Solutions of quasi-linear wave equations with small initial data

1989 ◽  
pp. 155-184 ◽  
Author(s):  
Fritz John
Keyword(s):  
Initial Data ◽  
Wave Equations ◽  
Linear Wave ◽  
Small Initial Data ◽  
Linear Wave Equations

LOCAL EXISTENCE FOR SEMILINEAR WAVE EQUATIONS AND APPLICATIONS TO YANG–MILLS EQUATIONS

2005 ◽  
Vol 02 (01) ◽  
pp. 61-76
Author(s):  
YUNG-FU FANG
Keyword(s):  
Initial Data ◽  
Wave Equations ◽  
Nonlinear Term ◽  
A Priori ◽  
Local Existence ◽  
A Priori Estimate ◽  
Linear Wave ◽  
Yang Mills ◽  
Linear Wave Equations ◽  
Local Result

In this work we are concerned with a local existence of certain semi-linear wave equations for which the initial data has minimal regularity. Assuming the initial data are in H1+∊ and H∊ for any ∊ > 0, we prove a local result by using a fixed point argument, the main ingredient being an a priori estimate for the quadratic nonlinear term uDu. The technique applies to the Yang–Mills equations in the Lorentz gauge.

scholarly journals SOLUTIONS OF QUASI-LINEAR WAVE EQUATIONS POLYHOMOGENEOUS AT NULL INFINITY IN HIGH DIMENSIONS

2011 ◽  
Vol 08 (02) ◽  
pp. 269-346 ◽  
Author(s):  
PIOTR T. CHRUŚCIEL ◽  
ROGER TAGNE WAFO
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Maxwell Equations ◽  
Wave Equations ◽  
Hyperbolic Systems ◽  
Small Data ◽  
Linear Wave ◽  
High Dimensions ◽  
Null Infinity ◽  
Linear Wave Equations

We prove propagation of weighted Sobolev regularity for solutions of the hyperboloidal Cauchy problem for a class of quasi-linear symmetric hyperbolic systems, under structure conditions compatible with the Einstein–Maxwell equations in space-time dimensions n + 1 ≥ 7. Similarly we prove propagation of polyhomogeneity in dimensions n + 1 ≥ 9. As a byproduct we obtain, in those last dimensions, polyhomogeneity at null infinity of small data solutions of vacuum Einstein, or Einstein–Maxwell equations evolving out of initial data which are stationary outside of a ball.

Remarks on nonlinear elastic waves with null conditions

2020 ◽  
Vol 26 ◽  
pp. 121
Author(s):  
Dongbing Zha ◽  
Weimin Peng
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Elastic Waves ◽  
Wave Equations ◽  
Alternative Proof ◽  
Classical Solutions ◽  
Small Initial Data ◽  
Hyperelastic Materials ◽  
Variational Structure ◽  
Nonlinear Elastic

For the Cauchy problem of nonlinear elastic wave equations for 3D isotropic, homogeneous and hyperelastic materials with null conditions, global existence of classical solutions with small initial data was proved in R. Agemi (Invent. Math. 142 (2000) 225–250) and T. C. Sideris (Ann. Math. 151 (2000) 849–874) independently. In this paper, we will give some remarks and an alternative proof for it. First, we give the explicit variational structure of nonlinear elastic waves. Thus we can identify whether materials satisfy the null condition by checking the stored energy function directly. Furthermore, by some careful analyses on the nonlinear structure, we show that the Helmholtz projection, which is usually considered to be ill-suited for nonlinear analysis, can be in fact used to show the global existence result. We also improve the amount of Sobolev regularity of initial data, which seems optimal in the framework of classical solutions.

scholarly journals POLYHOMOGENEOUS SOLUTIONS OF NONLINEAR WAVE EQUATIONS WITHOUT CORNER CONDITIONS

2006 ◽  
Vol 03 (01) ◽  
pp. 81-141 ◽  
Author(s):  
PIOTR T. CHRUŚCIEL ◽  
SZYMON ŁȨSKI
Keyword(s):  
Initial Data ◽  
Wave Equations ◽  
Space Time ◽  
Einstein Equations ◽  
Nonlinear Wave Equations ◽  
Linear Wave ◽  
Cauchy Problems ◽  
Time Dimension ◽  
Wave Maps ◽  
Corner Conditions

The study of Einstein equations leads naturally to Cauchy problems with initial data on hypersurfaces which closely resemble hyperboloids in Minkowski space-time, and with initial data with polyhomogeneous asymptotics, that is, with asymptotic expansions in terms of powers of ln r and inverse powers of r. Such expansions also arise in the conformal method for analysing wave equations in odd space-time dimension. In recent work it has been shown that for non-linear wave equations, or for wave maps, polyhomogeneous initial data lead to solutions which are also polyhomogeneous provided that an infinite hierarchy of corner conditions holds. In this paper we show that the result is true regardless of corner conditions.

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