Abstract
We present a technique for derivation of a priori bounds for Gevrey--Sobolev norms of space-periodic three-dimensional solutions to evolutionary partial differential equations of hydrodynamic type. It involves a transformation of the flow velocity in the Fourier space, which introduces a feedback between the index of the norm and the norm of the transformed solution, and results in emergence of a mildly dissipative term. We illustrate the technique, using it to derive finite-time bounds for Gevrey--Sobolev norms of solutions to the Euler and inviscid Burgers equations, and global-in-time bounds for the Voigt-type regularizations of the Euler and Navier--Stokes equation (assuming that the respective norm of the initial condition is bounded). The boundedness of the norms implies analyticity of the solutions in space.
Citation
Vladislav Zheligovsky. "A priori bounds for Gevrey--Sobolev norms of space-periodic three-dimensional solutions to equations of hydrodynamic type." Adv. Differential Equations 16 (9/10) 955 - 976, September/October 2011. https://doi.org/10.57262/ade/1355703183
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