Existence of bounded Kruzhkov entropy solutions

Normalization and smooth flux approximation. By [F1] it suffices to treat $\tilde{f}=f-f(0)$. Choose a radial mollifier and, for $m\ge 1$, set $f_m(s)=\chi (s)(\tilde{f}\ast {\rho }_{1\mathrm{/}m})(s)-(\tilde{f}\ast {\rho }_{1\mathrm{/}m})(0)$, where $\chi \in C_c^{\mathrm{\infty }}$ equals $1$ on $[-M-\frac{1}{2},M+\frac{1}{2}]$ and is supported in $[-M-1,M+1]$. Then each $f_m$ is $C^{\mathrm{\infty }}$ (hence $C^2$), $f_m(0)=0$, and on $[-M-\frac{1}{2},M+\frac{1}{2}]$ one has $f_m\to \tilde{f}$ and $f_m^{\mathrm{\prime }}={\tilde{f}}^{\mathrm{\prime }}\ast {\rho }_{1\mathrm{/}m}\to {\tilde{f}}^{\mathrm{\prime }}$ uniformly by [F4]; in particular ${\sup }_m{\sup }_{\mathrm{\mid }s\mathrm{\mid }\le M+1}\mathrm{\mid }{f}_m^{\mathrm{\prime }}(s)\mathrm{\mid }<\mathrm{\infty }$.

Smooth-datum case: extraction. Let $u_0\in C_c^{\mathrm{\infty }}({\mathbb{R}}^n)$ with $\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{\mathrm{\infty }}=M$, choose ${\epsilon }_m\downarrow 0$ with $0<{\epsilon }_m\le 1$, and let $u^m$ be the global classical solution with flux $f_m$ and datum $u_0$ given by [F2]. Then $\mathrm{\mid }{u}^m(t,x)\mathrm{\mid }\le \mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{\mathrm{\infty }}=M$ by the range bound of [F2], so the hypotheses of [F5] are met; passing to a subsequence (relabelled) there is $u\in L^{\mathrm{\infty }}\cap L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1({\mathrm{\Pi }}_T)$ with $\mathrm{\parallel }u{\mathrm{\parallel }}_{\mathrm{\infty }}\le M$, $u^m\to u$ in $L^1(K)$ for every compact $K⋐{\mathrm{\Pi }}_T$ and almost everywhere, and $u$ has a representative in $C([0,T];L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1)$ with $u(0)=u_0$ in $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1$. At every time $t$ this representative obeys $\mathrm{\mid }u(t,\cdot )\mathrm{\mid }\le M$ almost everywhere: for $t$ in the full-measure set where the construction of [F5] passes the bound, slicewise almost-everywhere convergence preserves it, and general $t$ follows by $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1$-continuity.

The weak equation passes to the limit. For $\phi \in C_c^{\mathrm{\infty }}({\mathrm{\Pi }}_T)$, testing the pointwise viscous equation of $u^m$ gives ${\int }_{{\mathrm{\Pi }}_T}(u^m{\phi }_t+f_m(u^m)\cdot \mathrm{\nabla }\phi )=-{\epsilon }_m{\int }_{{\mathrm{\Pi }}_T}{u}^m\mathrm{\Delta }\phi$, whose right side is bounded by ${\epsilon }_{m}M\mathrm{\parallel }\mathrm{\Delta }\phi {\mathrm{\parallel }}_1\to 0$. The left side converges to ${\int }_{{\mathrm{\Pi }}_T}(u{\phi }_t+\tilde{f}(u)\cdot \mathrm{\nabla }\phi )$ by [F7], because $u^m\to u$ almost everywhere with uniform bounds and $f_m\to \tilde{f}$ uniformly on $[-M,M]$; hence $u$ is a weak solution for $\tilde{f}$, and therefore for $f$ by [F1].

