Related papers: Uniform estimates for 2D quasilinear wave
This paper is devoted to the study of large time bounds for the Sobolev norms of the solutions of the following fractional cubic Schr{\"o}dinger equation on the torus :$$i \partial\_t u = |D|^\alpha u+|u|^2 u, \quad u(0, \cdot)=u\_0,$$where…
We show that a general class of quasilinear wave equations have global solutions for small initial data as we conjectured in an earlier paper.
By means of a direct and constructive method based on the theory of semiglobal $C^2$ solution, the local exact boundary observability is shown for nonautonomous 1-D quasilinear wave equations. The essential difference between nonautonomous…
For 3-D quadratic quasilinear wave equations with or without null conditions in exterior domains, when the compatible initial data and Dirichlet boundary values are given, the global existence or the maximal existence time of small data…
A key feature of $(1+1)$-dimensional nonlinear wave equations is that they admit left or right traveling waves, under appropriate algebraic conditions on the nonlinearities. In this paper, we prove global stability of such traveling wave…
This paper is devoted to the investigation of long-time behaviour of solutions to wave equations with quadratic nonlinearity and cubic Dirac equations with Hartree-type nonlinearity. We consider the nonlinearity here with enough simplicity…
Our goal in this paper is to apply a normal forms method to estimate the Sobolev norms of the solutions of the water waves equation. We construct a paradifferential change of unknown, without derivatives losses, which eliminates the part of…
The classical system of shallow-water (Saint--Venant) equations describes long surface waves in an inviscid incompressible fluid of a variable depth. Although shock waves are expected in this quasilinear hyperbolic system for a wide class…
We consider the Cauchy problem of systems of quasilinear wave equations in 2-dimensional space. We assume that the propagation speeds are distinct and that the nonlinearities contain quadratic and cubic terms of the first and second order…
In this manuscript we first give the explicit variational structure of the nonlinear elastic waves for isotropic, homogeneous, hyperelastic materials in 2-D. Based on this variational structure, we suggest a null condition which is a kind…
We consider quasilinear wave equations in $(1+3)$-dimensions where the nonlinearity $F(u,u',u")$ is permitted to depend on the solution rather than just its derivatives. For scalar equations, if $(\partial_u^2 F)(0,0,0)=0$, almost global…
We consider a system of quasilinear wave equations on the product space $\mathbb{R}^{1+3}\times \mathbb{S}^1$, which we want to see as a toy model for Einstein equations with additional compact dimensions. We show global existence for small…
In this paper, we prove the first asymptotic completeness result for a scalar quasilinear wave equation satisfying the weak null condition. The main tool we use in the study of this equation is the geometric reduced system introduced in…
We prove an extended lifespan result for the full gravity-capillary water waves system with a $2$ dimensional periodic interface: for initial data of sufficiently small size $\varepsilon$, smooth solutions exist up to times of the order of…
We undertake a systematic review of some results concerning local well-posedness of the Cauchy problem for certain systems of nonlinear wave equations, with minimal regularity assumptions on the initial data. Moreover we provide a…
Using suitable modified energies we study higher order Sobolev norms' growth in time for the nonlinear Schr\"odinger equation (NLS) on a generic $2d$ or $3d$ compact manifold. In $2d$ we extend earlier results that dealt only with cubic…
We consider a completely resonant nonlinear Schr\"odinger equation on the $d$-dimensional torus, for any $d\geq 1$, with polynomial nonlinearity of any degree $2p+1$, $p\geq1$, which is gauge and translation invariant. We study the…
This paper concerns the time growth of the highest-order energy of the systems of incompressible isotropic elastodynamics in two space dimensions. The global well-posedness of smooth solutions near equilibrium was first obtained by Lei [31]…
We consider a class of quasilinear wave equations in $3+1$ space-time dimensions that satisfy the "weak null condition" as defined by Lindblad and Rodnianski \cite{LR1}, and study the large time behavior of solutions to the Cauchy problem.…
We prove almost global existence for multiple speed quasilinear wave equations with quadratic nonlinearities in three spatial dimensions. We prove new results both for Minkowski space and also for nonlinear Dirichlet-wave equations outside…