On the vanishing viscosity approximation to the Cauchy problem for a \(2\times 2\) system of conservation laws
The author investigates the following quasilinear \(2 \times 2\) hyperbolic system of conservation laws: \[ u_ t + \bigl( (a + 1/2)u + f(v) \bigr)_ x = 0, \;v_ t + (uv)_ x = 0, \;x \in \mathbb{R},\;t>0, \;u(x,0) = u, \;v(x,0) = v, \tag{1} \] where \(a>1/2\), \(f \in C^ 2(\mathbb{R})\) is even, superlinear and for all \(v \neq 0\) satisfies \(f'(v)v>0\), \(f''(v)>0\). Under these assumptions the system (1) is not strictly hyperbolic (there is an umbilical point in (0,0)) and moreover (1) is not genuinely nonlinear along the \(u\) axis. First, existence of invariant domains, a priori estimates in \(L^ \infty\) and existence of infinitely many entropies are shown. Then existence of solutions of viscosity approximation to (1) and their strong convergence to the solution of (1) is proved using the method of compensated compactness.
scientific article; zbMATH DE number 554342
scientific article