Well-posedness of the modified dispersion-generalized Benjamin-Ono equations in modulation spaces

Critical exponent for evolution equations in modulation spaces

Navier–Stokes equations and nonlinear heat equations in modulation spaces with negative derivative indices

We study time-frequency localization operators of the form A a φ 1 , φ 2 A_a^{\varphi _1\!,\varphi _2} , where a a is the symbol of the operator and φ 1 , φ 2 \varphi _1 , \varphi _2 are the analysis and synthesis windows, respectively. It is shown in an earlier paper by the authors that a sufficient condition for A a φ 1 , φ 2 ∈ S p ( L 2 ( R d ) ) A_a^{\varphi _1,\varphi _2}\in S_p(L^2(\mathbb {R}^d)) , the Schatten class of order p p , is that a a belongs to the modulation space M p , ∞ ( R 2 d ) M^{p,\infty }(\mathbb {R}^{2d}) and the window functions to the modulation space M 1 M^1 . Here we prove a partial converse: if A a φ 1 , φ 2 ∈ S p ( L 2 ( R d ) ) A_a^{\varphi _1,\varphi _2}\in S_p(L^2(\mathbb {R}^d)) for every pair of window functions φ 1 , φ 2 ∈ S ( R 2 d ) \varphi _1,\varphi _2\in \mathcal {S}(\mathbb {R}^{2d}) with a uniform norm estimate, then the corresponding symbol a a must belong to the modulation space M p , ∞ ( R 2 d ) M^{p,\infty }(\mathbb {R}^{2d}) . In this sense, modulation spaces are optimal for the study of localization operators. The main ingredients in our proofs are frame theory and Gabor frames. For p = ∞ p=\infty and p = 2 p=2 , we recapture earlier results, which were obtained by different methods.

On the existence of global solutions of the Hartree equation for initial data in the modulation space [formula omitted

We give a method to estimate non-integer power function |u| k u in modulation space which is an open question in the study of modulation space. As an application, we can study Cauchy problem for the nonlinear Klein-Gordon equation with nonlinear term |u| k u in modulation space, where k is not an integer. Moreover, we also study the global solution with small initial value for the Klein-Gordon-Hartree equation. The results show some advantages of modulation space both in high and low regularity cases.

Remarks on Fourier multipliers and applications to the wave equation

We show the local well-posedness of the Cauchy problem for the cubic nonlinear Schrödinger equation on modulation spaces Msp;q(Rd) for d 2 N, 1 p; q 1 and s > d 1 􀀀 1 q for q > 1 or s 0 for q = 1. This improves [4, Theorem 1.1] by Bényi and Okoudjou where only the case q = 1 is considered. Our result is based on the algebra property of modulation spaces with indices as above for which we give an elementary proof via a new Hölder-like inequality for modulation spaces.

We improve our previous result ([L. Molinet and T. Tanaka, Unconditional well-posedness for some nonlinear periodic one-dimensional dispersive equations, J. Funct. Anal. 283 (2022) 109490]) on the Cauchy problem for one dimensional dispersive equations with a quite general nonlinearity in the periodic setting. Under the same hypotheses that the dispersive operator behaves for high frequencies as a Fourier multiplier by [Formula: see text] with [Formula: see text], and that the nonlinear term is of the form [Formula: see text] where [Formula: see text] is a real analytic function whose Taylor series around the origin has an infinite radius of convergence, we prove the unconditional LWP of the Cauchy problem in [Formula: see text] for [Formula: see text] with [Formula: see text]. It is worth noting that this result is optimal in the case [Formula: see text] (generalized KdV equation) in view of the restriction [Formula: see text] for the continuous injection of [Formula: see text] into [Formula: see text]. Our main new ingredient is the replacement of refined Strichartz estimates with refined bilinear estimates in the treatment of the worst resonant interactions. Such refined bilinear estimates already appeared in the work of Hani [A bilinear oscillatory integral estimate and bilinear refinements to Strichartz estimates on closed manifolds, Anal. PDE 5 (2012) 339–363] in the context of Schrödinger equations on a compact manifold. Finally, the main theorem yields global existence results for [Formula: see text].

