The well-posedness of the Cauchy problem, mentioned in title, is studied. The main result means that the solution of this problem is usual C∞ - function on the space argument, if the initial function is a real functional on the conjugate space to the space, containing the fundamental solution of the corresponding problem. The basic tool for the proof is the functional analysis technique.
In this paper, we consider the Cauchy problem for the 3D MHD fluid passing
through the porous medium, and provide some fundamental Serrin type
regularity criteria involving the velocity or its gradient, the pressure or
its gradient. This extends and improves [S. Rahman, Regularity criterion for
3D MHD fluid passing through the porous medium in terms of gradient pressure,
J. Comput. Appl. Math., 270 (2014), 88-99].
Abstract
In this paper, a Cauchy problem for the Laplace equation is considered. We develop a modified Tikhonov regularization method based on Hermite expansion to deal with the ill posed-ness of the problem. The regularization parameter is determined by a discrepancy principle. For various smoothness conditions, the solution process of the method is uniform and the convergence rate can be obtained self-adaptively. Numerical tests are also carried out to verify the effectiveness of the method.