Banach evolution triples and weak time derivatives

Let $V$ be a separable reflexive real Banach space, let $H$ be a real Hilbert space, and let $i:V\to H$ be an injective continuous linear map with dense range. Let $V^{\ast }$ be the continuous real dual of $V$ and write $⟨\cdot ,\cdot {⟩}_{V^{\ast },V}$ for its pairing with $V$. The canonical map $J:H\to V^{\ast }$ is $Jh(v)=(h,iv{)}_H(h\in H,v\in V)\mathrm{.}$ It is continuous because $\mathrm{\mid }Jh(v)\mathrm{\mid }\le \mathrm{\parallel }h{\mathrm{\parallel }}_H\mathrm{\parallel }i\mathrm{\parallel }\mathrm{\parallel }v{\mathrm{\parallel }}_V$, and injective because $i(V)$ is dense in $H$. A triple $(V,H,J)$ with these properties is a Banach evolution triple. If $V$ is Hilbert, this is the real evolution triple of Evolution triples and the parabolic energy space.

For $1<p<\mathrm{\infty }$, $L^p(0,T;V)$ denotes the Bochner space of strongly measurable $V$-valued functions with finite $p$th power norm, and $p^{\mathrm{\prime }}=p\mathrm{/}(p-1)$. For $u\in L^p(0,T;V)$ and $g\in L^{p^{\mathrm{\prime }}}(0,T;V^{\ast })$, say $g=u^{\mathrm{\prime }}$ in the distributional time sense if ${\int }_0^T(iu(t),iv{)}_H{\varphi }^{\mathrm{\prime }}(t)dt=-{\int }_0^T⟨g(t),v{⟩}_{V^{\ast },V}\varphi (t)dt$ for every $v\in V$ and $\varphi \in C_c^{\mathrm{\infty }}(0,T)$. Define $W_p(0,T;V)=\{u\in L^p(0,T;V):u^{\mathrm{\prime }}\in L^{p^{\mathrm{\prime }}}(0,T;V^{\ast })\}\mathrm{.}$ This definition assigns no endpoint values; a separate chain-rule theorem is needed before evaluating a function in this space at an endpoint.

The published parabolic FEM definition uses $p=2$ and a Hilbert energy space. This definition allows the reflexive Banach energy space needed for the $p$-Laplacian branch. The pairing convention is real and bilinear throughout.