Strichartz estimates for the Schrödinger equation with radial data

We prove an endpoint Strichartz estimate for radial solutions of the two-dimensional Schrödinger equation:

     

Strichartz estimates for the wave equation on manifolds with boundary

We prove certain mixed-norm Strichartz estimates on manifolds with boundary. Using them we are able to prove new results for the critical and subcritical wave equation in 4-dimensions with Dirichlet or Neumann boundary conditions. We obtain global existence in the subcritical case, as well as global existence for the critical equation with small data. We also can use our Strichartz estimates to prove scattering results for the critical wave equation with Dirichlet boundary conditions in 3-dimensions.     

Improved inhomogeneous Strichartz estimates for the Schrödinger equation

We study inhomogeneous Strichartz estimates for the Schrödinger equation for dimension n?3. Using a frequency localization, we obtain some improved range of Strichartz estimates for the solution of inhomogeneous Schrödinger equation except dimension n=3.     

Dispersion and Strichartz estimates for some finite rank perturbations of the Laplace operator

We consider the dispersion properties in L^p spaces of Schrödinger hamiltonians with a large number of obstacles modelled by rank one perturbations. We obtain both for the dispersion an Strichartz estimates nonperturbative results with respect to the coupling constants.     

Some new Strichartz estimates for the Schrödinger equation

We deal with fixed-time and Strichartz estimates for the Schrödinger propagator as an operator on Wiener amalgam spaces. We discuss the sharpness of the known estimates and we provide some new estimates which generalize the classical ones. As an application, we present a result on the wellposedness of the linear Schrödinger equation with a rough time-dependent potential.     

Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation

The aim of this article is twofold. First we consider the wave equation in the hyperbolic space and obtain the counterparts of the Strichartz type estimates in this context. Next we examine the relationship between semilinear hyperbolic equations in the Minkowski space and in the hyperbolic space. This leads to a simple proof of the recent result of Georgiev, Lindblad and Sogge on global existence for solutions to semilinear hyperbolic problems with small data. Shifting the space-time Strichartz estimates from the hyperbolic space to the Minkowski space yields weighted Strichartz estimates in which extend the ones of Georgiev, Lindblad, and Sogge.

     

Global Strichartz estimates for the wave equation with time-periodic potentials

We obtain global Strichartz estimates for the solutions u of the wave equation for time-periodic potentials V(t,x) with compact support with respect to x. Our analysis is based on the analytic properties of the cut-off resolvent Rχ(z)=χ(U⁻¹(T)−zI)ψ₁, where U(T)=U(T,0) is the monodromy operator and T>0 the period of V(t,x). We show that if Rχ(z) has no poles z∈C, |z|?1, then for n?3, odd, we have a exponential decal of local energy. For n?2, even, we obtain also an uniform decay of local energy assuming that Rχ(z) has no poles z∈C, |z|?1, and Rχ(z) remains bounded for z in a small neighborhood of 0.     

Dispersion and Strichartz estimates for the Liouville equation

We consider the Liouville equation associated to a metric g and we prove dispersion and Strichartz estimates for the solution of this equation in terms of the geometry of the trajectories associated to g. In particular, we obtain global Strichartz estimates in time for metrics where dispersion estimate is false even locally in time. We also study the analogy between Strichartz estimates obtained for the Liouville equation and the Schrödinger equation with variable coefficients. To cite this article: D. Salort, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Global Existence for a Kinetic Model of Chemotaxis via Dispersion and Strichartz Estimates

We investigate further the existence of solutions to kinetic models of chemotaxis. These are nonlinear transport-scattering equations with a quadratic nonlinearity which have been used to describe the motion of bacteria since the 80's when experimental observations have shown they move by a series of ‘run and tumble’. The existence of solutions has been obtained in several papers Chalub et al. (2004 Chalub , F. A. C. C. , Markowich , P. A. , Perthame , B. , Schmeiser , C. ( 2004 ). Kinetic models for chemotaxis and their drift-diffusion limits . Monatsh. Math. 142 : 123 – 141 .[Crossref], [Web of Science ®] , [Google Scholar]), Hwang et al. (2005a Hwang , H. J. , Kang , K. , Stevens , A. ( 2005a ). Global solutions of nonlinear transport equations for chemosensitive movement . SIAM J. Math. Anal. 36 ( 4 ): 1177 – 1199 . [Google Scholar] b Hwang , H. J. , Kang , K. , Stevens , A. ( 2005b ). Drift-diffusion limits of kinetic models for chemotaxis: a generalization . Discrete Contin. Dyn. Syst. Ser. B 5 ( 2 ): 319 – 334 . [Google Scholar]) using direct and strong dispersive effects.

