scholarly journals Long time approximations for solutions of wave equations via standing waves from quasimodes

2008 ◽  
Vol 90 (4) ◽  
pp. 387-411 ◽  
Author(s):  
Eugenia Pérez
Keyword(s):  
Wave Equations ◽  
Standing Waves ◽  
Solutions Of Wave Equations ◽  
Long Time

scholarly journals Quasi-stability and continuity of attractors for nonlinear system of wave equations

2021 ◽  
Vol 8 (1) ◽  
pp. 27-45
Author(s):  
M. M. Freitas ◽  
M. J. Dos Santos ◽  
A. J. A. Ramos ◽  
M. S. Vinhote ◽  
M. L. Santos
Keyword(s):  
Wave Equations ◽  
Coupled System ◽  
Global Attractors ◽  
Time Behavior ◽  
Exponential Attractors ◽  
Small Perturbations ◽  
Recent Approach ◽  
Nonlinear Coupled System ◽  
Long Time ◽  
External Forces

Abstract In this paper, we study the long-time behavior of a nonlinear coupled system of wave equations with damping terms and subjected to small perturbations of autonomous external forces. Using the recent approach by Chueshov and Lasiecka in [21], we prove that this dynamical system is quasi-stable by establishing a quasistability estimate, as consequence, the existence of global and exponential attractors is proved. Finally, we investigate the upper and lower semicontinuity of global attractors under autonomous perturbations.

scholarly journals Global small data solutions for semilinear waves with two dissipative terms

2021 ◽  
Author(s):  
Wenhui Chen ◽  
Marcello D’Abbicco ◽  
Giovanni Girardi
Keyword(s):  
Wave Equations ◽  
Space Dimension ◽  
Full Range ◽  
Small Data ◽  
Decay Estimates ◽  
Time Decay ◽  
Nonexistence Result ◽  
Derivative Type ◽  
Long Time ◽  
Dissipative Terms

AbstractIn this work, we prove the existence of global (in time) small data solutions for wave equations with two dissipative terms and with power nonlinearity $$|u|^p$$ | u | p or nonlinearity of derivative type $$|u_t|^p$$ | u t | p , in any space dimension $$n\geqslant 1$$ n ⩾ 1 , for supercritical powers $$p>{\bar{p}}$$ p > p ¯ . The presence of two dissipative terms strongly influences the nature of the problem, allowing us to derive $$L^r-L^q$$ L r - L q long time decay estimates for the solution in the full range $$1\leqslant r\leqslant q\leqslant \infty $$ 1 ⩽ r ⩽ q ⩽ ∞ . The optimality of the critical exponents is guaranteed by a nonexistence result for subcritical powers $$p<{\bar{p}}$$ p < p ¯ .

Path-integral solutions of wave equations with dissipation

1989 ◽  
Vol 62 (19) ◽  
pp. 2201-2204 ◽  
Author(s):  
Cecile DeWitt-Morette ◽  
See Kit Foong
Keyword(s):  
Path Integral ◽  
Wave Equations ◽  
Solutions Of Wave Equations ◽  
Integral Solutions

scholarly journals A conservative fully-discrete numerical method for the regularised shallow water wave equations

2021 ◽  
Author(s):  
Dimitrios Mitsotakis ◽  
Hendrik Ranocha ◽  
David I Ketcheson ◽  
Endre Süli
Keyword(s):  
Finite Element ◽  
Shallow Water ◽  
Numerical Method ◽  
Solitary Waves ◽  
Water Waves ◽  
Wave Equations ◽  
Mixed Finite Element ◽  
Boussinesq System ◽  
Fully Discrete ◽  
Long Time

The paper proposes a new, conservative fully-discrete scheme for the numerical solution of the regularised shallow water Boussinesq system of equations in the cases of periodic and reflective boundary conditions. The particular system is one of a class of equations derived recently and can be used in practical simulations to describe the propagation of weakly nonlinear and weakly dispersive long water waves, such as tsunamis. Studies of small-amplitude long waves usually require long-time simulations in order to investigate scenarios such as the overtaking collision of two solitary waves or the propagation of transoceanic tsunamis. For long-time simulations of non-dissipative waves such as solitary waves, the preservation of the total energy by the numerical method can be crucial in the quality of the approximation. The new conservative fully-discrete method consists of a Galerkin finite element method for spatial semidiscretisation and an explicit relaxation Runge--Kutta scheme for integration in time. The Galerkin method is expressed and implemented in the framework of mixed finite element methods. The paper provides an extended experimental study of the accuracy and convergence properties of the new numerical method. The experiments reveal a new convergence pattern compared to standard Galerkin methods.

scholarly journals Small-data shock formation in solutions to 3D quasilinear wave equations: An overview

2016 ◽  
Vol 13 (01) ◽  
pp. 1-105 ◽  
Author(s):  
Gustav Holzegel ◽  
Sergiu Klainerman ◽  
Jared Speck ◽  
Willie Wai-Yeung Wong
Keyword(s):  
Wave Equations ◽  
Initial Conditions ◽  
Large Body ◽  
Small Data ◽  
Shock Formation ◽  
Time Behavior ◽  
Long Time ◽  
Remarkable Result ◽  
Quasilinear Wave Equations ◽  
Three Space

In his 2007 monograph, Christodoulou proved a remarkable result giving a detailed description of shock formation, for small [Formula: see text]-initial conditions (with [Formula: see text] sufficiently large), in solutions to the relativistic Euler equations in three space dimensions. His work provided a significant advancement over a large body of prior work concerning the long-time behavior of solutions to higher-dimensional quasilinear wave equations, initiated by John in the mid 1970’s and continued by Klainerman, Sideris, Hörmander, Lindblad, Alinhac, and others. Our goal in this paper is to give an overview of his result, outline its main new ideas, and place it in the context of the above mentioned earlier work. We also introduce the recent work of Speck, which extends Christodoulou’s result to show that for two important classes of quasilinear wave equations in three space dimensions, small-data shock formation occurs precisely when the quadratic nonlinear terms fail to satisfy the classic null condition.

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