Abstract
In 1981, Foias, Guillopé and Temam proved a priori estimates for arbitrary-order space derivatives of solutions to the Navier–Stokes equation. Such bounds are instructive in the numerical investigation of intermittency that is often observed in simulations, e.g., numerical study of vorticity moments by Donzis et al. (2013) revealed depletion of nonlinearity that may be responsible for smoothness of solutions to the Navier–Stokes equation. We employ an original method to derive analogous estimates for space derivatives of three-dimensional space-periodic weak solutions to the evolutionary equations of diffusive magnetohydrodynamics. Construction relies on space analyticity of the solutions at almost all times. An auxiliary problem is introduced, and a Sobolev norm of its solutions bounds from below the size in C3 of the region of space analyticity of the solutions to the original problem. We recover the exponents obtained earlier for the hydrodynamic problem. Moreover, the same approach is followed here to derive and prove similar a priori bounds for arbitrary-order space derivatives of the first-order time derivative of the weak MHD solutions.
Highlights
A standing problem of the analytical study of turbulence is to derive from the basic equations of hydrodynamics, the Euler equation and the Navier–Stokes equation, the empirical relations characterising this phenomenon
To carry over the a priori bounds for arbitrary-order space derivatives of solutions to the Navier–Stokes equation to space-periodic solutions to the equations of diffusive magnetohydrodynamics; To derive similar a priori bounds for arbitrary-order space derivatives of the first-order time derivative of the Fourier–Galerkin approximants and to prove that the bounds are admitted by weak solutions to the equations of magnetohydrodynamics; To reveal a link between these bounds and space analyticity of the MHD solutions at almost all times
(The initial data for the Fourier–Galerkin system of Equations (14) should be used in the r.h.s. of (19) and (20), but we replace the norms in (19) and (20) by the norms of the initial data for the original problem (4), since the Hs (T3 ) norms of the truncated initial conditions monotonically increase with the resolution parameter N.) the Fourier–Galerkin approximants (7) of solutions to (4) obey an a priori bound that is independent of N
Summary
A standing problem of the analytical study of turbulence is to derive from the basic equations of hydrodynamics, the Euler equation and the Navier–Stokes equation, the empirical relations characterising this phenomenon. To carry over the a priori bounds for arbitrary-order space derivatives of solutions to the Navier–Stokes equation to space-periodic solutions to the equations of diffusive magnetohydrodynamics; To derive similar a priori bounds for arbitrary-order space derivatives of the first-order time derivative of the Fourier–Galerkin approximants and to prove that the bounds are admitted by weak solutions to the equations of magnetohydrodynamics; To reveal a link between these bounds and space analyticity of the MHD solutions at almost all times They are achieved by following an original approach [25] based on a transformation of coefficients in the expansion of the solutions in the Fourier series in spatial variables. This paper is dedicated to Professor Uriel Frisch on the occasion of his 80th anniversary as a sign of appreciation of the Scientist and the Teacher
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