Here, we use the weak dispersion estimates of Castella and Perthame (1996 Castella , F. , Perthame , B. ( 1996 ). Estimations de Strichartz pour les équations de transport cinétique. [Strichartz’ estimates for kinetic transport equations.] C. R. Acad. Sci. Paris Sér. I 322 ( 6 ): 535 – 540 . [Google Scholar]) to prove global existence in various situations depending on the turning kernel. In the most difficult cases, where both the velocities before and after tumbling appear, with the known methods, only Strichartz estimates can give a result, with a smallness assumption.     

Strichartz estimates for the Dunkl wave equation and application

In this paper, we prove dispersive and Strichartz estimates associated for the Dunkl wave equation.     

Endpoint Strichartz estimates for the magnetic Schrödinger equation

We prove Strichartz estimates for the Schrödinger equation with an electromagnetic potential, in dimension n?3. The decay and regularity assumptions on the potentials are almost critical, i.e., close to the Coulomb case. In addition, we require repulsivity and a nontrapping condition, which are expressed as smallness of suitable components of the potentials, while the potentials themselves can be large. The proof is based on smoothing estimates and new Sobolev embeddings for spaces associated to magnetic potentials.     

Strichartz estimates for the magnetic Schrödinger equation

We prove global, scale invariant Strichartz estimates for the linear magnetic Schrödinger equation with small time dependent magnetic field. This is done by constructing an appropriate parametrix. As an application, we show a global regularity type result for Schrödinger maps in dimensions n?6.     

Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients III

In an earlier work of the author it was proved that the Strichartz estimates for second order hyperbolic operators hold in full if the coefficients are of class . Here we strengthen this and show that the same holds if the coefficients have two derivatives in . Then we use this result to improve the local theory for second order nonlinear hyperbolic equations.

     

Strichartz estimates for dispersive equations and solvability of the Kawahara equation

We first establish a series of Strichartz estimates for a general class of linear dispersive equations by applying the theory of oscillatory integrals established by Kenig, Ponce and Vega. Next we use such estimates to study solvability of the Cauchy problem of the Kawahara equation in the class C(R,Hs(R)). Local existence is proved for s>1/4 and global existence is proved for s?2.     

Strichartz estimates for the Euler equations in the rotational framework

We consider the initial value problems of the incompressible Euler equations in the rotational framework. We obtain the optimal range of the Strichartz estimate for the linear group associated with the Coriolis forces. As an application, we prove that the lifespan of the solution can be taken arbitrarily large provided that the speed of rotation is sufficiently high.     

A counter example to Strichartz estimates for the inhomogeneous Schrödinger equation

We disprove Strichartz estimates for the solution of the inhomogeneous Schrödinger equation in a certain range of the Lebesgue exponents values.     

Strichartz type estimates for fractional heat equations

We obtain Strichartz estimates for the fractional heat equations by using both the abstract Strichartz estimates of Keel-Tao and the Hardy-Littlewood-Sobolev inequality. We also prove an endpoint homogeneous Strichartz estimate via replacing by BMOx(Rn) and a parabolic homogeneous Strichartz estimate. Meanwhile, we generalize the Strichartz estimates by replacing the Lebesgue spaces with either Besov spaces or Sobolev spaces. Moreover, we establish the Strichartz estimates for the fractional heat equations with a time dependent potential of an appropriate integrability. As an application, we prove the global existence and uniqueness of regular solutions in spatial variables for the generalized Navier-Stokes system with Lr(Rn) data.     

Strichartz Estimates for Wave Equations with Coefficients of Sobolev Regularity

Wave packet techniques provide an effective method for proving Strichartz estimates on solutions to wave equations whose coefficients are not smooth. We use such methods to show that the existing results for C ^{1, 1} and C ^{1, α} coefficients can be improved when the coefficients of the wave operator lie in a Sobolev space of sufficiently high order.     

Strichartz Estimates for Schr(?)dinger Equations with Non-degenerate Coefficients

In the present paper,the full range Strichartz estimates for homogeneous Schr(?)dinger equations with non-degenerate and non-smooth coefficients are proved.For inhomogeneous equation,the non-endpoint Strichartz estimates are also obtained.