The $L^{\mathrm{\infty }}$ maximum bound for entropy solutions

Given: Countable Choice, $n\ge 1$, $T>0$, a locally Lipschitz $C^1$ flux $f$, a bounded Kruzhkov entropy solution $u$ on ${\mathrm{\Pi }}_T$ with datum $u_0\in L^{\mathrm{\infty }}({\mathbb{R}}^n)$, and the essential bounds $m_0=\mathrm{ess\hspace{0.17em}inf}{u}_0$, $M_0=\mathrm{ess\hspace{0.17em}sup}{u}_0$, both finite.

Constant functions on ${\mathrm{\Pi }}_T$ are Kruzhkov entropy solutions with their own constant value as initial datum, for every flux: for $u\equiv c$ the weak equation is the equality ${\mathrm{\partial }}_{t}c+{\mathrm{div}}_{x}f(c)=0$, and for every $k$ the functions ${\eta }_k(c)=\mathrm{\mid }c-k\mathrm{\mid }$ and $q_k(c)=\mathrm{sgn}(c-k)(f(c)-f(k))$ are constant in $(t,x)$, so ${\mathrm{\partial }}_t{\eta }_k(c)+{\mathrm{div}}_x{q}_k(c)=0\le 0$ in distributions; the strong local $L^1$ trace of the constant $c$ is the constant $c$, with ${\int }_K\mathrm{\mid }c-c\mathrm{\mid }dx=0$ for every compact $K$ (Kruzhkov entropy solutions).

Order preservation: if two bounded Kruzhkov entropy solutions $v,w$ on ${\mathrm{\Pi }}_T$ have $\mathrm{\mid }v\mathrm{\mid },\mathrm{\mid }w\mathrm{\mid }\le M$ and $v_0\le w_0$ almost everywhere, then $v\le w$ almost everywhere on ${\mathrm{\Pi }}_T$ (Uniqueness, comparison and order preservation of entropy solutions).

The cited essential-supremum definition defines $\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{\mathrm{\infty }}$ using bounds on $\mathrm{\mid }{u}_0\mathrm{\mid }$ (The essential supremum of a measurable function with respect to a measure). Here define the signed extrema explicitly by $M_0=\inf \{b\in \mathbb{R}:u_0\le b\text{ a.e.}\}$ and $m_0=\sup \{a\in \mathbb{R}:a\le u_0\text{ a.e.}\}$. Since $u_0$ is essentially bounded on the nonnull space ${\mathbb{R}}^n$, these are finite. For each integer $j\ge 1$, the infimum property gives an essential upper bound below $M_0+1\mathrm{/}j$, so $u_0\le M_0+1\mathrm{/}j$ a.e.; the supremum property similarly gives $m_0-1\mathrm{/}j\le u_0$ a.e. Discarding the countable union of exceptional null sets and letting $j\to \mathrm{\infty }$ yields $m_0\le u_0\le M_0$ a.e. Thus $\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{\mathrm{\infty }}\le \max \{\mathrm{\mid }{m}_0\mathrm{\mid },\mathrm{\mid }{M}_0\mathrm{\mid }\}$. Conversely every essential absolute bound $B$ gives $m_0\ge -B$ and $M_0\le B$, so $\max \{\mathrm{\mid }{m}_0\mathrm{\mid },\mathrm{\mid }{M}_0\mathrm{\mid }\}\le B$; taking its infimum proves equality. Inequalities between $L^{\mathrm{\infty }}$ classes are a.e. (The space $L^p(\mu )$ as the quotient by null functions). Fubini transfers null sets to spatial slices for a.e. time (Fubini's theorem for L^1 functions on a sigma-finite product).