Related papers: The Cauchy Problem for Wave Maps on a Curved Backg…

In previous work, we developed a topological framework for solving Riemann initial-value problems for a system of conservation laws. Its core is a differentiable manifold, called the wave manifold, with points representing shock and…

We consider the inverse problem for the wave equation which consists of determining an unknown space-dependent force function acting on a vibrating structure from Cauchy boundary data. Since only boundary data are used as measurements, the…

In an influential 1964 article, P. Lax studied $2 \times 2$ genuinely nonlinear strictly hyperbolic PDE systems (in one spatial dimension). Using the method of Riemann invariants, he showed that a large set of smooth initial data lead to…

We prove local existence and uniqueness of the Cauchy problem for a large class of tensorial second order linear hyperbolic partial differential equations with coefficients of low regularity in a suitable class of generalized functions.

In recent years, the global existence of classical solutions to the Cauchy problem for 2D incompressible viscous MHD equations without magnetic diffusion has been proved in \cite{Ren,TZhang}, under the assumption that initial data is close…

This work studies solutions of the scalar wave equation \[\Box_g\phi=0\] on a fixed subextremal Reissner-Nordstr\"{o}m spacetime with non-vanishing charge $q$ and mass $M$. In a recent paper, Luk and Oh established that generic smooth and…

The paper provides a direct proof the uniqueness of solutions to the Camassa-Holm equation, based on characteristics. Given a conservative solution $u=u(t,x)$, an equation is introduced which singles out a unique characteristic curve…

We consider the following half wave Schr{\"o}dinger equation,$$\left(i \partial_{t}+\partial_{x }^2-\left|D_{y}\right|\right) U=|U|^{2} U$$on the plane $\mathbb{R}_x \times \mathbb{R}_y$. We prove the existence of modified wave operators…

In this paper we study the global existence of small data solutions to the Cauchy problem for the semilinear wave equation with scale-invariant damping. We obtain estimates for the solution and its energy with the same decay rate of the…

In this paper we study the existence of global-in-time energy solutions to the Cauchy problem for the Euler-Poisson-Darboux equation, with a power nonlinearity: $$u_{tt}-u_{xx} + \frac\mu{t}\,u_t = |u|^p \,, \quad t>t_0, \…

In this paper, we establish the existence and instability of standing wave for a system of nonlinear Schr\"{o}dinger equations arising in the two-wave model with quadratic interaction in higher space dimensions under mass resonance…

Our goal is to review the known theory on the one-dimensional obstacle problem for the wave equation, and to discuss some extensions. We introduce the setting established by Schatzman within which existence and uniqueness of solutions can…

In this article, we give a completely constructive proof of the observability/controllability of the wave equation on a compact manifold under optimal geometric conditions. This contrasts with the original proof of Bardos-Lebeau-Rauch,…

We consider the Cauchy problem with smooth and compactly supported initial data for the wave equation in a general class of spherically symmetric geometries which are globally smooth and asymptotically flat. Under certain mild conditions on…

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…

We prove observability estimates for oscillatory Cauchy data modulo a small kernel for $n$-dimensional wave equations with space and time dependent $C^2$ and $C^{1,1}$ coefficients using Gaussian beams. We assume the domains and…

We consider 1-equivariant wave maps from 1+2 dimensions to the 2-sphere. For wave maps of topological degree zero we prove global existence and scattering for energies below twice the energy of harmonic map, Q, given by stereographic…

Here we investigate the Cauchy problem for the inhomogeneous Navier-Stokes equations in the whole $n$-dimensional space. Under some smallness assumption on the data, we show the existence of global-in-time unique solutions in a critical…

In the paper [S. Alinhac, The null condition for quasilinear wave equations in two space dimensions I, Invent. Math. 145 (2001), no. 3, 597-618], S. Alinhac established the global existence of small data smooth solutions to the Cauchy…

We establish the existence of a conservative weak solution to the Cauchy problem for the nonlinear variational wave equation $u_{tt} - c(u)(c(u)u_x)_x=0$, for initial data of finite energy. Here $c(\cdot)$ is any smooth function with…