Vanishing-viscosity families are locally precompact in $L^1$

Uniform bounds and spatial modulus. By [F2], $\mathrm{\mid }{u}^j(t,x)\mathrm{\mid }\le M$ and $\mathrm{\parallel }{u}^j(t,\cdot ){\mathrm{\parallel }}_1\le \mathrm{\parallel }{u}_0{\mathrm{\parallel }}_1$ for all $j,t$; by [F3], $\mathrm{\parallel }{u}^j(t,\cdot +z)-u^j(t,\cdot ){\mathrm{\parallel }}_1\le {\omega }_0(\mathrm{\mid }z\mathrm{\mid })$ with ${\omega }_0(r)\downarrow 0$, uniformly in $j$ and $t$.

The local time modulus. Fix $R>0$, $0<\rho <1$, $j$, and $t,\tau$ with $t,t+\tau \in [0,T]$; put $w=u^j(t+\tau ,\cdot )-u^j(t,\cdot )$. Choose $\chi$ as in [F5] and an even mollifier ${\varrho }_h$ with $0<h<\rho \mathrm{/}4$; set $z={\varrho }_h\ast w$ and $g={\varrho }_h\ast (\chi \mathrm{sgn}z)\in C_c^{\mathrm{\infty }}(B_{R+\rho })$, so that $\mathrm{\parallel }{D}^{\alpha }g{\mathrm{\parallel }}_{\mathrm{\infty }}\le C{h}^{-\mathrm{\mid }\alpha \mathrm{\mid }}$ for $\mathrm{\mid }\alpha \mathrm{\mid }\le 2$ and ${\int }_{{\mathbb{R}}^n}gwdx={\int }_{{\mathbb{R}}^n}\chi \mathrm{\mid }z\mathrm{\mid }dx$ by [F4]. Assume $\tau >0$; negative increments follow by reversing the two times. Since $u^j$ solves the viscous equation pointwise at positive times and is $C^1$ there, $\frac{d}{ds}\int g{u}^j(t+s,\cdot )dx=\int \mathrm{\nabla }g\cdot f_j(u^j(t+s,\cdot ))dx+{\epsilon }_j\int \mathrm{\Delta }g{u}^j(t+s,\cdot )dx$ for $s\in [0,\tau ]$; integrating over $s$ and using $\mathrm{\parallel }{f}_j(v){\mathrm{\parallel }}_1\le L\mathrm{\parallel }v{\mathrm{\parallel }}_1$ for $\mathrm{\mid }v\mathrm{\mid }\le M$ and [F2], $\mid \int gw\mid \le {\int }_0^{\tau }(\mathrm{\parallel }\mathrm{\nabla }g{\mathrm{\parallel }}_{\mathrm{\infty }}L\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_1+{\epsilon }_j\mathrm{\parallel }\mathrm{\Delta }g{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_1)ds\le C_R\tau h^{-2}$.

The uniform modulus in $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1$. With $\chi ,w,z$ as in step 1.2 and [F4], ${\int }_{B_R}\mathrm{\mid }w\mathrm{\mid }\le \int \chi \mathrm{\mid }w\mathrm{\mid }\le \int \chi \mathrm{\mid }z\mathrm{\mid }+\int \chi \mathrm{\mid }w-z\mathrm{\mid }\le C_R\tau h^{-2}+2{\omega }_0(h)$, because $\int \chi \mathrm{\mid }z\mathrm{\mid }=\mid \int gw\mid$ and $\mathrm{\parallel }w-z{\mathrm{\parallel }}_{L^1(B_{R+\rho \mathrm{/}2})}\le 2{\omega }_0(h)$ by [F3] applied at the two times $t$ and $t+\tau$. Taking $h={\tau }^{1\mathrm{/}3}$ (for $0<\tau <(\rho \mathrm{/}4{)}^3$) gives ${\int }_{B_R}\mathrm{\mid }{u}^j(t+\tau )-u^j(t)\mathrm{\mid }\le C_R({\omega }_0({\tau }^{1\mathrm{/}3})+{\tau }^{1\mathrm{/}3})=:{\eta }_R(\tau )$ with ${\eta }_R(\tau )\to 0$ as $\tau \downarrow 0$, uniformly in $j$ and in $t\in [0,T-\tau ]$; the endpoint $t=0$ follows by first integrating from a positive time and then using the $L^1$ continuity $u^j(t)\to u_0$.

