Strichartz Type Research Articles

Strichartz and localized energy estimates for the wave equation in strictly concave domains

We prove localized energy estimates for the wave equation in domains with a strictly concave boundary when homogeneous Dirichlet or Neumann conditions are imposed. By restricting the solution to small, frequency dependent, space time collars of the boundary, it is seen that a stronger gain in regularity can be obtained relative to the usual energy estimates. Mixed norm estimates of Strichartz and square function type follow as a result, using the energy estimates to control error terms which arise in a wave packet parametrix construction. While the latter estimates are not new for Dirichlet conditions, the present approach provides an avenue for treating these estimates when Neumann conditions are imposed. The method also treats Schr{\"o}dinger equations with time independent coefficients.

Infinite energy solutions for critical wave equation with fractional damping in unbounded domains

This work is devoted to infinite-energy solutions of semi-linear wave equations in unbounded smooth domains of R3 with fractional damping of the form (−Δx+1)12∂tu. The work extends previously known results for bounded domains in finite energy case. Furthermore, well-posedness and existence of locally-compact smooth attractors for the critical quintic non-linearity are obtained under less restrictive assumptions on non-linearity, relaxing some artificial technical conditions used before. This is achieved by virtue of new type Lyapunov functional that allows to establish extra space–time regularity of solutions of Strichartz type.

Existence and Regularity of Propagators for Multi-Particle Schrödinger Equations in External Fields

We consider Schrodinger equations for N number of particles in (classical) electro-magnetic fields that are interacting with each other via time dependent inter-particle potentials. We prove that they uniquely generate unitary propagators $${\{U(t,s), t,s \in \mathbb{R}\}}$$ on the state space $${\mathcal{H}}$$ under the conditions that fields are spatially smooth and do not grow too rapidly at infinity so that propagators for single particles satisfy Strichartz estimates locally in time, and that local singularities of inter-particle potentials are not too strong that time frozen Hamiltonians define natural selfadjoint realizations in $${\mathcal{H}}$$ . We also show, under very mild additional assumptions on the time derivative of inter-particle potentials, that propagators possess the domain of definition of the quantum harmonic oscillator $${\Sigma(2)}$$ as an invariant subspace such that, for initial states in $${\Sigma(2)}$$ , solutions are C 1 functions of the time variable with values in $${\mathcal{H}}$$ . New estimates of Strichartz type for propagators for N independent particles in the field will be proved and used in the proof.

Quantum hydrodynamics with nonlinear interactions

In this paper we prove the global existence of large amplitude finite energy solutions for a system describing Quantum Fluids with nonlinear nonlocal interaction terms. The system may also (but not necessarily) include dissipation terms which do not provide any help to get the global existence. The method is based on the polar factorization of the wave function (which somehow generalizes the WKB method), the construction of approximate solutions via a fractional step argument and the deduction of Strichartz type estimates for the approximate solutions. Finally local smoothing and a compactness argument of Lions Aubin type allow to show the convergence.

Strichartz estimates and smooth attractors for a sub-quintic wave equation with fractional damping in bounded domains

The work is devoted to Dirichlet problem for sub-quintic semi-linear wave equation with damping term of the form $(-{\Delta_x})^\alpha{\partial_t}u$, $\alpha\in(0,\frac{1}{2})$, in bounded smooth domains of $\mathbb R^3$. It appears that to prove well-posedness and develop smooth attractor theory for the problem, we need additional regularity of the solutions, which does not follow from the energy estimate. Considering the original problem as perturbation of the linear one the task is reduced to derivation of Strichartz type estimate for the linear wave equation with fractional damping, which is the main feature of the work. Existence of smooth exponential attractor for the natural dynamical system associated with the problem is also established.

A characterization of band-limited <italic>L</italic><sup><italic>p</italic></sup> function and its applications

Using the ideas of Shannon-Nyquist sampling theorem, we give a characterization for the band-limited L p functions. As applications, we obtain some new estimates for the Schrodinger equation such as Strichartz type estimate and reverse Strichartz type estimate, and simple proof for some known estimates.

The Fourier restriction norm method for the Zakharov-Kuznetsov equation

The Cauchy problem for the Zakharov-Kuznetsov equation is shown to be locallywell-posed in $H^s(\mathbb{R}^2)$ for all $s>\frac{1}{2}$ by using the Fourier restriction norm method and bilinear refinements of Strichartz type inequalities.

Recent progress in attractors for quintic wave equations

We report on new results concerning the global well-posedness, dissipativity and attractors for the quintic wave equations in bounded domains of $\mathbb R^3$ with damping terms of the form $(-\Delta _x)^\theta \partial _t u$, where $\theta =0$ or $\theta =1/2$. The main ingredient of the work is the hidden extra regularity of solutions that does not follow from energy estimates. Due to the extra regularity of solutions existence of a smooth attractor then follows from the smoothing property when $\theta =1/2$. For $\theta =0$ existence of smooth attractors is more complicated and follows from Strichartz type estimates.

Strichartz Estimates for Some 2D Water Wave Models

In this paper we establish Strichartz type estimates associated with a class of semigroup operators in Rn, which for n = 2 correspond to some 2D water wave models. We also establish a nonlinear scattering result for solutions of the generalized Benney-Luke equation for higher order nonlinearity and small data initial in the energy space.

Local well-posedness for the nonlinear Dirac equation in two space dimensions

The Cauchy problem for the cubic nonlinear Dirac equation in two space dimensions is locally well-posed for data in $H^s$ for $ s > 1/2$. The proof given in spaces of Bourgain-Klainerman-Machedon type relies on the null structure of the nonlinearity as used by d'Ancona-Foschi-Selberg for the Dirac-Klein-Gordon system before and bilinear Strichartz type estimates for the wave equation by Selberg and Foschi-Klainerman.

