Temporal Regularity of Symmetric Stochastic p-Stokes Systems

Abstract

We study the symmetric stochastic p-Stokes system, p∈(1,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p \\in (1,\\infty )$$\\end{document}, in a bounded domain. The results are two-fold: First, we show that in the context of analytically weak solutions, the stochastic pressure—related to non-divergence free stochastic forces—enjoys almost -1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-1/2$$\\end{document} temporal derivatives on a Besov scale. Second, we verify that the velocity u of strong solutions obeys 1/2 temporal derivatives in an exponential Nikolskii space. Moreover, we prove that the non-linear symmetric gradient V(ϵu)=(κ+ϵu)(p-2)/2ϵu\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V(\\mathbb {\\epsilon } u) = (\\kappa + \\left| \\mathbb {\\epsilon } u\\right| )^{(p-2)/2} \\mathbb {\\epsilon } u$$\\end{document}, κ≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa \\ge 0$$\\end{document}, which measures the ellipticity of the p-Stokes system, has 1/2 temporal derivatives in a Nikolskii space.