On Regularity of the Time Derivative for Degenerate Parabolic Systems

Abstract

Abstract. We prove regularity estimates for time derivatives of a large class of nonlinear parabolic partial differential systems. This includes the instationary (symmetric) p-Laplace system and models for non Newtonien fluids of powerlaw or Carreau type. By the use of special weak different quotients, adapted to the variational structure we bound fractional derivatives of ut in time and space direction.Although the estimates presented here are valid under very general assumptions they are a novelty even for the parabolic p-Laplace equation.

Evolutionary, symmetric $p$-Laplacian. Interior regularity of time derivatives and its consequences

“…The main difference between our results and these of [29] concerns localization. We derive space-time local estimates and actually the biggest difficulty that we face, compare Subsection 6.1, follows from spacetime mismatch in lower-order terms that stem from using a cutoff function.…”

“…Results for generic boundary-value problems, developed for the full p-Navier-Stokes system, are of course available for the symmetric p-Laplacian. In particular, one has smoothness of solutions to basic initial-boundary value problems in 2d case, see Kaplický,Málek & Stará [39] and Kaplický [37], [38] as well as existence of strong solutions for 3d case, compare [40] by Málek, Nečas & Růžička and also [7], [8] by Beirão da Veiga and [11] by Beirão da Veiga, Kaplický & Růžička. Let us mention also here a recent local regularity study for the p-Laplacian, symmetric p-Laplacian and p-Stokes type problems by Frehse & Schwarzacher [29]. Since their results corresponds strongly to ours, we compare them in a more detailed manner in Subsection 5.3.…”

Self-improving property of degenerate parabolic equations of porous medium-type

“…These include (without any ambition of completion) the full L q theory and beyond [AM07,Sch13], as well as partial regularity results [BZM13], or pointwise estimates via potential theory for equations [KM14/1]. Summarizing, many different ways to show regularity for the gradient of solutions to the p-Laplacian are available which, among other benefits, has a natural impact on the regularity of the time derivative, as it was recently shown in [FSch15].…”

Second-Order Regularity for Parabolic p-Laplace Problems

“…and let {u k } be the corresponding sequence of weak solutions to problems (2.2). A global in time version of [FrSch,Proposition 4.1] tells us that u k ∈ L ∞ ((0, T ); W 1,p (Ω)), (u k ) t ∈ L 2 (Ω T ) and…”

“…Inequality (3.3) holds in the whole interval [0, T ], namely up to t = 0, instead of just locally, as in the result of [FrSch], thanks to the present assumption that ψ belongs to W 1,p 0 (Ω). The proof is completely analogous to (in fact, simpler than) that of [FrSch,Proposition 4.1], and will be omitted. One clearly has that L ∞ ((0, T ); W 1,p 0 (Ω)) → L p (Ω T ).…”