Strichartz Type Estimates Research Articles

Strichartz type estimates and the well-posedness of an energy critical 2D wave equation in a bounded domain

We study the well-posedness of the Cauchy problem with Dirichlet or Neumann boundary conditions associated to an H 1 -critical semilinear wave equation on a smooth bounded domain Ω ⊂ R 2 . First, we prove an appropriate Strichartz type estimate using the L q spectral projector estimates of the Laplace operator. Our proof follows Burq, Lebeau and Planchon (2008) [4]. Then, we show the global well-posedness when the energy is below or at the threshold given by the sharp Moser–Trudinger inequality. Finally, in the supercritical case, we prove an instability result using the finite speed of propagation and a quantitative study of the associated ODE with oscillatory data.

Strichartz estimates for water waves

In this paper we investigate the dispersive properties of the solutions of the two dimensional water-waves system with surface tension. First we prove Strichartz type estimates with loss of derivatives at the same low level of regularity we were able to construct the solutions in [3]. On the other hand, for smoother initial data, we prove that the solutions enjoy the optimal Strichartz estimates (i.e, without loss of regularity compared to the system linearized at (η=0,ψ=0)).

Strichartz Estimates for the Water-Wave Problem with Surface Tension

Strichartz-type estimates for one-dimensional surface water-waves under surface tension are studied, based on the formulation of the problem as a nonlinear dispersive equation. We establish a family of dispersion estimates on time scales depending on the size of the frequencies. We infer that a solution u of the dispersive equation we introduce satisfies local-in-time Strichartz estimates with loss in derivative: where C depends on T and on the norms of the H s -norm of the initial data. The proof uses the frequency analysis and semiclassical Strichartz estimates for the linealized water-wave operator.

L 2-Stability Theory of the Boltzmann Equation near a Global Maxwellian

We present three a priori L2-stability estimates for classical solutions to the Boltzmann equation with a cut-off inverse power law potential, when initial datum is a perturbation of a global Maxwellian. We show that L2-stability estimates of classical solutions depend on Strichartz type estimates of perturbations and the non-positive definiteness of the linearized collision operator. Several well known classical solutions to the Boltzmann equation fit our L2-stability framework.

A nonlinear Schrödinger equation with two symmetric point interactions in one dimension

We consider a time-dependent one-dimensional nonlinear Schrödinger equation with a symmetric double-well potential represented by two Dirac's δ. Among our results we give an explicit formula for the integral kernel of the unitary semigroup associated with the linear part of the Hamiltonian. Then we establish the corresponding Strichartz-type estimate and we prove local existence and uniqueness of the solution to the originalnonlinear problem.

Introduction to scattering for radial 3 D NLKG below energy norm

We prove scattering for some radial 3 D semilinear Klein–Gordon equations with rough data. First we prove Strichartz-type estimates in mixed norm spaces. Then by using these decays we establish some local bounds. By combining these results to a Morawetz-type estimate and a radial Sobolev inequality we control the variation of an almost conserved quantity on arbitrary large intervals. Once we have showed that this quantity is controlled, we prove that some of these local bounds can be upgraded to global bounds. This is enough to establish scattering. All the estimates involved require a delicate analysis due to the nature of the nonlinearity and the lack of scaling.

Instability of solitary wave solutions for the generalized BO–ZK equation

In this paper we study the generalized BO–ZK equation in two space dimensions u t + u p u x + α H u x x + ε u x y y = 0 . We review the existence theory for solitary waves and prove that they are nonlinearly unstable if p belongs to the range 4 / 3 < p < 4 . We also establish Strichartz-type estimates.

UNIFORM L2-STABILITY ESTIMATES FOR THE RELATIVISTIC BOLTZMANN EQUATION

In this paper, we derive an a prioriL2-stability estimate for classical solutions to the relativistic Boltzmann equation, when the initial datum is a small perturbation of a global relativistic Maxwellian. For the stability estimate, we use the dissipative property of the linearized collision operator and a Strichartz type estimate for classical solutions. As a direct application of our stability estimates, we establish that classical solutions in Glassey–Strauss and Hsiao–Yu's frameworks satisfy a uniform L2-stability estimate.

On KP-II type equations on cylinders

In this article we study the generalized dispersion version of the Kadomtsev–Petviashvili II equation, on T×R and T×R2. We start by proving bilinear Strichartz type estimates, dependent only on the dimension of the domain but not on the dispersion. Their analogues in terms of Bourgain spaces are then used as the main tool for the proof of bilinear estimates of the nonlinear terms of the equation and consequently of local well-posedness for the Cauchy problem.

Équation des ondes sur les espaces symétriques riemanniens

We prove that the solutions of the homogeneous wave equation on Riemannian symmetric spaces have dispersion properties and we deduce Strichartz type estimates for these solutions. To cite this article: A. Hassani, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Rotating Fluids with Small Viscosity

We study a rotating fluid in the whole space , in the case where there are no viscosity in the vertical direction and a small viscosity in the horizontal direction (of size e α with 0 0). Using Strichartz-type estimates, we prove global existence of strong solutions when the Rossby number e is small enough.