Orthogonal time frequency space (OTFS) modulations are robust to time-varying frequency-selective fading channels. OTFS modulations operate in the delay-Doppler domain whereas two-dimensional (2D) orthogonal frequency division multiplexing (OFDM) modulations operate in the time-frequency domain. For 2D OFDM modulations in time-varying frequency-selective fading channels, we investigate the pilot pattern design problem, which minimizes the mean square error (MSE) of channel estimation. The MSE lower bound (LB) is theoretically derived to provide the design criterion. Based on the criterion, we show that the LB achieving design can be found by exhaustive 2D pilot location search. Exhaustive 2D search has high computational complexity. To reduce the complexity, sufficient conditions on the existence of LB achieving design are provided to decouple the 2D problem into two one-dimensional (1D) problems. For the 1D problem, we propose the LB achieving design when the number of possible pilot locations is divisible by the number of pilots. Simulation results illustrate that our proposed LB achieving design is superior to the random pilot pattern design and the bilinear interpolation channel estimation method.

Global well-posedness for fractional Hartree equation on modulation spaces and Fourier algebra

We consider the Cauchy problem of Navier-Stokes equations in weak Morrey spaces. We first define a class of weak Morrey type spaces Mp,λ*(ℝn) on the basis of Lorentz space Lp,∞ = Lp*(ℝn) (in particular, Mp,0*(ℝn) = Lp,∞, if p > 1), and study some fundamental properties of them; Second, we prove that the heat operator U(t) = etΔ. and Calderon-Zygmund singular integral operators are bounded linear operators on weak Morrey spaces, and establish the bilinear estimate in weak Morrey spaces. Finally, by means of Kato’s method and the contraction mapping principle, we prove that the Cauchy problem of Navier-Stokes equations in weak Morrey spaces Mp,λ*(ℝn) (1 < p ⩽ n) is time-global well-posed, provided that the initial data are sufficiently small. Moreover, we also obtain the existence and uniqueness of the self-similar solution for Navier-Stokes equations in these spaces, because the weak Morrey space Mp,n−p*(ℝn) can admit the singular initial data with a self-similar structure. Hence this paper generalizes Kato’s results.

In this paper, we study the regularity of solutions of nonlinear stochastic partial differential equations (SPDEs) with multiplicative noises in the framework of Hilbert scales. Then we apply our abstract result to several typical nonlinear SPDEs such as stochastic Burgers and Ginzburg-Landau equations on the real line, stochastic 2D Navier-Stokes equations (SNSEs) in the whole space and a stochastic tamed 3D Navier-Stokes equation in the whole space, and obtain the existence of their smooth solutions respectively. In particular, we also get the existence of local smooth solutions for 3D SNSEs.

The purpose of this dissertation is to establish, in a certain infinite dimensional setting, some existence and uniqueness results pertaining to the type of exponential martingale problem previously considered by Strook and Varadhan in the finite dimensional setting. The setting is among those considered by K. Ito in his work on stochastic differential equations in infinite dimensional spaces. Briefly, the martingale problem takes the following form: to show the existence and uniqueness of a measure m on the space S of continuous functions defined on the nonnegative real lien with values in some state space, having the property that exp(f(t,w)) is an m-martingale, f being a function of type, with t denoting a time parameter, and w denoting a generic element of S. The measure m is then called a solution of the martingale problem. In this dissertation, the state spaces under consideration are subspaces of the space of tempered distributions on the real line. The existence of solutions is established under certain continuity conditions by constructing a family of measures satisfying a Prokhorov criterion for relative compactness with respect to weak convergence, and then showing that a weak limit point of that family has the desired martingale property. Secondly, under somewhat stronger assumptions, a uniqueness result for a previously constructed solution is proved in a manner similar to that in which the uniqueness of a related stochastic differential equation is established. Finally, the relation of the martingale problem to certain infinite dimensional stochastic integrals and stochastic equations is pointed out.

The problem of existence and uniqueness of solutions defined on the whole real line and satisfying given initial point data for general abstract linear functional differential equations is considered. The equation is not assumed to be of the delay type. The essence of the method presented here consists in the representation of a solution in the form analogous to the variation of constants formula known for linear ordinary differential equations. It is shown that such an approach can be effectively applied to the problem of existence and uniqueness of solutions satisfying an exponential growth estimate, provided that the deviation of the argument is sufficiently small. The proofs are based on the Banach fixed point principle. Detailed comparison and discussion of the hypotheses ensuring the existence and uniqueness of solutions are presented.

The purpose of this note is to show that solutions of rst and second order Cauchy problems with almost periodic inhomogeneity are almost periodic on the real line whenever the spectrum of the underlying operator is discrete.