Entropy inequalities for $\mathrm{\mid }k\mathrm{\mid }\le M$. Fix $k\in [-M,M]$ and $\delta >0$, and put ${\eta }_{\delta }(r)=\sqrt{r^2+{\delta }^2}-\delta$, a convex $C^2$ function with $0\le {\eta }_{\delta }\le \mathrm{\mid }\cdot \mathrm{\mid }$, $\mathrm{\mid }{\eta }_{\delta }(r)-\mathrm{\mid }r\mathrm{\mid }\mathrm{\mid }\le \delta$, and ${\eta }_{\delta }^{\mathrm{\prime }\mathrm{\prime }}\ge 0$; let $q_{\delta ,k,m}(s)={\int }_k^s{\eta }_{\delta }^{\mathrm{\prime }}(r-k)f_m^{\mathrm{\prime }}(r)dr$, so $q_{\delta ,k,m}^{\mathrm{\prime }}={\eta }_{\delta }^{\mathrm{\prime }}(\cdot -k)f_m^{\mathrm{\prime }}$. Let $\phi \in C_c^{\mathrm{\infty }}({\mathbb{R}}^n\times [0,T))$ be nonnegative. Multiplying the exact viscous balance [F3] for the pair $({\eta }_{\delta }(\cdot -k),q_{\delta ,k,m})$ by $\phi$, integrating over ${\mathrm{\Pi }}_T$ and integrating the Laplacian by parts gives $I_{m,\delta }:={\int }_{{\mathrm{\Pi }}_T}({\eta }_{\delta }(u^m-k){\phi }_t+q_{\delta ,k,m}(u^m)\cdot \mathrm{\nabla }\phi )+{\int }_{{\mathbb{R}}^n}{\eta }_{\delta }(u_0-k)\phi (x,0)dx=-{\epsilon }_m{\int }_{{\mathrm{\Pi }}_T}{\eta }_{\delta }(u^m-k)\mathrm{\Delta }\phi +{\epsilon }_m{\int }_{{\mathrm{\Pi }}_T}{\eta }_{\delta }^{\mathrm{\prime }\mathrm{\prime }}(u^m-k)\mathrm{\mid }\mathrm{\nabla }{u}^m{\mathrm{\mid }}^2\phi$ — the only boundary term is the one at $t=0$, displayed with the initial datum; the last term is nonnegative and the first is bounded by ${\epsilon }_m\cdot 2M\mathrm{\parallel }\mathrm{\Delta }\phi {\mathrm{\parallel }}_1\to 0$, so ${\mathrm{lim\hspace{0.17em}inf}}_m{I}_{m,\delta }\ge 0$. On the other hand $I_{m,\delta }$ converges by [F7]: $u^m\to u$ almost everywhere, $\mathrm{\mid }{\eta }_{\delta }(u^m-k)\mathrm{\mid }\le 2M$, and $q_{\delta ,k,m}\to q_{\delta ,k}$ uniformly on $[-M,M]$, where $q_{\delta ,k}(s)={\int }_k^s{\eta }_{\delta }^{\mathrm{\prime }}(r-k){\tilde{f}}^{\mathrm{\prime }}(r)dr$, so the limit obeys $I_{\delta }={\int }_{{\mathrm{\Pi }}_T}({\eta }_{\delta }(u-k){\phi }_t+q_{\delta ,k}(u)\cdot \mathrm{\nabla }\phi )+{\int }_{{\mathbb{R}}^n}{\eta }_{\delta }(u_0-k)\phi (x,0)dx\ge 0$.

The Kruzhkov inequalities. Letting $\delta \downarrow 0$ in step 2.2, ${\eta }_{\delta }(r-k)\to \mathrm{\mid }r-k\mathrm{\mid }$ uniformly on $[-M,M]$ and $q_{\delta ,k}(s)\to {\int }_k^s\mathrm{sgn}(r-k){\tilde{f}}^{\mathrm{\prime }}(r)dr=\mathrm{sgn}(s-k)(\tilde{f}(s)-\tilde{f}(k))=q_k(s)$ uniformly on $[-M,M]$ by dominated convergence, since $\mathrm{\mid }{\eta }_{\delta }^{\mathrm{\prime }}\mathrm{\mid }\le 1$ and ${\tilde{f}}^{\mathrm{\prime }}$ is continuous there; hence dominated convergence gives ${\int }_{{\mathrm{\Pi }}_T}(\mathrm{\mid }u-k\mathrm{\mid }{\phi }_t+q_k(u)\cdot \mathrm{\nabla }\phi )+\int \mathrm{\mid }{u}_0-k\mathrm{\mid }\phi (x,0)dx\ge 0$ for every nonnegative $\phi \in C_c^{\mathrm{\infty }}({\mathbb{R}}^n\times [0,T))$ and every $k\in [-M,M]$. If $k>M$, then $u-k<0$ almost everywhere, ${\eta }_k$ is affine on the range $[-M,M]$, and $q_k(s)=\tilde{f}(k)-\tilde{f}(s)$, so the identity ${\mathrm{\partial }}_t{\eta }_k(u)+{\mathrm{div}}_x{q}_k(u)=-(u_t+{\mathrm{div}}_x\tilde{f}(u))=0$ holds by step 2.1, and similarly for $k<-M$; thus all Kruzhkov inequalities hold and, with the trace of step 1.2, $u$ is a bounded Kruzhkov entropy solution with datum $u_0$.

