\(L^q\)-\(L^\infty\) Hölder continuity for quasilinear parabolic equations associated to Sobolev derivations (Q697446)

This paper introduces a general analytic-algebraic point of view in analyzing the \(L^p\) regularity properties of a class of abstract nonlinear evolution equations governed by a \(p\)-Laplacian-like operator constructed from a closed derivation with values in a general Hilbert \(C^*\)-monomodule, for which an appropriate Sobolev inequality holds. \(L^q\)-\(L^\infty\) Hölder continuity like \[ \|u(t)- v(t)\|_\infty\leq Ct^{-\beta}\|u(0)- v(0)\|^\gamma_q \] is obtained for two solutions \(u\), \(v\). The model example is the evolution equation driven by the Euclidean \(p\)-Laplacian: \(\partial_t u= \text{div}(|\nabla u|^{p-2}\nabla u)\) with \(2\leq p< d\) on a bounded domain \(\Omega\) in \(\mathbb{R}^d\) with the homogeneous Dirichlet boundary condition. Other concrete examples are nonlinear evolution equations governed by the \(p\)-Laplacian on manifolds, by subelliptic \(p\)-Laplacians constructed in terms of suitable vector fields on Euclidean domains or Lie groups, and by the sub-Riemannian \(p\)-Laplacian associated to a sub-Riemannian structure on a manifold.