Global smooth solutions of 3-D null-form wave equations in exterior domains with Neumann boundary conditions

The paper is devoted to investigating long time behavior of smooth small data solutions to 3-D quasilinear wave equations outside of compact convex obstacles with Neumann boundary conditions. Concretely speaking, when the surface of a 3-D compact convex obstacle is smooth and the quasilinear wave equation fulfills the null condition, we prove that the smooth small data solution exists globally provided that the Neumann boundary condition on the exterior domain is given. One of the main ingredients in the current paper is the establishment of local energy decay estimates of the solution itself. As an application of the main result, the global stability to 3-D static compressible Chaplygin gases in exterior domain is shown under the initial irrotational perturbation with small amplitude.     

Global existence of null-form wave equations on small asymptotically Euclidean manifolds

We prove the global existence of the small solutions to the Cauchy problem for quasilinear wave equations satisfying the null condition on (R³,g)$({\mathbb{R}}^3,\mathfrak{g})$, where the metric g$\mathfrak{g}$ is a small perturbation of the flat metric and approaches the Euclidean metric like (1+|x|²)^−ρ/2${(1+{|x|}^2)}^{-\rho /2}$ with ρ>1$\rho >1$. Global and almost global existence for systems without the null condition are also discussed for certain small time-dependent perturbations of the flat metric in Appendix?A.     

Global solutions for nonlinear wave equations with localized dissipations in exterior domains

The Cauchy problem for nonlinear wave equations with localized dissipation is considered in exterior domains outside of compact obstacles in three spatial dimensions. Under the null conditions for the quadratic nonlinear terms, the global solutions are shown for sufficiently small data. The solutions which have different propagation speeds are considered.     

Global existence results for Oldroyd-B fluids in exterior domains

In this paper we consider the set of equations describing Oldroyd-B fluids in exterior domains. It is shown that these equations admit a unique, global solution defined in a certain function space provided the initial data and the coupling constant are small enough.     

$L^p$ estimates for the linear wave equation and global existence for semilinear wave equations in exterior domains

We consider the initial-boundary value problem for the semilinear wave equation where is an exterior domain in , is a dissipative term which is effective only near the 'critical part' of the boundary. We first give some estimates for the linear equation by combining the results of the local energy decay and estimates for the Cauchy problem in the whole space. Next, on the basis of these estimates we prove global existence of small amplitude solutions for semilinear equations when is odd dimensional domain . When our result is applied if . We note that no geometrical condition on the boundary is imposed. Received April 13, 2000 / Revised July 6, 2000 / Published online February 5, 2001     

Global existence for defocusing cubic NLS and Gross–Pitaevskii equations in three dimensional exterior domains

We prove global wellposedness in the energy space of the defocusing cubic nonlinear Schrödinger and Gross–Pitaevskii equations on the exterior of a nontrapping domain in dimension 3. The main ingredient is a Strichartz estimate obtained combining a semi-classical Strichartz estimate [R. Anton, Strichartz inequalities for Lipschitz metrics on manifolds and nonlinear Schrödinger equation on domains, arxiv:math.AP/0512639, Bull. Soc. Math. France, submitted for publication] with a smoothing effect on exterior domains [N. Burq, P. Gérard, N. Tzvetkov, On nonlinear Schrödinger equations in exterior domains, Ann. I.H.P. (2004) 295–318].     

Decay estimates for dissipative wave equations in exterior domains

Consider the initial boundary value problem for the linear dissipative wave equation (□+∂_t)u=0 in an exterior domain . Using the so-called cut-off method together with local energy decay and L² decays in the whole space, we study decay estimates of the solutions. In particular, when N?3, we derive L^p decays with p?1 of the solutions. Next, as an application of the decay estimates for the linear equation, we consider the global solvability problem for the semilinear dissipative wave equations (□+∂_t)u=f(u) with f(u)=|u|^α+1,|u|^αu in an exterior domain.     

