Scalar Conservation Laws
Summary
We being our treatment of scalar conservation laws:
$$ u_t + (F(u))_x = 0\quad \text{in}~\mathbb{R}^n\times(0,\infty) $$ for $u=u(x,t)$, $u\colon \mathbb{R}\times(0,\infty)\to \mathbb{R}$. Here $F\colon \mathbb{R}\to \mathbb{R}$ is given. Our focus is the initial-value problem (IVP)
$$ \left\{ \begin{alignedat}{2} u_t + (F(u))_x &= 0 \quad&&\text{in}~\mathbb{R}\times(0,\infty)\\ u&=g \quad &&\text{on}~\mathbb{R}\times\{t=0\} \end{alignedat} \right. $$
We first observe that if $(F’\circ g)(\zeta)<0$ for some $\zeta\in\mathbb{R}$, the projected characteristic curves intersect at finite time, which result in formation of discontinuities and gradient catastrophe. We then introduce the notion of a weak solution (an integral solution) to allow for discontinuities. We then cover Rankine-Hugoniot condition, which imposes restrictions on the behavior of a weak solution at a curve of discontinuity.
We then illustrate non-uniqueness of weak solutions, state the entropy condition and provide a glimpse of Lax-Oleinik theory, arriving at the definition of a shock and the notion of an entropy solution. We provide some examples as well.