On the spreading of characteristics for non-convex conservation laws (Q2764661)

The Cauchy problem for the scalar conservation law \(u_t +f(u)_x =0\) is considered. If \(f^{''}(u)\geq c\), the Oleinik estimate says that \(u(y,t)-u(x,t)\leq\) \(\frac{y-x}{ct}\), \(x\leq y\), \(t>0.\) A more accurate analysis shows that NEWLINE\[NEWLINEf^{'}(u(y,t))-f^{'}(u(x,t))\leq \frac{y-x}{t}. \tag{1}NEWLINE\]NEWLINE However, when \(f\) is not a convex inequality (1) fails due to the fact that the solution \(u\) may exhibit contact discontinuities. The authors provide a suitable weakened form of (1) that is also valid in the non-convex case. The assumptions are \(f(0)=\) \(f^{'}(0)=\) \(f^{''}(0)=0,\) \(uf^{''}(u)<0\) for \(u\neq 0.\) It is proved that inequality (1) is still valid away from contact discontinuities, while near such a discontinuity the function \(f^{'}(u(x,t))\) satisfies an estimate like (1), with the right-hand side replaced by \(c\sqrt {y-x}\) for some constant \(c\). As another generalization, it is proved that \(f^{'}(u(y,t))-\) \(f^{'}(u(x,t))\leq\) \(\frac{y-x}{t}+\) \(cTVu_0 -\) \(cTVu(\cdot ,t),\) \(x\leq y ,\) \(t>0.\)