The Riemann problem for a class of nonlinear degenerate equations (Q439240)

The authors consider the system of conservation laws NEWLINE\[NEWLINE v_t -u_x=0, \;\;\;\;u_t + p(v)_x=0 NEWLINE\]NEWLINE with Riemann initial data \((v,u)|_{t=0}=(v_r,u_r)\) for \(x>0\), \((v,u)|_{t=0}=(v_l,u_l)\) for \(x<0\), where the unknowns \(v\) and \(u\) are the strain and velocity, respectively, and \(p(v)\) is the pressure. When the eigenvalues coalesce the system is degenerated. Here the function \(p(v)\) is piecewise smooth and monotone decreasing as it is equal to constant in some interval \([\alpha , \beta ]\). This case corresponds to the Maxwell line in the van der Waals gas state equation. Actually, \(p(v)\) satisfies the conditions:NEWLINENEWLINE(1) \( \;\;\) \( p'(v)<0\), \(v\in (-\infty ,\alpha )\cup (\beta ,+\infty ) \); \( p'(v)=0\), \(v\in [\alpha , \beta ]\),NEWLINENEWLINE(2) \( \;\;\) \( p''(v)>0\), \(v\in (-\infty ,\alpha )\); \( p''(v)<0\), \(v\in (\beta ,+\infty ) \); \( p''(v)=0\), \(v\in [\alpha , \beta ]\).NEWLINENEWLINEHere the system is strictly hyperbolic, except for \(v\in [\alpha , \beta ]\), where the two eigenvalues coalesce. The Riemann invariants along to characteristic fields are NEWLINE\[NEWLINE\omega =u-\int\limits_{\alpha }^{v}\sqrt{-p'(s)}ds \;\text{and} \;z =u+\int\limits_{\alpha }^{v}\sqrt{-p'(s)}ds.NEWLINE\]NEWLINE It is shown that there exist shock waves satisfying the Liu-entropy condition. Also the Riemann solutions of twelve regions in the \(v\)-\(u\) plane are completely constructed. The main result is that the set \(\Sigma =\{(v,u): \;c_1\leq \omega \leq c_2, \;c_1\leq z \leq c_2\}\) is an invariant region for the Rieman problem under consideration, i.e. if the Riemann data belong to \(\Sigma \), then the Riemann solutions belong to \(\Sigma \) too.