Kruskov's theorem for a scalar conservation law

The scalar Cauchy problem

$$u_t + f(u)_x = 0, f \in C^1({\mathbb{R}})$$ (1)

with initial condition

$$u(0,x) = u_o(x), u_o \in L^\infty({\mathbb{R}})$$ (2)

has a unique entropy solution

$$u \in L^\infty({\mathbb{R}}_+ \times {\mathbb{R}})$$

which fulfills (important for numerics)

1. $\vert\vert u(t, \cdot)\vert\vert _{L^\infty} \le \vert\vert u_o \vert\vert _{L^\infty}$, a.e. in $t \in {\mathbb{R}}_+$ (Stability)
2. if $u_o \ge v_o$ a.e. in ${\mathbb{R}}$, then
   
   $$u(t, \cdot) \ge v(t, \cdot) \textrm{ a.e. in } {\mathbb{R}}, \textrm{ a.e. in t} \in {\mathbb{R}}_+$$
   
   
   (monotone)
3. if $u_o \in BV({\mathbb{R}})$ then
   
   $$u(t, \cdot) \in BV({\mathbb{R}}) \textrm{ and } TV(u(t, \cdot)) \le TV(u_o)$$
   
   
   (TV-Diminishing)
4. if $u_o \in L^1({\mathbb{R}})$ then
   
   $$\int_{{\mathbb{R}}} u(t,x) \textrm{d}x = \int_{\mathbb{R}}u_o(x) \textrm{d}x, \textrm{ a.e. in } t \in {\mathbb{R}}_+$$
   
   
   (conservation)
5. if $u$, $v$ are two entropy solutions, $u_o, v_o \in L^\infty$ and
   
   $$M = \max\{ \vert f^\prime(\phi) \vert : \vert \phi \vert \le... ...ert\vert _{L^\infty}, \vert\vert v_o\vert\vert _{L^\infty}) \}$$
   
   
   then
   
   $$\int_{\vert x\vert \le R} \vert u(t,x) - v(t,x) \vert \textr... ...ert x\vert \le R + Mt} \vert u_o(x) - v_o(x) \vert \textrm{d}x$$
   
   
   (finite domain of dependence)