Decay of solutions of the wave equation outside nontrapping obstacles

Morawetz, Cathleen S.; Ralston, James; Strauss, Walter A.

doi:10.1002/cpa.3160300405

Cited by 185 publications

(122 citation statements)

References 22 publications

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“…This follows from the nontrapping assumption and certain resolvent estimates (see Burq [2], Theorem 3). Of course (2.13) is essentially equivalent to the classical local decay estimates in [16] and [18].…”

Section: Main Estimatesmentioning

confidence: 99%

Concerning the Strauss Conjecture and Almost Global Existence for Nonlinear Dirichlet-Wave Equations in 4-Dimensions

Du¹,

Metcalfe²,

Sogge³

et al. 2008

Communications in Partial Differential Equations

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Section: Main Estimatesmentioning

confidence: 99%

Concerning the Strauss Conjecture and Almost Global Existence for Nonlinear Dirichlet-Wave Equations in 4-Dimensions

Du¹,

Metcalfe²,

Sogge³

et al. 2008

Communications in Partial Differential Equations

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“…The case where the matrices A and B are linearized around zero velocities corresponds to the wave equation and has been studied extensively; see, e.g., [9]. The equations are In [7] and [8] it is proved, for example, that the energy in a bounded domain decays like / "2 if the wave equation is solved in the exterior of a star-shaped domain for the Neumann and Dirichlet problems.…”

Section: Ax V 'mentioning

confidence: 99%

Steady State Computations for Wave Propagation Problems

Engquist¹,

Gustafsson²

1987

Mathematics of Computation

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Abstract. The behavior of difference approximations of hyperbolic partial differential equations as time t -* oo is studied. The rate of convergence to steady state is analyzed theoretically and expe ¡mentally for the advection equation and the linearized Euler equations. The choice of difference formulas and boundary conditions strongly influences the rate of convergence in practical steady state calculations. In particular it is shown that upwind difference methods and characteristic boundary conditions have very attractive convergence properties.1. Introduction. For computing steady state solutions to problems in fluid mechanics, the time-dependent formulation is often used. There are several mechanisms that drive the solution to a steady state. In this paper we shall concentrate on the dissipation effect due to the boundary conditions, and not to the effect of friction and viscosity. Therefore we shall study hyperbolic partial differential equations where the boundary effects are dominant. The results are also valid for more general classes of differential equations of essentially hyperbolic character, as for example the Navier-Stokes equations for high Reynolds numbers.The purpose of this paper is to analyze the convergence properties to steady state both for the continuous problem and the corresponding discrete approximation. The basis for this study is the behavior of the spectrum of the differential and the difference operators, respectively. Two model problems are considered, the scalar advection problem and the isentropic Euler equation problem. The latter problem is studied in a two-dimensional geometry corresponding to channel flow. It is shown that the choice of boundary conditions radically affects the convergence rate to steady state as time increases. The asymptotic rate may actually change from exponential to algebraic, and for certain sets of boundary conditions there may be no convergence at all.The convergence rate for time-marching procedures has been discussed by others. For example, in [4] Giles investigated the eigenmodes of the solution of the one-dimensional Euler equations under various boundary conditions. Eriksson and Rizzi in [2] studied the convergence rate for a time-marching centered finite-volume

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“…For the spherical case, it dates back to Watson's work on the scattering of electromagnetic wave by the earth [33]. Other notable works include Lax-Phillips [14]- [16], Babich-Grigoreva [3], Filippov-Zayaev [8], Morawetz-Ralston-Strauss [21], Melrose [19], Lebeau [17], Bardos-Lebeau-Rauch [4], Popov [22], Hargé-Lebeau [9], Sjöstrand [24], Sjöstrand-Zworski [26]- [31] and Stefanov [25]. See [20], [34] for surveys on this topic and other related settings.…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%