Finite energy well-posedness for nonlinear Schrödinger equations with non-vanishing conditions at infinity

The authors consider the well-posedness in energy space of the critical non-linear system of wave equations with Hamiltonian structure $$ \left\{ \begin{gathered} u_{tt} - \Delta u = - F_1 (|u|^2 ,|v|^2 )u, \hfill \\ v_{tt} - \Delta v = - F_2 (|u|^2 ,|v|^2 )v, \hfill \\ \end{gathered} \right. $$ where there exists a function F(λ, µ) such that $$ \frac{{\partial F(\lambda ,\mu )}} {{\partial \lambda }} = F_1 (\lambda ,\mu ),\frac{{\partial F(\lambda ,\mu )}} {{\partial \mu }} = F_2 (\lambda ,\mu ). $$ By showing that the energy and dilation identities hold for weak solution under some assumptions on the non-linearities, we prove the global well-posedness in energy space by a similar argument to that for global regularity as shown in “Shatah and Struwe’s paper, Ann. of Math. 138, 503–518 (1993)”.

The Euler-Korteweg equations are a modification of the Euler equations that takes into account capillary effects. In the general case they form a quasi-linear system that can be recast as a degenerate Schr\"odinger type equation. Local well-posedness (in subcritical Sobolev spaces) was obtained by Benzoni-Danchin-Descombes in any space dimension, however, except in some special case (semi-linear with particular pressure) no global well-posedness is known. We prove here that under a natural stability condition on the pressure, global well-posedness holds in dimension $d\geq 3$ for small irrotational initial data. The proof is based on a modified energy estimate, standard dispersive properties if $d\geq 5$, and a careful study of the nonlinear structure of the quadratic terms in dimension $3$ and $4$ involving the theory of space time resonance.

In this paper, we prove the local well-posedness of 3-D axi-symmetric Navier–Stokes system with initial data in the critical Lebesgue spaces. We also obtain the global well-posedness result with small initial data. Furthermore, with the initial swirl component of the velocity being sufficiently small in the almost critical spaces, we can still prove the global well-posedness of the system.

New low regularity well-posedness results for the generalized Benjamin-Ono equations with quartic or higher nonlinearity and periodic boundary conditions are shown. We use the short-time Fourier transform restriction method and modified energies to overcome the derivative loss. Previously, Molinet–Ribaud established local well-posedness in $H^1(\mathbb{T},\mathbb{R})$ via gauge transforms. We show local existence and a priori estimates in $H^s(\mathbb{T},\mathbb{R}),$ $s>1/2$, and local well-posedness in $H^s(\mathbb{T},\mathbb{R})$, $s\ge3/4$ without using gauge transforms. In case of quartic nonlinearity we prove global existence of solutions conditional upon small initial data.

This paper is dedicated to the study of both viscous compressible barotropic fluids and Navier-Stokes equations with dependent density, when the viscosity coefficients are variable, in dimension $d\geq2$. We aim at proving the local and global well-posedness for respectively large and {small initial data having critical Besov regularity, and more precisely we are interested in extending the class of initial-data velocity when we consider the shallow-water system, improving the results in [17] and [15]. Our result relies on the fact that the velocity $u$ can be written as the sum of the solution $u_{L}$ of the associated linear system and a remainder velocity term $\bar{u}$; then in the specific case of the shallow-water system the remainder term $\bar{u}$ is more regular than $u_{L}$ by taking into account the regularizing effects induced on the bilinear convection term. In particular we are able to deal with initial velocity in $\dot{H}^{\N-1}$ as Fujita and Kato for the incompressible Navier-Stokes equations (see [12]) with an additional condition of type $u_{0}\in B^{-1}_{\infty,1}$. We would like to point out that this type of result is of particular interest when we want to deal with the problem of the convergence of the solution of compressible system to the incompressible system when the Mach number goes to $0$.

The objective of this paper is two-fold: firstly, we develop a local and global (in time) wellposedness theory for a system describing the motion of two fluids with different densities under capillary-gravity waves in a deep water flow (namely, a Schrdinger-Benjamin-Ono system) for lowregularity initial data in both periodic and continuous cases; secondly, a family of new periodic traveling waves for the Schrdinger-Benjamin-Ono system is given: by fixing a minimal period we obtain, via the implicit function theorem, a smooth branch of periodic solutions bifurcating a Jacobian elliptic function called dnoidal, and, moreover, we prove that all these periodic traveling waves are nonlinearly stable by perturbations with the same wavelength.

