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An \(L^{2}\)-maximal regularity result for the evolutionary Stokes-Fourier system. (English) Zbl 1209.35102

Summary: We establish an \(L^{2}\)-regularity result for a weak solution of the evolutionary Stokes-Fourier system. Although this system does not contain the convective terms, the fact that the viscosity depends on the temperature makes the considered system of partial differential equations nonlinear. The result holds for a class of the viscosities that includes the Arrhenius formula as a special case. For simplicity, we restrict ourselves to a spatially periodic setting in this study.

MSC:

35Q35 PDEs in connection with fluid mechanics
35B10 Periodic solutions to PDEs
35B65 Smoothness and regularity of solutions to PDEs
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
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[1] Ladyzhenskaya OA, 2nd English Edition, Revised and Enlarged. (Translated from the Russian Richard A. Silverman and John Chu. Mathematics and its Applications 2 (1969)
[2] Solonnikov VA, Trudy Mat. Inst. Steklov. 70 pp 213– (1964)
[3] DOI: 10.1016/j.nonrwa.2007.11.018 · Zbl 1167.76316 · doi:10.1016/j.nonrwa.2007.11.018
[4] DOI: 10.1137/07069540X · Zbl 1195.35239 · doi:10.1137/07069540X
[5] Ladyženskaja OA, (Translated from the Russian by S. Smith. Translations of Mathematical Monographs 23 (1967)
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