General datum: approximation and Cauchy property. Now let $u_0\in L^{\mathrm{\infty }}\cap L^1$ be arbitrary. By [F4] choose $u_0^r\in C_c^{\mathrm{\infty }}$ with $\mathrm{\parallel }{u}_0^r{\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{\mathrm{\infty }}$ and $u_0^r\to u_0$ in $L^1$ (truncate $u_0$ to a large ball and mollify). Steps 1.1–3.1 applied to each smooth datum $u_0^r$ give bounded Kruzhkov entropy solutions $u^r$ for the flux $f$, with $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1$-continuous representatives and $\mathrm{\parallel }{u}^r(t,\cdot ){\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\parallel }{u}_0^r{\mathrm{\parallel }}_{\mathrm{\infty }}\le M$. By [F6], for every $t\in [0,T]$, $\mathrm{\parallel }{u}^r(t)-u^{\mathrm{\ell }}(t){\mathrm{\parallel }}_1\le \mathrm{\parallel }{u}_0^r-u_0^{\mathrm{\ell }}{\mathrm{\parallel }}_1$, so ${\sup }_{t\in [0,T]}\mathrm{\parallel }{u}^r(t)-u^{\mathrm{\ell }}(t){\mathrm{\parallel }}_{L^1(K)}\le \mathrm{\parallel }{u}_0^r-u_0^{\mathrm{\ell }}{\mathrm{\parallel }}_1$ for every compact $K$, and the right side tends to $0$ as $r,\mathrm{\ell }\to \mathrm{\infty }$.

The limit for general datum. By completeness of $L^1(K)$ [F7] and the uniform-in-time contraction in step 4.1, $u^r$ converges in $C([0,T];L^1(K))$ for each compact ball $K$, consistently on nested balls, to $u\in C([0,T];L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1({\mathbb{R}}^n))$. Since $u^r(0)=u_0^r\to u_0$ in $L^1$, this representative has initial trace $u_0$. For every $t$ and ball $K$, the inequality $(\mathrm{\mid }u(t)\mathrm{\mid }-M{)}_{+}\le \mathrm{\mid }u(t)-u^r(t)\mathrm{\mid }+(\mathrm{\mid }{u}^r(t)\mathrm{\mid }-M{)}_{+}$ and $\mathrm{\parallel }{u}^r(t){\mathrm{\parallel }}_{\mathrm{\infty }}\le M$ show, after integration on $K$ and passage to the $L^1(K)$ limit, that $(\mathrm{\mid }u(t)\mathrm{\mid }-M{)}_{+}=0$ almost everywhere there. Thus $\mathrm{\parallel }u(t){\mathrm{\parallel }}_{\mathrm{\infty }}\le M$ for every $t$. The weak equation passes to the limit because $f$ is Lipschitz on $[-M,M]$ and $u^r\to u$ in local $L^1$. For each fixed $k\in \mathbb{R}$, $s\mapsto \mathrm{\mid }s-k\mathrm{\mid }$ is $1$-Lipschitz and $q_k(s)=\mathrm{sgn}(s-k)(f(s)-f(k))$ is Lipschitz on $[-M,M]$ with constant at most ${\sup }_{\mathrm{\mid }s\mathrm{\mid }\le M}\mathrm{\mid }{f}^{\mathrm{\prime }}(s)\mathrm{\mid }$. Therefore the entropy and flux terms in the inequality of step 3.1 converge in $L^1$ on every test support; the initial entropy term converges by $u_0^r\to u_0$ in $L^1$ and the same Lipschitz bound for $\mathrm{\mid }\cdot -k\mathrm{\mid }$. Passing to the limit proves every Kruzhkov inequality without requiring an almost-everywhere subsequence for the general-data approximation. Hence $u$ is a bounded Kruzhkov entropy solution with datum $u_0$.

Uniqueness and conclusion. If $v$ is another bounded Kruzhkov entropy solution with the same datum $u_0$, then $u=v$ almost everywhere on ${\mathrm{\Pi }}_T$ by [F6], so the constructed solution is the unique bounded Kruzhkov entropy solution with this datum; this completes the proof.