Wellposedness and singularities of the water wave equations

A class of water wave problems concerns the dynamics of the free interface separating an inviscid, incompressible and irrotational fluid, under the influence of gravity, from a zero-density region. In this note, we present some recent methods and ideas developed concerning the local and global wellposedness of these problems, the focus is on the structural aspect of the equations.Comment: This is the lecture notes for a short course given at the Newton Institute, Cambridge in August 201

Recent advances on the global regularity for irrotational water waves

We review recent progress on the long-time regularity of solutions of the Cauchy problem for the water waves equations, in two and three dimensions. We begin by introducing the free boundary Euler equations and discussing the local existence of solutions using the paradifferential approach, as in [7, 1, 2]. We then describe in a unified framework, using the Eulerian formulation, global existence results for three dimensional and two dimensional gravity waves, see [70, 146, 145, 87, 5, 6, 79, 80, 136], and our joint result with Deng and Pausader [60] on global regularity for the 3D gravity-capillary model. We conclude this review with a short discussion about the formation of singularities, and give a few additional references to other interesting topics in the theory.Comment: 28 pages, 5 figures, 154 references. To appear in Philos. Trans. Roy. Soc. A. arXiv admin note: text overlap with arXiv:1601.0568

On the global behavior of weak null quasilinear wave equations

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. The prototype for the class of equations considered is $-\partial_t^2 u + (1+u) \Delta u = 0$. Global solutions for such equations have been constructed by Lindblad \cite{Lindblad1,Lindblad2} and Alinhac \cite{Alinhac1}. Our main results are the derivation of a precise asymptotic system with good error bounds, and a detailed description of the behavior of solutions close to the light cone, including the blow-up at infinity.Comment: 48 page

Global analysis of quasilinear wave equations on asymptotically Kerr-de Sitter spaces

We extend the semilinear framework developed by the two authors and the non-trapping quasilinear theory developed by the first author to solve quasilinear wave equations with normally hyperbolic trapping. The most well-known example that fits into our general framework is wave-type equations on Kerr-de Sitter space. The key advance is an adaptation of the Nash-Moser iteration to our framework.Comment: 55 pages, 5 figures. v2 is the published version, with an extended introduction, additional figures, and many minor corrections throughou

Towards the KPP-Problem and log t-Front shift for Higher-Order Nonlinear PDEs II. Quasilinear Bi- and Tri-Harmonic Equations

It is shows that some aspects of classic KPP-problem (1937) can be extended to some fourth and sixth-order quasilinear parabolic equations.Comment: 26 pages, 15 figure

Uniform Bound of the Highest Energy for the 3D Incompressible Elastodynamics

This article concerns the time growth of Sobolev norms of classical solutions to the 3D incompressible isotropic elastodynamics with small initial displacements.Comment: arXiv admin note: text overlap with arXiv:1212.639

The class of second order quasilinear equations: models, solutions and background of classification

The paper is concerned with the unsteady solutions to the model of mutually penetrating continua and quasilinear hyperbolic modification of the Burgers equation (QHMB). The studies were focused on the peculiar solutions of models in question. On the base of these models and their solutions, the ideas of second order quasilinear models classification were developed.Comment: 11 pages, 6 figure

Global Existence for Systems of Nonlinear Wave Equations in 3D with Multiple Speeds

Global smooth solutions to the initial value problem for systems of nonlinear wave equations with multiple propagation speeds will be constructed in the case of small initial data and nonlinearities satisfying the null condition

Formation of finite-time singularities for nonlinear elastodynamics with small initial disturbances

This article concerns the formation of finite-time singularities in solutions to quasilinear hyperbolic systems with small initial data. By constructing a special test function, we first present a simpler proof of the main result in Sideris' "Formation of singularities in three-dimensional compressible fluids": the global classical solution is non-existent for compressible Euler equation even for some small initial data. Then we apply this approach to nonlinear elastodynamics and magnetohydrodynamics, showing that the classical solutions to these equations can still blow up in finite time even if the initial data is small enough

On the quasilinear wave equations in time dependent inhomogeneous media

We consider the problem of small data global existence for quasilinear wave equations with null condition on a class of Lorentzian manifolds $(\mathbb{R}^{3+1}, g)$ with \textbf{time dependent} inhomogeneous metric. We show that sufficiently small data give rise to a unique global solution for metric which is merely $C^1$ close to the Minkowski metric inside some large cylinder $\{\left.(t, x)\right||x|\leq R\}$ and approaches the Minkowski metric weakly as $|x|\rightarrow \infty$. Based on this result, we give weak but sufficient conditions on a given large solution of quasilinear wave equations such that the solution is globally stable under perturbations of initial data.Comment: 46page