Uniform L-infinity, mass and energy bounds for the viscous approximations

Supremum bound. Statement (i) is exactly the range bound of the construction theorem [F1]: $\mathrm{\parallel }{u}^{\epsilon }(t){\mathrm{\parallel }}_{\mathrm{\infty }}\le \max \{\mathrm{\mid }{\sup }_x{u}_0\mathrm{\mid },\mathrm{\mid }{\inf }_x{u}_0\mathrm{\mid }\}=\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{\mathrm{\infty }}$.

Mass identity with cutoffs. Multiply the pointwise equation by a cutoff ${\chi }_R$ of [F3], integrate first from a positive lower time, and then pass that time to zero by $C([0,T];L^1)$ continuity: since ${\chi }_R$ vanishes outside a compact set, integration by parts is legitimate and gives $\int u^{\epsilon }(t){\chi }_R-\int u_0{\chi }_R={\int }_0^t\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\int (f(u^{\epsilon })\cdot \mathrm{\nabla }{\chi }_R+\epsilon u^{\epsilon }\mathrm{\Delta }{\chi }_R)$; by [F1] the right side is bounded in absolute value by $(C{L}_M{R}^{-1}+\epsilon C{R}^{-2}){\int }_0^t\mathrm{\parallel }{u}^{\epsilon }(s){\mathrm{\parallel }}_{1}ds\le C_T(R^{-1}+\epsilon R^{-2})$.

Positive-time energy identity. Put $M=\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{\mathrm{\infty }}$. Fix $0<s<t\le T$, take the convex entropy $\eta (r)=r^2\mathrm{/}2$ and its flux $q(r)={\int }_0^{r}a{f}^{\mathrm{\prime }}(a)da$, and integrate the balance [F2] against ${\chi }_R$ over $[s,t]\times {\mathbb{R}}^n$. This is legitimate on each compact support because $u^{\epsilon }\in C^{1,2}$ for positive times. The resulting identity is $\frac{1}{2}\int \mathrm{\mid }{u}^{\epsilon }(t){\mathrm{\mid }}^2{\chi }_R+\epsilon {\int }_s^t\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\int \mathrm{\mid }\mathrm{\nabla }{u}^{\epsilon }{\mathrm{\mid }}^2{\chi }_R=\frac{1}{2}\int \mathrm{\mid }{u}^{\epsilon }(s){\mathrm{\mid }}^2{\chi }_R+{\int }_s^t\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\int q(u^{\epsilon })\cdot \mathrm{\nabla }{\chi }_R+\epsilon {\int }_s^t\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\int \eta (u^{\epsilon })\mathrm{\Delta }{\chi }_R\mathrm{.}$ On the solution range, $\mathrm{\mid }q(u^{\epsilon })\mathrm{\mid }\le L_{M}M\mathrm{\mid }{u}^{\epsilon }\mathrm{\mid }\mathrm{/}2$ and $\eta (u^{\epsilon })\le M\mathrm{\mid }{u}^{\epsilon }\mathrm{\mid }\mathrm{/}2$, so both cutoff errors tend to zero by [F1, F3]; the endpoint energies converge by dominated convergence. Since ${\chi }_R\uparrow 1$, monotone convergence applies to the nonnegative dissipation and gives $\frac{1}{2}\mathrm{\parallel }{u}^{\epsilon }(t){\mathrm{\parallel }}_2^2+\epsilon {\int }_s^t\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\int \mathrm{\mid }\mathrm{\nabla }{u}^{\epsilon }{\mathrm{\mid }}^2=\frac{1}{2}\mathrm{\parallel }{u}^{\epsilon }(s){\mathrm{\parallel }}_2^2\mathrm{.}$ In particular the dissipation is finite on every positive-time interval.

Signed mass and $L^1$ bound. In step 1.2 the right side tends to $0$ when $R\to \mathrm{\infty }$, while $u^{\epsilon }(t){\chi }_R\to u^{\epsilon }(t)$ and $u_0{\chi }_R\to u_0$ pointwise with $\mathrm{\mid }{u}^{\epsilon }(t)\mathrm{\mid }{\chi }_R\le \mathrm{\mid }{u}^{\epsilon }(t)\mathrm{\mid }$ and $\mathrm{\mid }{u}_0\mathrm{\mid }{\chi }_R\le \mathrm{\mid }{u}_0\mathrm{\mid }$, and both majorants are integrable by [F1]; dominated convergence gives $\int u^{\epsilon }(t)=\int u_0$, which is signed-mass conservation. For the $L^1$ bound let ${\eta }_{\delta }(s)=\sqrt{s^2+{\delta }^2}-\delta$, a convex $C^2$ function with $0\le {\eta }_{\delta }\le \mathrm{\mid }\cdot \mathrm{\mid }$, ${\eta }_{\delta }\uparrow \mathrm{\mid }\cdot \mathrm{\mid }$ as $\delta \downarrow 0$, and let $q_{\delta }^{\mathrm{\prime }}={\eta }_{\delta }^{\mathrm{\prime }}{f}^{\mathrm{\prime }}$ with $q_{\delta }(0)=0$. Testing the balance [F2] with ${\chi }_R$, integrating first on a positive-time interval and passing its lower endpoint to zero by the Lipschitz entropy and $L^1$ continuity and using $\mathrm{\mid }{q}_{\delta }(s)\mathrm{\mid }\le L_M\mathrm{\mid }s\mathrm{\mid }$ and ${\eta }_{\delta }(s)\le \mathrm{\mid }s\mathrm{\mid }$ on the range of [F1], the cutoff terms vanish in the limit $R\to \mathrm{\infty }$ exactly as in step 1.2, and dropping the nonpositive dissipation gives $\int {\eta }_{\delta }(u^{\epsilon }(t))\le \int {\eta }_{\delta }(u_0)$; monotone convergence in $\delta \downarrow 0$ yields $\mathrm{\parallel }{u}^{\epsilon }(t){\mathrm{\parallel }}_1\le \mathrm{\parallel }{u}_0{\mathrm{\parallel }}_1$.

Passage to the initial time. The construction gives $u^{\epsilon }\in C([0,T];L^1)$ and $\mathrm{\parallel }{u}^{\epsilon }(s){\mathrm{\parallel }}_{\mathrm{\infty }}\le M=\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{\mathrm{\infty }}$. Hence $\mathrm{\parallel }{u}^{\epsilon }(s)-u_0{\mathrm{\parallel }}_2^2\le 2M\mathrm{\parallel }{u}^{\epsilon }(s)-u_0{\mathrm{\parallel }}_1⟶0(s\downarrow 0),$ so the endpoint energy in step 1.3 converges to $\frac{1}{2}\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_2^2$. Letting $s\downarrow 0$, monotone convergence for the nonnegative space-time dissipation gives the identity in (iii) for every $t>0$; at $t=0$ it is immediate. Dropping the nonnegative final energy yields the stated uniform bound for $0<\epsilon \le 1$.