Dispersive Estimates for Manifolds with One Trapped Orbit

For a large class of complete, non-compact Riemannian manifolds, (M, g), with boundary, we prove high energy resolvent estimates in the case where there is one trapped hyperbolic geodesic. As an application, we have the following local smoothing estimate for the Schrödinger propagator: where ρ s (x) ∈ 𝒞∞(M) satisfies ρ s = ⟨ dist g (x,x 0) ⟩−s , , and V ∈ 𝒞∞(M), 0 ≤ V ≤ C satisfies |∇ V| ≤ C ⟨ dist(x,x 0) ⟩−1−δ for some δ > 0. From the local smoothing estimate, we deduce a family of Strichartz-type estimates, which are used to prove two well-posedness results for the nonlinear Schrödinger equation. As a second application, we prove the following sub-exponential local energy decay estimate for solutions to the wave equation when dim M = n ≥ 3 is odd and M is equal to ℝ n outside a compact set: where ψ ∈ 𝒞∞(M), ψ ≡ e −|x|2 outside a compact set.

On Schrödinger Propagator for the Special Hermite Operator

We establish a Strichartz type estimate for the Schrodinger propagator e itℒ for the special Hermite operator ℒ on ℂ n . Our method relies on a regularization technique. We show that no admissibility condition is required on (q,p) when 1≤q≤2.

Resolvent Estimates and Matrix-Valued Schrödinger Operator with Eigenvalue Crossings; Application to Strichartz Estimates

We consider a semi-classical Schrödinger operator with a matrix-valued potential presenting eigenvalue crossings on isolated points. We obtain estimates for the boundary values of the resolvent under a generalized non-trapping assumption. As a consequence, we prove the smoothing effect of this operator, derive Strichartz type estimate for the propagator and get an existence theorem for a system of non-linear Schrödinger equations.

The Cauchy Problem of the Nonlinear Schrödinger Equations in ℝ1+1

In this paper, we consider the local and global solution for the nonlinear Schrodinger equation with data in the homogeneous and nonhomogeneous Besov space and the scattering result for small data. The techniques to be used are adapted from the Strichartz type estimate, Kato’s smoothing effect and the maximal function (in time) estimate for the free Schrodinger operator.

Well-posedness for Semi-relativistic Hartree Equations of Critical Type

We prove local and global well-posedness for semi-relativistic, nonlinear Schrödinger equations $i \partial_t u = \sqrt{-\Delta + m^2} u + F(u)$ with initial data in H s (ℝ3), $s \geqslant 1/2$ . Here F(u) is a critical Hartree nonlinearity that corresponds to Coulomb or Yukawa type self-interactions. For focusing F(u), which arise in the quantum theory of boson stars, we derive global-in-time existence for small initial data, where the smallness condition is expressed in terms of the L 2-norm of solitary wave ground states. Our proof of well-posedness does not rely on Strichartz type estimates. As a major benefit from this, our method enables us to consider external potentials of a quite general class.

A DISPERSIVE APPROACH TO THE ARTIFICIAL COMPRESSIBILITY APPROXIMATIONS OF THE NAVIER–STOKES EQUATIONS IN 3D

In this paper we study how to approximate the Leray weak solutions of the incompressible Navier–Stokes equations. In particular we describe an hyperbolic version of the so-called artificial compressibility method investigated by J. L. Lions and Temam. By exploiting the wave equation structure of the pressure of the approximating system we achieve the convergence of the approximating sequences by means of dispersive estimates of Strichartz type. We prove that the projection of the approximating velocity fields on the divergence free vectors is relatively compact and converges to a Leray weak solution of the incompressible Navier–Stokes equation.

Packing spheres and fractal Strichartz estimates in $\mathbb {R}^d$ for $d\geq 3$

We prove an estimate for the spherical average operator in R d if d > 3. This leads to a lower bound for the Hausdorff dimension of unions of certain collections of spheres and to a Strichartz-type estimate for solutions of the wave equation.

A class on non-local linear operators for vorticity waves

A steady longitudinal current in the nearshore can, in some conditions, support oscillations known as vorticity waves or shear waves. In this article, we consider a family of nonlinear evolution equations derived by Shrira and Voronovitch to describe the dynamics of vorticity waves near the coastal line and make the study of the dispersion and smoothing properties of the associated nonlocal free problems. More precisely, after establishing long and short time uniform estimates for a certain class of oscillatory integrals, we derive “L p −L q ” and Strichartz-type estimates for the solutions of the linearized equations.

Weak solutions for the Falk model system of shape memory alloys in energy class

AbstractWe show the unique global existence of energy class solutions for the Falk model system of shape memory alloys under the general non‐linearity as well as considered in Aiki (Math. Meth. Appl. Sci. 2000; 23: 299). Our main tools of the proofs are the Strichartz type estimate for the Boussinesq type equation and the maximal regularity estimate for the heat equation. Copyright © 2005 John Wiley &amp; Sons, Ltd.