On almost global existence for nonrelativistic wave equations in 3D

1996/03/01 by Sergiu Klainerman, Sergiù Klainerman, Thomas C. Sideris · 197 citations
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Applied mathematics #Computational Fluid Dynamics and Aerodynamics #Energy (signal processing) #Energy method #Hyperelastic material #Invariant (physics) #Mathematical analysis #Mathematical physics #Mathematics #Nonlinear system #Physics #Quadratic growth #Quantum mechanics #Stability and Controllability of Differential Equations #Wave equation

paper · doi:10.1002/(sici)1097-0312(199603)49:3<307::aid-cpa4>3.0.co;2-h

published in Communications on Pure and Applied Mathematics 49(3), 307-321 (Wiley)

openalex publication_date 1996/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/25

Abstract

Almost global solutions are constructed to three-dimensional, quadratically nonlinear wave equations. The proof relies on generalized energy estimates and a new decay estimate. The method applies to equations that are only classically invariant, such as the nonlinear system of hyperelasticity. © 1996 John Wiley & Sons, Inc.

Citations

• Uniform decay estimates and the lorentz invariance of the classical wave equation

Cited by

• Global Well‐Posedness of Incompressible Elastodynamics in Two Dimensions
• On the energy method for global well–posedness to 2D simplified inertial Ericksen–Leslie's system near equilibrium
• Global Well-posedness for the Three Dimensional Simplified Inertial Ericksen-Leslie Systems Near Equilibrium
• Almost global well-posedness of Ericksen-Leslie's hyperbolic liquid crystal model for small data in two dimensions
• Almost global existence for quasilinear wave equations in waveguides with Neumann boundary conditions
• Global existence and stability of near-affine solutions of compressible elastodynamics
• Global smooth solutions of 3-D null-form wave equations in exterior domains with Neumann boundary conditions
• Global existence for a system of multiple-speed wave equations violating the null condition
• Global existence for null-form wave equations with data in a Sobolev space of lower regularity and weight
• Global existence of quasilinear, nonrelativistic wave equations satisfying the null condition
• Global Well-posedness of Incompressible Elastodynamics in Two Dimensions
• Global stability of solutions to nonlinear wave equations
• Global existence of null-form wave equations in exterior domains
• Low regularity ill-posedness for elastic waves driven by shock formation
• A Morawetz type energy estimate for wave equation in ℝ² and application to elastic waves
• Global multidimensional shock waves for 2-D and 3-D unsteady potential flow equations
• Decay estimates of wave equations in un-isotropic media
• Space-Time resonances and the null condition for (first order) systems of wave equations
• Global solutions of nonlinear wave equations in time dependent inhomogeneous media
• Global Existence with Small Initial Data for Three-Dimensional Incompressible Isotropic Viscoelastic Materials
• Global low regularity solutions for nonlinear elastic waves
• Almost global existence for some semilinear wave equations
• Remarks on Nonlinear Elastic Waves in the Radial Symmetry in 2-D
• Estimates for the Dirichlet-wave equation and applications to nonlinear wave equations
• Almost global existence for quasilinear wave equations in three space dimensions
• Global solutions of nonlinear wave equations with large energy
• The global existence theorem for quasi-linear wave equations with\n multiple speeds, II
• A vector field method for non-trapping, radiating spacetimes
• Global Solutions for Incompressible Viscoelastic Fluids

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• Almost global existence for some semilinear wave equations
• Almost global existence for quasilinear wave equations in three space dimensions
• Global Existence for Systems of Nonlinear Wave Equations in 3D with Multiple Speeds
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