In this paper we obtain some new Strichartz estimates for the wave propagator $e^{it\sqrt{-Δ}}$ in the context of Wiener amalgam spaces. While it is well understood for the Schrödinger case, nothing is known about the wave propagator. This is because there is no such thing as an explicit formula for the integral kernel of the propagator unlike the Schrödinger case. To overcome this lack, we instead approach the kernel by rephrasing it as an oscillatory integral involving Bessel functions and then by carefully making use of cancellation in such integrals based on the asymptotic expansion of Bessel functions. Our approach can be applied to the Schrödinger case as well. We also obtain some corresponding retarded estimates to give applications to nonlinear wave equations where Wiener amalgam spaces as solution spaces can lead to a finer analysis of the local and global behavior of the solution.
In this paper we study the Strichartz estimates for the Schrödinger propagator in the context of Wiener amalgam spaces which, unlike the Lebesgue spaces, control the local regularity of a function and its decay at infinity separately. This separability makes it possible to perform a finer analysis of the local and global behavior of the propagator. Our results improve some of the classical ones in the case of large time.
In this paper we obtain some new Strichartz estimates for the Dirac flow in the context of Wiener amalgam spaces which control the local regularity of a function and its decay at infinity separately unlike $L^p$ spaces. While it is well understood recently for some flows such as the Schrödinger and wave flows that work in the non-relativistic regime, nothing is known about the Dirac flow which governs a physical system in the case of relativistic fields.
We study the strong ill-posedness (norm inflation with infinite loss of regularity) for the nonlinear wave equation at every initial data in Wiener amalgam and Fourier amalgam spaces with negative regularity. In particular these spaces contain Fourier-Lebesgue, Sobolev and some modulation spaces. The equations are posed on $\mathbb R^d$ and on torus $\mathbb T^d$ and involve a smooth power nonlinearity. Our results are sharp with respect to well-posedness results of Bényi and Okoudjou (2009) and Cordero and Nicola (2009) in the Wiener amalgam and modulation space cases. In particular, we also …
We consider Benjamin-Bona-Mahony (BBM) equation of the form $$ u_t+u_x+uu_x-u_{xxt}=0, \quad (x, t)\in \mathcal{M}\times \mathbb R $$ where $\mathcal{M}= \mathbb T$ or $\mathbb R.$ We establish norm inflation (NI) with infinite loss of regularity at general initial data in Fourier amalgam and Wiener amalgam spaces with negative regularity. This strengthen several known NI results at zero initial data in $H^s(\mathbb T)$ established by Bona-Dai (2017) and ill-posedness result established by Bona-Tzvetkov (2008) and Panthee (2011) in $H^s(\mathbb R).$ Our result is sharp with respect to local we…
Orthonormal Strichartz estimates for the Schrödinger equations with Hamiltonian on Wiener amalgam spaces
National Institute of Science Education and Research（国家科学教育研究中心）；Homi Bhabha National Institute（霍米·巴巴国立研究所）；Indian Statistical Institute Kolkata（印度统计学院加尔各答）
本文在 Wiener 调和空间上建立薛定谔方程的正交 Strichartz 估计，改进时间可积性指数并推广至正交族，应用于无穷粒子 Hartree 方程的适定性。
We establish existence of solutions in a scale of classes weaker than the finite energy Leray class and stronger than the infinite energy Lemarié-Rieusset class. The new classes are based on the $L^2$ Wiener amalgam spaces. Solutions in the classes closer to the Leray class are shown to satisfy some properties known in the Leray class but not the Lemarié-Rieusset class, namely eventual regularity and long time estimates on the growth of the local energy. In this sense, these solutions bridge the gap between Leray's original solutions and Lemarié-Rieusset's solutions and help identify scalings …
For a complex function $F$ on $\mathbb C$, we study the associated composition operator $T_{F}(f):=F\circ f= F(f)$ on Wiener amalgam $W^{p,q}(\mathbb R^d) \ (1\leq p< \infty, 1\leq q<2).$ We have shown $T_{F} $ maps $W^{p, 1}(\mathbb R^d)$ to $W^{p,q}(\mathbb R^d)$ if and only if $F$ is real analytic on $\mathbb R^2$ and $F(0)=0.$ Similar result is proved in the case of modulation spaces $M^{p,q}(\mathbb R^d).$ In particular, this gives an affirmative answer to the open question proposed by Bhimani-Ratnakumar (J. Funct. Anal. 270 (2016), p.621-648).
The objective of this paper is to report on recent progress on Strichartz estimates for the Schrödinger equation and to present the state-of-the-art. These estimates have been obtained in Lebesgue spaces, Sobolev spaces and, recently, in Wiener amalgam and modulation spaces. We present and compare the different technicalities. Then, we illustrate applications to well-posedness.