Relative compactness on cylinders. Fix $l\ge 1$ and cutoffs ${\psi }_l,{\theta }_l$ as in [F5] equal to $1$ on the projections of the cylinder $K_l=\bar{B}(0,l)\times [1\mathrm{/}l,T-1\mathrm{/}l]$, and set $F_l^j(t,x)={\theta }_l(t){\psi }_l(x)u^j(t,x)$, extended by zero. The family $(F_l^j{)}_j$ is uniformly bounded in $L^1({\mathbb{R}}^{n+1})$ by $T\mathrm{\parallel }{\theta }_l{\psi }_l{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_1$, has common compact support (uniform tails), and is uniformly translation-continuous: for a shift $(y,s)$ one has $\mathrm{\parallel }{F}_l^j(\cdot +(y,s))-F_l^j{\mathrm{\parallel }}_1\le \mathrm{\parallel }{F}_l^j(t+s,x+y)-F_l^j(t,x+y){\mathrm{\parallel }}_1+\mathrm{\parallel }{F}_l^j(t,x+y)-F_l^j(t,x){\mathrm{\parallel }}_1$, where the spatial part is bounded by $T\mathrm{\parallel }({\theta }_l{\psi }_l)(\cdot +y)-({\theta }_l{\psi }_l){\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_1+T\mathrm{\parallel }{\theta }_l{\psi }_l{\mathrm{\parallel }}_{\mathrm{\infty }}{\omega }_0(\mathrm{\mid }y\mathrm{\mid })$ and the temporal part by $T\mathrm{\parallel }{\theta }_l(\cdot +s)-{\theta }_l{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }{\psi }_l{\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_1+T\mathrm{\parallel }{\theta }_l{\psi }_l{\mathrm{\parallel }}_{\mathrm{\infty }}{\eta }_{l^{\mathrm{\prime }}}(\mathrm{\mid }s\mathrm{\mid })$ for a radius $l^{\mathrm{\prime }}$ containing $\mathrm{supp}{\psi }_l$, both tending to $0$ as $\mathrm{\mid }(y,s)\mathrm{\mid }\to 0$ uniformly in $j$ by step 1.1 and step 2.1. By [F6] every subsequence of $(F_l^j{)}_j$ has a further subsequence converging in $L^1({\mathbb{R}}^{n+1})$, hence, after diagonal extraction over the countably many $l\ge 1$ (Dependent Choice), some subsequence of $(u^j)$ converges in $L^1(K_l)$ for every $l$, and therefore in $L^1(K)$ for every compact $K⋐{\mathrm{\Pi }}_T$, since each such $K$ lies in some $K_l$.

Continuous representative and the initial trace. Pass to a further subsequence converging almost everywhere on ${\mathrm{\Pi }}_T$ by [F6]. Since $\mathrm{\mid }{u}^{j_k}\mathrm{\mid }\le M$, this gives $\mathrm{\mid }u\mathrm{\mid }\le M$ almost everywhere; by Fubini and dominated convergence, there is a common full-measure set $S\subset (0,T)$ on which $u^{j_k}(t)\to u(t)$ in $L^1(B_m)$ for every integer $m\ge 1$. Thus $S$ is dense. Fix $m$ and choose an integer $m^{\mathrm{\prime }}>m$. For $s,t\in S$ sufficiently close that step 2.1 applies, lower semicontinuity on the local ball, followed by its time-modulus estimate on the larger ball, gives $\mathrm{\parallel }u(t)-u(s){\mathrm{\parallel }}_{L^1(B_m)}\le {\mathrm{lim\hspace{0.17em}inf}}_k\mathrm{\parallel }{u}^{j_k}(t)-u^{j_k}(s){\mathrm{\parallel }}_{L^1(B_m)}\le {\eta }_{m^{\mathrm{\prime }}}(\mathrm{\mid }t-s\mathrm{\mid })\mathrm{.}$ For $t\in S$ sufficiently close to $0$, the same local comparison with $u^{j_k}(0,\cdot )=u_0$ gives $\mathrm{\parallel }u(t)-u_0{\mathrm{\parallel }}_{L^1(B_m)}\le {\eta }_{m^{\mathrm{\prime }}}(t)$. This modulus tending to zero at zero makes the map on $S$ uniformly continuous; completeness of $L^1(B_m)$ extends it uniquely to a continuous map on $[0,T]$, with value $u_0$ at $t=0$. These extensions agree on nested balls because they agree on the dense set $S$. Consequently $u$ has a representative in $C([0,T];L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1({\mathbb{R}}^n))$ with $u(0)=u_0$ in $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1$.

Limit properties. The diagonal limit of step 3.1 lies in $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1({\mathrm{\Pi }}_T)$ and the selected subsequence converges in $L^1(K)$ for every compact $K⋐{\mathrm{\Pi }}_T$; the further almost-everywhere subsequence in step 3.2 preserves these convergences and passes the uniform bound $\mathrm{\mid }{u}^{j_k}\mathrm{\mid }\le M$ to $u$. This establishes the asserted limit and almost-everywhere convergence.

Applicability and hypotheses. If $f_j\to f$ in $C^1$ on compact sets with ${\sup }_j{\sup }_{\mathrm{\mid }s\mathrm{\mid }\le M+1}\mathrm{\mid }{f}_j^{\mathrm{\prime }}(s)\mathrm{\mid }<\mathrm{\infty }$ — a $C^1$-convergent smooth approximation of one flux — then the hypotheses above hold, so the conclusions apply. The extraction used Dependent Choice in the diagonal step 3.1 (and Countable Choice inside [F6]); the uniform energy dissipation ${\epsilon }_j{\int }_0^T\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\int \mathrm{\mid }\mathrm{\nabla }{u}^j{\mathrm{\mid }}^2\le \frac{1}{2}\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_2^2$ supplies only a uniform gradient bound and would not by itself give the compactness in $L^1$ obtained from the uniform bounds and translation moduli of steps 1.1 and 2.1.