp − c(ε) λ

h T i

k (u

, u

) k

!

with ε > 0.

Remark 2.8. Note that the estimate (2.10) differs slightly from (2.8). The proof presented here will indeed differ from the one in [13] which appears in the context of Lebesgue spaces and relies on a series representation for z(t), a method that we decided not to use and present an alernative method. However, by applying the method of Oh-Pocovnicu one can prove a similar estimate.

Proof. The proof follows closely the one in [13, 14, 15] as only the parameter r has been added in the analysis. Nonetheless the proof is given in quite extensive details for reader’s convenience.

We will only give the proof in the case U = R

since the computations are almost the same in T

. It is indeed only the randomization setup which differs but in both cases they satisfy the same smoothing properties, see Appendix A. We will also assume that z

(t) = z(t), other cases could be treated in the exact same way.

Step 1. Assume s = 0, q

< ∞ and q

< ∞ . Without loss of generality we can assume that r < ∞ thanks to the inequality k z k

. k z k

and (2.7). Assume first that r > 2. We will prove that for p > max { q

, q

, r } one has

(2.11) k z k

. √ pT

k (u

, u

) k

.

Assume that (2.11) is proved, then the Markov inequality (with the function λ 7→ λ

) gives

P

k z k

> λ

6 λ

k z k