Existence of variational solutions to doubly nonlinear systems in nondecreasing domains
Leah Schätzler, Christoph Scheven, Jarkko Siltakoski, Calvin Stanko
TL;DR
The paper establishes the existence of variational solutions to a class of doubly nonlinear parabolic systems in noncylindrical, time-nondecreasing domains, under convexity and $p$-coercivity assumptions on the variational integrand $f$ with no upper growth bound. A nonlinear minimizing movements scheme in time, supported by energy estimates and compactness arguments, yields a limit $u$ in $C^{0}([0,T);L^{q+1}(\Omega))\cap V^p_q(E)$ that satisfies the associated variational inequality. Under additional $p$-growth and domain regularity, the work proves the distributional time derivative of $|u|^{q-1}u$ exists in the dual of the parabolic Sobolev space, providing a quantitative time-derivative bound. This extends prior theory from cylindrical to noncylindrical domains, enabling variational methods for nonlinear diffusion-type systems on evolving geometries with zero-measure boundaries.
Abstract
For $q \in (0, \infty)$, we consider the Cauchy-Dirichlet problem to doubly nonlinear systems of the form \begin{align*} \partial_t \big( |u|^{q-1}u \big) - \operatorname{div} \big( D_ξf(x,u,Du) \big) = - D_u f(x,u,Du) \end{align*} in a bounded noncylindrical domain $E \subset \mathbb{R}^{n+1}$. We assume that $x \mapsto f(x,u,ξ)$ is integrable, that $(u,ξ) \mapsto f(x,u,ξ)$ is convex, and that $f$ satisfies a $p$-coercivity condition for some $p \in (1,\infty)$. However, we do not impose any specific growth condition from above on $f$. For nondecreasing domains that merely satisfy $\mathcal{L}^{n+1}(\partial E) = 0$, we prove the existence of variational solutions $u \in C^{0}([0,T];L^{q+1}(E,\mathbb{R}^N))$ via a nonlinear version of the method of minimizing movements. Moreover, under additional assumptions on $E$ and a $p$-growth condition on $f$, we show that $|u|^{q-1}u$ admits a weak time derivative in the dual $(V^{p,0}(E))^{\prime}$ of the subspace $V^{p,0}(E) \subset L^p(0,T;W^{1,p}(Ω,\mathbb{R}^N))$ that encodes zero boundary values.