Global Regularity for the Quadratic Klein-Gordon Equation in R 1+2

We consider 2-D Klein-Gordon equation with quadratic nonlinearity and prove Strichartz type dispersive estimates for the global solution with small initial data in the Sobolev space H 1+ϵ.

Nonlinear Schrödinger equation for the twisted Laplacian

We establish the local well posedness of solution to the nonlinear Schrödinger equation associated to the twisted Laplacian on Cn in certain first order Sobolev space. Our approach is based on Strichartz type estimates, and is valid for a general class of nonlinearities including power type. The case n=1 represents the magnetic Schrödinger equation in the plane with magnetic potential A(z)=iz, z∈C.

Global solutions for the Navier-Stokes equations in the rotational framework

The existence of global unique solutions to the Navier-Stokes equations with the Coriolis force is established in the homogeneous Sobolev spaces \(\dot{H}^s (\mathbb R ^3)^3\) for \(1/2 < s < 3/4\) if the speed of rotation is sufficiently large. This phenomenon is so-called the global regularity. The relationship between the size of initial datum and the speed of rotation is also derived. The proof is based on the space time estimates of the Strichartz type for the semigroup associated with the linearized equations. In the scaling critical space \(\dot{H}^{\frac{1}{2}} (\mathbb R ^3)^3\), the global regularity is also shown.

THE ARTIFICIAL COMPRESSIBILITY APPROXIMATION FOR MHD EQUATIONS IN UNBOUNDED DOMAIN

We analyze a method of approximation for the weak solutions of the incompressible magnetohydrodynamic equations (MHD) in unbounded domains. In particular we describe an hyperbolic version of the so-called artificial compressibility method adapted to the MHD system. By exploiting the wave equation structure of the approximating system we achieve the convergence of the approximating sequences by means of dispersive estimate of Strichartz type. We prove that the soleinoidal component of the approximating velocity and magnetic fields is relatively compact and converges strongly to a weak solution of the MHD equation.

Generalized Besov Spaces and Their Applications

We define and study the Bessel potential and inhomogeneous Besov spaces associated with the Dunkl operators on $\mathbf{R}^d$. As applications on these spaces we construct the Sobolev type embedding theorem and the paraproduct operators associated with the Dunkl operators, as similar to those defined by Bony. We also establish Strichartz type estimates for the Dunkl-Schrodinger equation and finally we study the problem of well posedness of the generalized heat equation.

On well-posedness for nonlinear Schrödinger equations with power nonlinearity in fractional order Sobolev spaces

We study the well-posedness for the nonlinear Schrödinger equation (NLS) i∂tu+12Δu=λ|u|p−1u in R1+n, where p>1,λ∈C, and prove that (NLS) is locally well-posed in Hs if 2<s<4 and s/2<p<1+4/(n−2s)+. To obtain a good lower bound for p, we systematically use Strichartz type estimates in fractional order Besov spaces for the time variable.

Nonlinear instability of linearly unstable standing waves for nonlinear Schrödinger equations

We study the instability of standing waves for nonlinear Schrodinger equations. Under a general assumption on nonlinearity, we prove that linear instability implies orbital instability in any dimension. For that purpose, we establish a Strichartz type estimate for the propagator generated by the linearized operator around standing wave.

Conditional stability theorem for the one dimensional Klein-Gordon equation

The paper addresses the conditional non-linear stability of the steady state solutions of the one-dimensional Klein-Gordon equation for large time. We explicitly construct the center-stable manifold for the steady state solutions using the modulation method of Soffer and Weinstein and Strichartz type estimates. The main difficulty in the one-dimensional case is that the required decay of the Klein-Gordon semigroup does not follow from Strichartz estimates alone. We resolve this issue by proving an additional weighted decay estimate and further refinement of the function spaces, which allows us to close the argument in spaces with very little time decay.

Global parametrices and dispersive estimates for variable coefficient wave equations

In this article we consider variable coefficient time dependent wave equations in \({\mathbb {R} \times \mathbb {R}^n}\) . Using phase space methods we construct outgoing parametrices and prove Strichartz type estimates globally in time. This is done in the context of C 2 metrics which satisfy a weak asymptotic flatness condition at infinity.

The Klein–Gordon equation in the Anti-de Sitter cosmology

This paper deals with the Klein–Gordon equation on the Poincaré chart of the 5-dimensional Anti-de Sitter universe. When the mass μ is larger than − 1 4 , the Cauchy problem is well-posed despite the loss of global hyperbolicity due to the time-like horizon. We express the finite energy solutions in the form of a continuous Kaluza–Klein tower and we deduce a uniform decay as | t | − 3 2 . We investigate the case μ = ν 2 − 1 2 , ν ∈ N ⁎ , which encompasses the gravitational fluctuations, ν = 4 , and the electromagnetic waves, ν = 2 . The propagation of the wave front set shows that the horizon acts like a perfect mirror. We establish that the smooth solutions decay as | t | − 2 − μ + 1 4 , and we get global L p estimates of Strichartz type. When ν is even, there appears a lacuna and the equipartition of the energy occurs at finite time for the compactly supported initial data, although the Huygens principle fails. We address the cosmological model of the negative-tension Minkowski brane, on which a Robin boundary condition is imposed. We prove the hyperbolic mixed problem is well-posed and the normalizable solutions can be expanded into a discrete Kaluza–Klein tower. We establish some L 2 − L ∞ estimates in suitable weighted Sobolev spaces.