In the paper “Space analyticity and bounds for derivatives of solutions to the evolutionary equations of diffusive magnetohydrodynamics”, V. Zheligovsky has given solutions to the three following mutually related problems:
– to carry over the known 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 of the MHD solutions 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.
A standing problem of the analytical study of turbulence is to derive from the basic equations of hydrodynamics, the Euler and Navier–Stokes equations, the empirical relations characterising this phenomenon. Pure mathematical problems such as to identify the class of functions, in which existence and uniqueness of solutions is guaranteed, or to answer the related question whether singularities can develop at a finite time in the solutions still remain unsolved. A progress in tackling these issues requires gaining a profound understanding of the behaviour of small-scale structures in flows.
A possible approach to addressing this problem consists of obtaining information on norms of high-order derivatives of the solutions: the higher the order, the more the respective norms are controlled by the small-scale components of the solutions.
In 1981, Foias, Guillopé and Temam proved a priori estimates for arbitrary-order space derivatives of solutions to the Navier–Stokes equation. In the present work, V. Zheligovsky has employed an original method to derive analogous estimates for space derivatives of three-dimensional space-periodic weak solutions to the evolutionary equations of diffusive magnetohydrodynamics. The 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. The exponents αs=2/(2s–1) obtained ibid. for the Hs norms of flows in the hydrodynamic problem are recovered in the MHD problem (due to the similarity of the quadratic non-linear terms in the Navier–Stokes and magnetic induction equations). The author followed same approach to derive a priori bounds for arbitrary-order space derivatives of the first-order time derivative of the weak MHD solutions. He has proven that weak solutions constructed as limits of the Fourier–Galerkin approximants obey these bounds. The proof is obtained by studying the (assumed) time singularity sets of the weak solutions and showing that the approximants converge uniformly in the norm H1 on any closed interval in the complement to the singularity set.
Source: Zheligovsky V. Space analyticity and bounds for derivatives of solutions to the evolutionary equations of diffusive magnetohydrodynamics // Mathematics, 9, 2021, 1789 [arxiv.org/abs/2108.02746]