Global Calderòn & Zygmund theory for nonlinear parabolic systems

Abstract

We establish a global Calderón & Zygmund theory for solutions of a huge class of nonlinear parabolic systems whose model is the inhomogeneous parabolic $p$-Laplacian system \begin{equation*} \left\{\begin{array}{cc} \partial_t u - \Div (|Du|^{p-2}Du) = \Div (|F|^{p-2}F) &\mbox{in ${\Omega }_T:=\Omega \times (0,T)$} \\[5pt] u=g &\mbox{on $\partial \Omega \times (0,T)\cup \bar{\Omega }\times \{0\}$} \end{array}\right. \end{equation*} with given functions $F$ and $g$. Our main result states that the spatial gradient of the solution is as integrable as the data $F$ and $g$ up to the lateral boundary of ${\Omega }_T$, i.e. \begin{equation*} F,Dg\in L^q(Ω_T),\ \ \partial_t g\in L^{\frac{q(n+2)}{p(n+2)-n}}(Ω_T) \quad\Rightarrow \quad Du\in L^q(Ω\times(δ,T)) \end{equation*} for any $q>p$ and $\delta \in (0,T)$, together with quantitative estimates. This result is proved in a much more general setting, i.e. for asymptotically regular parabolic systems.

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• Global Existence of Weak Solutions to the Barotropic Compressible Navier-Stokes Flows with Degenerate Viscosities
• Well posedness of nonlinear parabolic systems beyond duality
• On global $L^q$ estimates for systems with $p$-growth in rough domains
• Sharp gradient estimates for quasilinear elliptic equations with $p(x)$ growth on nonsmooth domains
• Global existence of weak solutions to the compressible quantum Navier-Stokes equations with degenerate viscosity
• Lorentz improving estimates for the $p$-Laplace equations with mixed data
• Global regularity for degenerate/singular parabolic equations involving measure data
• $L^q$-regularity for nonlinear elliptic equations with Schrödinger-type lower order terms