Global existence of small radially symmetric solutions to quadratic nonlinear wave equations in an exterior domain

We study the initial boundary value problem for the nonlinear wave equation: (*) $$\left\{ \begin{gathered} \partial _t^2 u - (\partial _r^2 + \frac{{n - 1}}{r}\partial _r )u = F(\partial _t u,\partial _t^2 u),t \in \mathbb{R}^ + ,R< r< \infty , \hfill \\ u(0,r) = \in _0 u_0 (r),\partial _t u(0,r) = \in _0 u_1 (r),R< r< \infty , \hfill \\ u(t,R) = 0,t \in \mathbb{R}^ + , \hfill \\ \end{gathered} \right.$$ wheren=4,5,u ₀,u ₁ are real valued functions and ∈₀ is a sufficiently small positive constant. In this paper we shall show small solutions to (*) exist globally in time under the condition that the nonlinear termF:?²→? is quadratic with respect to ? _t u and ? _t ² u.     

General quasilinear wave equations with localized dissipation in exterior domains

We show that the method of commuting vector fields can be applied to quasilinear wave equations with localized dissipations in general exterior domains. This allows us to show long time existence for general quasilinear wave equations with quadratic nonlinearities. Moreover, by assuming that the dissipation is effective in a certain neighborhood of the boundary, we need not place any assumption on the geometry of the domain.     

Global existence of weak solutions for two-dimensional semilinear wave equations with strong damping in an exterior domain

We consider two-dimensional mixed problems in an exterior domain for a semilinear strongly damped wave equation with a power-type nonlinearity |u|^p${\left|u\right|}^p$. If the initial data have a small weighted energy, we shall derive a global existence and energy decay results in the case when the power p$p$ of the nonlinear term satisfies p>6$p>6$.     

On uniqueness and existence of viscosity solutions to Hessian equations in exterior domains

In this paper, we obtain the uniqueness and existence of viscosity solutions with prescribed asymptotic behavior at infinity to Hessian equations in exterior domains.     

Global solvability for the Kirchhoff equations in exterior domains of dimension three

We consider the initial (boundary) value problem for the Kirchhoff equations in exterior domains or in the whole space of dimension three, and show that these problems admit time-global solutions, provided the norms of the initial data in the usual Sobolev spaces of appropriate order are sufficiently small. We obtain uniform estimates of the L¹(R) norms with respect to time variable at each point in the domain, of solutions of initial (boundary) value problem for the linear wave equations. We then show that the estimates above yield the unique global solvability for the Kirchhoff equations.     

Global existence of strong solutions to micropolar equations in cylindrical domains

The micropolar equations are a useful generalization of the classical Navier–Stokes model for fluids with microstructure. We prove the existence of global and strong solutions to these equations in cylindrical domains in . We do not impose any restrictions on the magnitude of the initial and external data, but we require that they cannot change in the x₃‐direction too fast. Copyright © 2014 John Wiley & Sons, Ltd.     

Global existence of solutions for semilinear damped wave equation in 2-D exterior domain

We consider a mixed problem of a damped wave equation u_tt−Δu+u_t=|u|^p in the two dimensional exterior domain case. Small global in time solutions can be constructed in the case when the power p on the nonlinear term |u|^p satisfies p∗=2<p<+∞. For this purpose we shall deal with a radially symmetric solution in the exterior domain. A new device developed in Ikehata-Matsuyama (Sci. Math. Japon. 55 (2002) 33) plays an effective role.     

Global solvability for the Kirchhoff equations in exterior domains of dimension larger than three

We consider the unique global solvability of initial (boundary) value problem for the Kirchhoff equations in exterior domains or in the whole Euclidean space for dimension larger than three. The following sufficient condition is known: initial data is sufficiently small in some weighted Sobolev spaces for the whole space case; the generalized Fourier transform of the initial data is sufficiently small in some weighted Sobolev spaces for the exterior domain case. The purpose of this paper is to give sufficient conditions on the usual Sobolev norm of the initial data, by showing that the global solvability for this equation follows from a time decay estimate of the solution of the linear wave equation. Copyright © 2004 John Wiley & Sons, Ltd.