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

Abstract

We consider the evolutionary symmetric p-Laplacian with safety 1. By symmetric we mean that the full gradient of p-Laplacian is replaced by its symmetric part, which causes breakdown of the Uhlenbeck structure. We derive the interior regularity of time derivatives of its local weak solution. To circumvent the space-time growth mismatch, we devise a new local regularity technique of iterations in Nikolskii-Bochner spaces. It is interesting by itself, as it may be modified to provide new regularity results for the full-gradient p-Laplacian case with lower-order dependencies. Finally, having the regularity result for time derivatives, we obtain respective regularity of the main part. The Appendix on Nikolskii-Bochner spaces, that includes theorems on their embeddings and interpolations, may be of independent interest.

Parabolic p-Laplacian revisited: Global regularity and fractional smoothness

“…Theorem 3 and Corollary 5 are new contributions for the regularity of time derivatives of p-Laplacian-like evolutive equations. Indeed, there have been few works in this direction, e.g., [8] and the references therein, where the authors prove related results for non-degenerate systems of more general operators. Despite that we employ different techniques, the additional information that our results give rely mainly on the gain of global regularity in space and time, as well as = 0 is allowed.…”

“…Despite that we employ different techniques, the additional information that our results give rely mainly on the gain of global regularity in space and time, as well as = 0 is allowed. In other words, these results are also a partial contribution since when contrasted to [8] we treat the full degenerate case, while when contrasting to [16], we do not assume additional regularity to the solutions in order to obtain extra time regularity. Nevertheless, we consider the scalar degenerate case what enables us to use additional elliptic regularity which is still not known to the vectorial or singular case.…”

$\mathcal{A}$-caloric approximation and partial regularity for parabolic systems with Orlicz growth

“…In [19], Diening, Scharle, and Schwarzacher assume t ϕ ′′ (t) and ϕ ′ (t) to be comparable, while in [32], only a doubling property is needed to obtain the boundedness of u. For additional regularity and higher integrability results, where an Uhlenbeck or similar structure is assumed, we also mention [2,3,8,20]. Without such a structural assumption, higher integrability was established by Hästö and Ok in [31].…”

Higher differentiability for solutions of stationary p‐Stokes systems

“…All of the mentioned results are restricted to dimension . Finally, we note that for corresponding evolutionary problems, related results on higher differentiability in the space and time direction have been established in [9, 10, 17]. However, the analysis in these works does not cover the case of systems with convective term.…”