2013

Persistency of Analyticity for Nonlinear Wave Equations: An Energy-like Approach

Guo, Yanqiu, Titi, Edriss S.

Understand

We study the persistence of the Gevrey class regularity of solutions to nonlinear wave equations with real analytic nonlinearity.

Specifically, it is proven that the solution remains in a Gevrey class, with respect to some of its spatial variables, during its whole life-span, provided the initial data is from the same Gevrey class with respect to these spatial variables.
In addition, for the special Gevrey class of analytic functions, we find a lower bound for the radius of the spatial analyticity of the solution that might shrink either algebraically or exponentially, in time, depending on the structure of the nonlinearity.
The standard $L^2$ theory for the Gevrey class regularity is employed; we also employ energy-like methods for a generalized version of Gevrey classes based on the $\ell^1$ norm of Fourier transforms (Wiener algebra).

Built on

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Then

A. Biswas and D. Swanson, Gevrey regularity of solutions to the 3-D Navier-Stokes equations with weighted l p l_{p} initial data, Indiana University Mathematics Journal 56 (2007), 1157-1188
2007
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2009
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I. Kukavica and V. Vicol, On the radius of analyticity of solutions to the three-dimensional Euler equations, Proceedings of the American Mathematical Society 137 (2009), 669-677
2009
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A. Larios and E. S. Titi, On the higher-order global regularity of the inviscid Voigt-regularization of three-dimensional hydrodynamic models, Discrete and Continuous Dynamical Systems Series B 14 (2010), 603-627
2010
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2011
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M. Struwe, A ‘super-critical’ nonlinear wave equation in 2 space dimensions, Milan Journal of Mathematics 79 (2011), 129-143
2011
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S. Kuksin and N. Nadirashvili, Analyticity of solutions for quasilinear wave equations and other quasilinear systems, preprint (2012), arXiv:1205.5926
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2012
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A. Biswas, M. S. Jolly, V. Martinez, and E. S. Titi, Smallest scale estimates for the Navier-Stokes equations in the Wiener algebra, preprint (2013)
2013
Closest in time.
Patrick Gérard, Yanqiu Guo, and Edriss S. Titi, On the radius of analyticity of solutions to the cubic Szegő equation, preprint (2013), arXiv:1303.6148
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2013
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