An alternate relaxation approximation to conservation laws
For the scalar conservation law \[ \partial_t u + \partial_x f(u)=0, \;(x,t) \in\mathbb R \times (0; \infty),\;u(x,0)=u_0(x),\;x \in \mathbb R, \] where \(u_0, f\) are some smooth functions, the author proposes a new way of regularization: \[ \begin{aligned} \partial_tu_\varepsilon + \partial_x v_\varepsilon &= v_\varepsilon - f(u_\varepsilon),\;\partial_t v_\varepsilon + a^2 \partial_x u_\varepsilon = -\frac{1}{\varepsilon} \left(v_\varepsilon-f(u_\varepsilon)\right),\;(x,t) \in \mathbb R \times (0; \infty),\\ u_\varepsilon(x,0)&=u_0(x),\;v_\varepsilon(x,0)=f\left(u_0(x)\right),\;x \in \mathbb{R}.\end{aligned}\tag{1} \] A nice feature of (1) is that it satisfies the conservation law \[ \partial_\varepsilon\left(u_\varepsilon+\varepsilon v_\varepsilon\right)+\partial_x(v_\varepsilon+ \varepsilon a^2 u_\varepsilon)=0. \] Based on these assumptions the author proves the theorem: Assume \(f(u)\) is uniformly Lipschitz with \(| f'(u)| <a\) for all \(u\) and \(f(0)=0\) then for any \(N_0>0\), \(\varepsilon >0\) with \(a\varepsilon <1\) there exist an unique, globally bounded solution \((u_\varepsilon, v_\varepsilon)\) to (1) in \(C([0; \infty); L_{loc}^1(\mathbb{R}^2))\) with the estimates \(\| u_\varepsilon \pm a v_\varepsilon\| _{L^\infty\left(\mathbb{R}\times (0; \infty)\right)} \leq 2N_0 \frac{1+a}{1-a\varepsilon}\). A convex entropy is also constructed.
scientific article; zbMATH DE number 5151712