We first prove decay estimates and spacetime integral bounds for Stokes flows in amalgam spaces $E^r_q$ which connect the classical Lebesgue spaces to the spaces of uniformly locally $r$-integrable functions. Using these estimates, we construct mild solutions of the Navier-Stokes equations in the amalgam spaces satisfying the corresponding spacetime integral bounds. Time-global solutions are constructed for small data in $E^3_q$, $1\le q \le 3$. Our results provide new bounds for the strong solutions classically constructed by Kato and the more recent solutions in uniformly local spaces constr…
We completely characterize the boundedness on $L^p$ spaces and on Wiener amalgam spaces of the short-time Fourier transform (STFT) and of a special class of pseudodifferential operators, called localization operators. Precisely, a well-known STFT boundedness result on $L^p$ spaces is proved to be sharp. Then, sufficient conditions for the STFT to be bounded on the Wiener amalgam spaces $W(L^p,L^q)$ are given and their sharpness is shown. Localization operators are treated similarly. Using different techniques from those employed in the literature, we relax the known sufficient boundedness cond…
We study the local-in-time regularity of the Brownian motion with respect to localized variants of modulation spaces M^{p, q}_s and Wiener amalgam spaces W^{p, q}_s. We show that the periodic Brownian motion belongs locally in time to M^{p, q}_s (T) and W^{p, q}_s (T) for (s-1)q < -1, and the condition on the indices is optimal. Moreover, with the Wiener measure μon T, we show that (M^{p, q}_s (T), μ) and (W^{p, q}_s (T), μ) form abstract Wiener spaces for the same range of indices, yielding large deviation estimates. We also establish the endpoint regularity of the periodic Brownian motion wi…
We consider a randomization of a function on $\mathbb R^d$ that is naturally associated to the Wiener decomposition and, intrinsically, to the modulation spaces. Such randomized functions enjoy better integrability, thus allowing us to improve the Strichartz estimates for the Schrödinger equation. As an example, we also show that the energy-critical cubic nonlinear Schrödinger equation on $\mathbb R^4$ is almost surely locally well-posed with respect to randomized initial data below the energy space.
We establish some new properties of the Dunkl-Wiener amalgam spaces defined on the real line. These results allow us to obtain the boundedness of Dunkl-type fractional integral and fractional maximal operators in the Dunkl-Fofana spaces.
The local solvability of the Cauchy problem for the nonlinear vibrating plate equation is showed in the framework of modulation spaces. In the opposite direction, it is proved that there is no local wellposedness in Wiener amalgam spaces even for the solution to the homogeneous vibrating plate equation.
The purpose of this paper is to investigate several issues concerning the Dirac equation from a time-frequency analysis perspective. More precisely, we provide estimates in weighted modulation and Wiener amalgam spaces for the solutions of the Dirac equation with rough potentials. We focus in particular on bounded perturbations, arising as the Weyl quantization of suitable time-dependent symbols, as well as on quadratic and sub-quadratic non-smooth functions, hence generalizing the results in a recent paper by Kato and Naumkin. We then prove local well-posedness on the same function spaces for…
We consider the Cauchy problem for a generalized KdV equation \begin{eqnarray*} u_{t}+\partial_{x}^{3}u+u^{7}u_{x}=0, \end{eqnarray*} with random data on \R. Kenig, Ponce, Vega(Comm. Pure Appl. Math.46(1993), 527-620)proved that the problem is globally well-posed in H^{s}(\R)$ with s> s_{crit}=\frac{3}{14}, which is the scaling critical regularity indices. Birnir, Kenig, Ponce, Svanstedt, Vega(J. London Math. Soc. 53 (1996), 551-559.) proved that the problem is ill-posed in the sense that the time of existence T and the continuous dependence cannot be expressed in terms of the size of the data…
It is known that Fourier integral operators arising when solving Schrödinger-type operators are bounded on the modulation spaces $\cM^{p,q}$, for $1\leq p=q\leq\infty$, provided their symbols belong to the Sjöstrand class $M^{\infty,1}$. However, they generally fail to be bounded on $\cM^{p,q}$ for $p\not=q$. In this paper we study several additional conditions, to be imposed on the phase or on the symbol, which guarantee the boundedness on $\cM^{p,q}$ for $p\not=q$, and between $\cM^{p,q}\to\cM^{q,p}$, $1\leq q< p\leq\infty$. We also study similar problems for operators acting on Wiener amalg…
We study the Hermite operator $H=-Δ+|x|^2$ in $\mathbb{R}^d$ and its fractional powers $H^β$, $β>0$ in phase space. Namely, we represent functions $f$ via the so-called short-time Fourier, alias Fourier-Wigner or Bargmann transform $V_g f$ ($g$ being a fixed window function), and we measure their regularity and decay by means of mixed Lebesgue norms in phase space of $V_g f$, that is in terms of membership to modulation spaces $M^{p,q}$, $0< p,q\leq \infty$. We prove the complete range of fixed-time estimates for the semigroup $e^{-tH^β}$ when acting on $M^{p,q}$, for every $0< p,q\leq \infty$…
Consider the H^{1/2}-critical Schroedinger equation with a cubic nonlinearity in R^3, i \partial_t ψ+ Δψ+ |ψ|^2 ψ= 0. It admits an eight-dimensional manifold of periodic solutions called solitons e^{i(Γ+ vx - t|v|^2 + α^2 t)} ϕ(x-2tv-D, α), where ϕ(x, α) is a positive ground state solution of the semilinear elliptic equation -Δϕ+ α^2ϕ= ϕ^3. We prove that in the neighborhood of the soliton manifold there exists a H^{1/2} real analytic manifold N of asymptotically stable solutions of the Schroedinger equation, meaning they are the sum of a moving soliton and a dispersive term. Furthermore, a sol…
