Note on time-regularity for weak solutions to parabolic systems of 𝑝-Laplace type

Bortz, Simon; Egert, Moritz; Saari, Olli

doi:10.1090/proc/15344

Note on time-regularity for weak solutions to parabolic systems of $p$-Laplace type
HTML articles powered by AMS MathViewer

by Simon Bortz, Moritz Egert and Olli Saari PDF

Proc. Amer. Math. Soc. 149 (2021), 1677-1685 Request permission

Abstract:

We show that local weak solutions to parabolic systems of $p$-Laplace type are Hölder continuous in time with values in a spatial Lebesgue space and Hölder continuous on almost every time line. We provide an elementary and self-contained proof building on the local higher integrability result of Kinnunen and Lewis.

References

Pascal Auscher, Simon Bortz, Moritz Egert, and Olli Saari, On regularity of weak solutions to linear parabolic systems with measurable coefficients, J. Math. Pures Appl. (9) 121 (2019), 216–243 (English, with English and French summaries). MR 3906170, DOI 10.1016/j.matpur.2018.08.002
Wolfgang Arendt, Charles J. K. Batty, Matthias Hieber, and Frank Neubrander, Vector-valued Laplace transforms and Cauchy problems, 2nd ed., Monographs in Mathematics, vol. 96, Birkhäuser/Springer Basel AG, Basel, 2011. MR 2798103, DOI 10.1007/978-3-0348-0087-7
Robert Denk and Mario Kaip, General parabolic mixed order systems in ${L_p}$ and applications, Operator Theory: Advances and Applications, vol. 239, Birkhäuser/Springer, Cham, 2013. MR 3134525, DOI 10.1007/978-3-319-02000-6
Mariano Giaquinta and Michael Struwe, On the partial regularity of weak solutions of nonlinear parabolic systems, Math. Z. 179 (1982), no. 4, 437–451. MR 652852, DOI 10.1007/BF01215058
Juha Kinnunen and John L. Lewis, Higher integrability for parabolic systems of $p$-Laplacian type, Duke Math. J. 102 (2000), no. 2, 253–271. MR 1749438, DOI 10.1215/S0012-7094-00-10223-2
J.-L. Lions, Sur les problèmes mixtes pour certains systèmes paraboliques dans des ouverts non cylindriques, Ann. Inst. Fourier (Grenoble) 7 (1957), 143–182 (French). MR 102666, DOI 10.5802/aif.69
Camil Muscalu and Wilhelm Schlag, Classical and multilinear harmonic analysis. Vol. I, Cambridge Studies in Advanced Mathematics, vol. 137, Cambridge University Press, Cambridge, 2013. MR 3052498
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
Wiktoria Zatoń, Tent space well-posedness for parabolic Cauchy problems with rough coefficients, J. Differential Equations 269 (2020), no. 12, 11086–11164. MR 4150368, DOI 10.1016/j.jde.2020.07.033