We prove the global wellposedness for the nonlinear Schrodinger equation for the magnetic Laplacian corresponding to the constant magnetic field. The magnetic Laplacian in this case can be identified with the twisted Laplacian on \(\mathbb {C}^n\) and our approach is via spectral theory. The local existence is established in certain first order Sobolev space, naturally associated to the twisted Laplacian, the energy space. The global well posedness is then deduced via a blowup analysis, with the help of the mass and the energy conservation established here. The general class of nonlinearities considered includes in particular, the power type nonlinearities, focussing as well as defocussing cases. The results in this paper extends the local wellposedness established in Ratnakumar and Sohani (J Funct Anal 265(1):1–27, (2013).

An initial-boundary-value problem of 2D nonlinear Schrödinger equation in a strip

• Small data theory: (f, g) are small, and F is treated as a perturbation. The main questions are local and global well-posedness, the existence of conserved quantities (energy), their relation to the basic symmetries of the equation (especially the dilation symmetry). The choice of spaces in which to solve can be very challenging, and algebraic properties of F may be essential in order to obtain well-posedness. Specifically, nonlinearities exhibiting a null-form structure appear in geometric/physical equations. The dynamics of small data are typically simple, with the associated wave becoming asymptotically free (this is referred to as “scattering”). • Large data theory: For local-in-time existence, energy subcritical problems are easier since the time of existence depends only on the energy norm of the data, so one can then time-step to obtain global existence. The problem with this approach lies with the absence of information on the long-term dynamics such as scattering. Finite-time breakdown (blowup) of solutions may occur as well, and then the problem of classification of possible blowup dynamics poses itself. In general, large data theory is concerned with the classification of all possible types of dynamics that solutions may exhibit at large energies. The structure here is much richer, with the underlying

We prove existence of global solutions to a semilinear massless Dirac system with small initial data. We study solutions in generalised Sobolev spaces suggested by S. Klainerman. Our approach is based on using conservation law of charge together with Sobolev type weighted estimates for the spinor field. Our result seems to be sharp in a view of blowing-up results obtained by F. John (see [7]). We also study decay properties of the spinor field. With similar methods we prove global existence for a nonlinear wave equation in three space dimension. The same equation was studied by T. Sideris [14] and H. Takamura [15]. They proved global existence for spherically symmetrical initial data. In this work we remove this condition on the initial data.

This paper is concerned with adaptive stabilization of an undamped non-linear string with disturbed outputs by boundary feedback control. The adaptive controller is constructed by the concept of high-gain adaptive feedback and the estimation mechanism for the unknown parameters of the measurement noise. The local existence and uniqueness of the solution of the closed-loop system are justified. Moreover, global existence and boundedness of the solution are shown for smooth and small initial data. The stability of the closed-loop system is proved such that the convergence of the system state to zero and the convergence of the estimated parameter to the unknown parameter are guaranteed for the smooth and small initial data.

ABSTRACTThis paper is dedicated to study the Cauchy problem of the nonlinear Schrödinger equation without gauge invariance where and . When , we first prove local well-posedness of the equation in . If in addition, , we prove the global well-posedness with small initial data in . Under a suitable condition on the initial data and , we prove that the -norm of the solution would blow up in finite time although the initial data are arbitrarily small. Meanwhile, we also give a large initial data blow-up result when in . Finally, we show the non-existence of local weak solution for some -data with when .

We investigate the initial value problem for a semi-linear Schrodinger equation with exponential-growth nonlinearity. We establish global well-posedness and scattering in the energy space.

We investigate the initial value problem for a semilinear damped Schrodinger equation with exponential growth nonlinearity in two space dimensions. We obtain global well-posedness in the energy space. Copyright © 2013 John Wiley & Sons, Ltd.

The Schroedinger equation with the nonlinearity concentrated at a single point proves to be an interesting and important model for the analysis of long-time behavior of solutions, such as the asymptotic stability of solitary waves and properties of weak global attractors. In this note, we prove global well-posedness of this system in the energy space $H\sp 1$.