Asymptotic behavior of solutions to scalar conservation laws and optimal convergence orders to \(N\)-waves
The author of this interesting paper develops a new technique to obtain better asymptotic estimates for scalar conservation laws, which are hyperbolic partial differential equations of first order, \( u_t+f(u)_x=0\). The Cauchy problem, \(u(x,0)=u_0(x)\), \(x\in \mathbb R\), \(t>0\) is considered with integrable initial value \(u_0\) defined on a compact support, \(u_0\in L^1(\mathbb R)\), supp \(u_0\subset [-L,L]\), \(L\in \mathbb R \). It is assumed that the flux \(f(u)\) is convex, \(f(0)=f^{\prime }(0)=0\), \(f^{\prime\prime }\geq 0\), and has an algebraic growth of order \(\gamma >1\) near the zero state \(u=0\) in the sense that \(\lim\limits_{u\to 0}uf^{\prime }(u)/f(u)=\gamma \), \(\gamma >1\). It is shown that under suitable conditions on the initial value, the solution of the considered equation converges to an \(N-\)wave in \(L^1\) norm with optimal convergence order of \(O(1/t)\). The technique applied here is to enclose the solution with two rarefaction waves. A uniform convergence order in the sense of graphs is shown as well. An interesting numerical example is given to illustrate the theory and considered phenomenon.
scientific article; zbMATH DE number 1957263