Global classical solutions of nonlinear wave equations

Brenner, Philip; Wahl, Wolf von

doi:10.1007/bf01258907

Cited by 103 publications

(58 citation statements)

References 10 publications

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“…In practice, the typical nonlinearity is the pure power case, i.e. f (u) = λu p+1 with p ≥ 0 and λ ∈ R [9,10,[17][18][19][20][31][32][33][34]40,42,44,47]. In fact, the above KG equation is also known as the relativistic version of the Schrödinger equation under proper non-dimensionalization [31][32][33] and it is used to describe the motion of a spinless particle (see, e.g.…”

Section: Introductionmentioning

confidence: 99%

Analysis and comparison of numerical methods for the Klein–Gordon equation in the nonrelativistic limit regime

2011

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We analyze rigourously error estimates and compare numerically temporal/ spatial resolution of various numerical methods for solving the Klein-Gordon (KG) equation in the nonrelativistic limit regime, involving a small parameter 0 < ε 1 which is inversely proportional to the speed of light. In this regime, the solution is highly oscillating in time, i.e. there are propagating waves with wavelength of O(ε 2 ) and O(1) in time and space, respectively. We begin with four frequently used finite difference time domain (FDTD) methods and obtain their rigorous error estimates in the nonrelativistic limit regime by paying particularly attention to how error bounds depend explicitly on mesh size h and time step τ as well as the small parameter ε. Based on the error bounds, in order to compute 'correct' solutions when 0 < ε 1, the four FDTD methods share the same ε-scalability: τ = O(ε 3 ). Then we propose new numerical methods by using either Fourier pseudospectral or finite difference approximation for spatial derivatives combined with the Gautschi-type exponential integrator for temporal derivatives to discretize the KG equation. The new methods are unconditionally stable and their ε-scalability is improved to τ = O(1) and τ = O(ε 2 ) for linear and nonlinear KG equations, respectively, when 0 < ε 1. Numerical results are reported to support our error estimates.

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Section: Introductionmentioning

confidence: 99%

Analysis and comparison of numerical methods for the Klein–Gordon equation in the nonrelativistic limit regime

2011

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“…They obtained the result by using estimates of the nonlinearity in fractional order Besov spaces developed by P. Brenner and W. Wahl [5], the nonlinear interpolation theorem obtained by W. Wahl [19][20][21], and the inequality of H. Brezis and T. Gallouët [6] (see also H. Brezis and S. Wainger [7]). …”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

The Cauchy problem for the coupled Klein–Gordon–Schrödinger system

Miao

Zhu

2006

Ann. Polon. Math.

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Abstract. We consider the Cauchy problem for a generalized Klein-Gordon-Schrö-dinger system with Yukawa coupling. We prove the existence of global weak solutions by the compactness method and, through a special choice of the admissible pairs to match two types of equations, we prove the uniqueness of those solutions by an approach similar to the method presented by J. Ginibre and G. Velo for the pure Klein-Gordon equation or pure Schrödinger equation. Though it is very simple in form, the method has an unnatural restriction on the power of interactions. In the last part of this paper, we use special admissible pairs and Strichartz estimates to remove the restriction, thereby generalizing previous results and obtaining the well-posedness of the system.

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“…Next, we shall prove an (H s p Y H s H p H )-version of the following well-known H s -existence result (for a proof see [5] and [9]):…”

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On wave equations with supercritical nonlinearities

Brenner

Kumlin

2000

Arch. Math.

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We prove that the solution operators e t 0Y y for the nonlinear wave equations with supercritical nonlinearities are not Lipschitz mappings from a subset of the finite-energy space H 1 L &1 Â L 2 to H s q H for t j 0, and 0 % s % 1Y n 1a1a2 À 1aq H 1. This is in contrast to the subcritical case, where the corresponding operators are Lipschitz mappings ([3], [6]). Here e t 0Y y uÁY t, where u is a solution ofwhere n^4Y m^0 and & b & Ã n 2an À 2 in the supercritical case.

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Global classical solutions of nonlinear wave equations

Cited by 103 publications

References 10 publications

Analysis and comparison of numerical methods for the Klein–Gordon equation in the nonrelativistic limit regime

Analysis and comparison of numerical methods for the Klein–Gordon equation in the nonrelativistic limit regime

The Cauchy problem for the coupled Klein–Gordon–Schrödinger system

On wave equations with supercritical nonlinearities

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