Localwell-posedness for the nonlinear Schrödinger equation in modulation spaces Ms p;q(Rd)

Remarks on Fourier multipliers and applications to the wave equation

We show some new local smoothing estimates of the fractional Schrödinger equations with initial data in $$\alpha $$ -modulation spaces via decoupling inequalities. Furthermore, our necessary conditions show that the local smoothing estimates are sharp in some cases. As applications, the local smoothing estimates could show some new local well-posedness on modulation spaces of the fourth-order nonlinear Schrödinger equations on the line.

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.

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

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

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

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

Validity of the Cahn–Hilliard approximation for modulations of slightly unstable pattern in the real Ginzburg–Landau equation

We study the Cauchy problem for the generalized Korteweg–de Vries (KdV) and one‐dimensional fourth‐order derivative nonlinear Schrödinger equations, for which the global well‐posedness of solutions with small rough data in certain scaling limit of modulation spaces is shown, which contain some data with infinite L2 norm.

In this paper we show the local and global well-posedness for the Cauchy problem associated with a special class of 1D-Boussinesq systems that emerges in the study of the evolution of long water waves with small amplitude in the presence of surface tension. We also show the existence of solitons (finite energy travelling wave solutions) in the case of wave speed \(0 0\).

This paper is dedicated to the study of a one-dimensional congestion model, consisting of two different phases. In the congested phase, the pressure is free and the dynamics is incompressible, whereas in the non-congested phase, the fluid obeys a pressureless compressible dynamics.We investigate the Cauchy problem for initial data which are small perturbations in the non-congested zone of traveling wave profiles. We prove two different results. First, we show that for arbitrarily large perturbations, the Cauchy problem is locally well-posed in weighted Sobolev spaces. The solution we obtain takes the form (v s ,u s )(t,x-x ˜(t)), where x<x ˜(t) is the congested zone and x>x ˜(t) is the non-congested zone. The set {x=x ˜(t)} is a free boundary, whose evolution is coupled with the one of the solution. Second, we prove that if the initial perturbation is sufficiently small, then the solution is global. This stability result relies on coercivity properties of the linearized operator around a traveling wave, and on the introduction of a new unknown which satisfies better estimates than the original one. In this case, we also prove that traveling waves are asymptotically stable.

We prove the local well-posedness for the Cauchy problem of the Korteweg--de Vries equation in a quasi-periodic function space. The function space contains functions such that $f=f_1+f_2+\cdots+f_N$ where $f_j$ is in the Sobolev space of order $s>-1/2N$ of $2\pi\alpha^{-1}_j$ periodic functions. Note that $f$ is not a periodic function when the ratio of periods $\alpha_i/\alpha_j$ is irrational. The main tool of the proof is the Fourier restriction norm method introduced by Bourgain. We also prove an ill-posedness result in the sense that the flow map (if it exists) is not $C^2$, which is related to the Diophantine problem.

Consideration in this paper is the generalized fractional Camassa–Holm equation. The local well-posedness is established in Besov space $$B^{s_0}_{2,1}$$ with $$s_0=2\nu -\frac{1}{2}$$ for $$\nu >\frac{3}{2} $$ and $$s_0=\frac{5}{2}$$ for $$1<\nu \le \frac{3}{2} $$ . Then, with a given analytic initial data, the analyticity of the solutions in both variables, globally in space and locally in time, is established. Finally, a blow-up criterion is presented.

Nonlinear Schrödinger equation with a point defect

Relevant physical phenomena are described by nonlinear Schrödinger equations with non-vanishing conditions at infinity. This paper investigates the respective 2D and 3D Cauchy problems. Local well-posedness in the (curved) energy space, for energy-subcritical nonlinearities merely satisfying Kato-type assumptions, is proven, providing the analogue of the well-established local H^{1} -theory for solutions vanishing at infinity. The critical nonlinearity will be simply a byproduct of our analysis and the existing literature. Under an assumption that prevents the onset of a Benjamin–Feir type instability, global well-posedness in the energy space is proven for: (a) non-negative Hamiltonians, (b) sign-indefinite Hamiltonians under additional assumptions on the zeros of the nonlinearity, (c) generic nonlinearities and small initial data. The cases (b) and (c) only concern the 3D case.