Coercive monotone p-growth diffusion fluxes

Assume the Axiom of Choice (The Axiom of Choice).

Let $d\ge 1$, let $\mathrm{\Omega }\subset {\mathbb{R}}^d$ be a bounded open set, and let $1<p<\mathrm{\infty }$ satisfy $p>\max (1,2d\mathrm{/}(d+2))$. Put $V=W_0^{1,p}(\mathrm{\Omega };\mathbb{R})$ with norm $\mathrm{\parallel }v{\mathrm{\parallel }}_V=\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_{L^p(\mathrm{\Omega })}$, $H=L^2(\mathrm{\Omega };\mathbb{R})$, and $p^{\mathrm{\prime }}=p\mathrm{/}(p-1)$. The Poincaré inequality makes this norm equivalent to the full $W^{1,p}$ norm. The zero-boundary W1p space is separable and reflexive gives separability and reflexivity, while The subcritical W0 one p embedding into L2 gives the dense continuous compact inclusion $V⋐H$; these are the hypotheses of the evolution triple in Banach evolution triples and weak time derivatives. The compact-inclusion result uses the stated threshold also when $d=1$, where it reduces to $p>1$.

A measurable flux $a:\mathrm{\Omega }\times {\mathbb{R}}^d\to {\mathbb{R}}^d$ is an admissible coercive monotone diffusion flux of p-growth if it is Carathéodory (for each $\xi$, $x\mapsto a(x,\xi )$ is measurable, and for almost every $x$, $\xi \mapsto a(x,\xi )$ is continuous), and there are constants $C<\mathrm{\infty }$, $\alpha >0$, nonnegative functions $g\in L^{p^{\mathrm{\prime }}}(\mathrm{\Omega })$ and $h\in L^1(\mathrm{\Omega })$, and one null set outside which, for all $\xi ,\eta \in {\mathbb{R}}^d$, $\mathrm{\mid }a(x,\xi )\mathrm{\mid }\le C(g(x)+\mathrm{\mid }\xi {\mathrm{\mid }}^{p-1}),a(x,\xi )\cdot \xi \ge \alpha \mathrm{\mid }\xi {\mathrm{\mid }}^p-h(x),$ $(a(x,\xi )-a(x,\eta ))\cdot (\xi -\eta )\ge 0.$ The induced operator $A:V\to V^{\ast }$ is $⟨A(v),w{⟩}_{V^{\ast },V}={\int }_{\mathrm{\Omega }}a(x,\mathrm{\nabla }v(x))\cdot \mathrm{\nabla }w(x)dx\mathrm{.}$ The growth bound and Hölder's inequality make this pairing finite and give $\mathrm{\parallel }A(v){\mathrm{\parallel }}_{V^{\ast }}\le C_1(\mathrm{\parallel }g{\mathrm{\parallel }}_{p^{\mathrm{\prime }}}+\mathrm{\parallel }v{\mathrm{\parallel }}_V^{p-1})$ for a constant $C_1$. The finite-dimensional test-coordinate continuity used in the backward Euler existence argument follows from the stated Carathéodory and growth conditions by dominated convergence, as proved in Nonlinear backward Euler Galerkin existence and energy bounds ↗. No additional global operator-continuity hypothesis is imposed.

For $f\in L^{p^{\mathrm{\prime }}}(0,T;V^{\ast })$, a weak evolution solution is a function $u\in W_p(0,T;V)\cap L^{\mathrm{\infty }}(0,T;H)$ with a continuous $H$ representative such that $u^{\mathrm{\prime }}+A(u)=f$ in $L^{p^{\mathrm{\prime }}}(0,T;V^{\ast })$ and $u(0)=u_0\in H$.

The $L^2$ state space is available exactly on the stated strict subcritical branch. The growth, coercivity, and monotonicity conditions are explicit; no smoothness or error rate for degenerate fronts is implicit. No boundary regularity is assumed: the Poincaré and compact embedding suppliers used above hold on bounded open sets.

For $v\in V$, apply A Caratheodory integrand composed with measurable functions is measurable to each component of $a$ (independent of its scalar argument) and the measurable map $\mathrm{\nabla }v$; this makes $x\mapsto a(x,\mathrm{\nabla }v(x))$ measurable. The growth condition gives $\mathrm{\parallel }a(\cdot ,\mathrm{\nabla }v){\mathrm{\parallel }}_{L^{p^{\mathrm{\prime }}}(\mathrm{\Omega })}\le C(\mathrm{\parallel }g{\mathrm{\parallel }}_{L^{p^{\mathrm{\prime }}}(\mathrm{\Omega })}+\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_{L^p(\mathrm{\Omega })}^{p-1}),$ because $(p-1)p^{\mathrm{\prime }}=p$. Thus Hölder's inequality makes the displayed pairing a bounded linear functional of $w$, so $A(v)\in V^{\ast }$ with the stated norm bound.

The choice assumptions of the Sobolev suppliers are met under the stated Axiom of Choice: it implies Countable and Dependent Choice by AC supplies the countable and dependent choices used in Banach integration, the ultrafilter lemma by The ultrafilter lemma, from the Axiom of Choice: every filter extends to an ultrafilter, and Hahn–Banach by Hahn-Banach dominated extension theorem for real vector spaces.