Conservation laws with continuous flux functions
The authors investigate the uniqueness problem for generalized entropy solutions (in the sense of S. Kruzhkov) of the stationary problem \[ u+ \text{div}_x\varphi(u)=f \quad \text{on } {\mathbb R}^n \tag{E} \] and the corresponding Cauchy problem \[ u_t+ \text{div}_x\varphi(u)=f \quad \text{on } (0,T)\times{\mathbb R}^n, \qquad u(0,x)=u_0(x). \tag{CP} \] The flux vector \(\varphi(u)=(\varphi_1(u),\ldots,\varphi_n(u))\) is assumed to be only continuous. In this case the effect of infinite speed of propagation generally appears which can lead even to nonuniqueness of generalized entropy solutions (g.e.s.). The authors prove the uniqueness of g.e.s. under the following anisotropic condition on the flux functions: \[ \liminf_{r\to 0} r^{1-n}\prod_{i=1}^n \omega_i(r)<\infty, \] where \(\omega_i(r)\), \(i=1,\ldots,n\) are subadditive increasing functions on \([0,\infty)\) such that for all \(u,v\) \(|\varphi_i(u)-\varphi_i(v)|\leq\omega_i(|u-v|)\). Moreover it is proved that under the condition above \[ \int_{{\mathbb R}^n}(u-v)^+ dx\leq \int_{\{u>v\}}(f-g) dx+ \int_{\{u=v\}}(f-g)^+ dx, \] where \(u\) is a generalized entropy sub-solution, \(v\) is a generalized entropy super-solution of the problem (E) with the source functions \(f\), \(g\) respectively. The similar result is proved for the problem (CP) as well. This property is also established without any restrictions on the flux vector if one of the function \(u\) or \(v\) doesn't depend on \(x\). In this case it implies estimates of \(L^p\)-norms \(\|u\|_p\leq \|f\|_p\) and \(\|u(t,\cdot)\|_p\leq \|u_0\|_p+\int_0^t\|f(\tau,\cdot)\|_p d \tau\) for g.e.s. of the problems (E) and (CP) respectively for all \(1\leq p\leq\infty\). The authors also establish uniqueness of g.e.s. for the special case when \(\varphi_i(u)=|u|^{\alpha_i-1}u\), \(\alpha_i>0\), \(i=1,\ldots,n\), and the input functions are integrable with respect to spatial variables.
generalized entropy sub- and super-solutions
scientific article; zbMATH DE number 952684