Convergence rate for a regularized scalar conservation law (Q6541360)

This paper deals with the behavior, as \(\ell\) tends to zero, of solutions to the regularized scalar conservation laws \N\[\Nu^\ell_t +f(u^\ell)_x+\ell^2 P^\ell_x =0, \quad P^\ell -\ell^2 P^\ell = \frac{1}{2}f'' (u^\ell) (u^\ell_x)^2, \quad u^\ell(0,\cdot)=u_0. \N\]\NThe authors assume that the flux \(f\) satisfies \N\[\Nf \in C^4 (\mathbb{R}),\qquad 0<c_1 \le f''(\cdot)\le c_2 <\infty, \N\]\Nfor some given positive constants \(c_1\) and \(c_2\). On the initial datum they assume \N\[\Nu_0\in H^1(\mathbb{R}),\qquad u_0'\in L^1(\mathbb{R}),\qquad \sup_{\mathbb{R}}u_0'<\infty. \N\]\NThey prove that, for every \(T>0\), \(\ell\in(0,1]\), and \(p\in [1,\infty)\) \N\[\N\| u^\ell - u\|_{L^\infty ([0,T ];L^p (\mathbb{R}))}\le C\ell^{1/(2\ell)} \N\]\Nfor some constant \(C\), where \(u\) is the unique entropy solution of \N\[\Nu_t +f(u)_x =0, \quad u(0,\cdot)=u_0.\N\]