# singular shock solutions for strictly hyperbolic system of conservation law

Shallow water equation $$\begin{eqnarray} \rho_t+q_x=0\\ q_t + \left(q^2 +\frac{\rho ^2}{2} \right)_x=0 \end{eqnarray}$$ being a strictly hyperbolic equation does not admit delta shock solutions.

Where as some strictly hyperbolic systems such as Chaplign gas equation given by $$\begin{eqnarray} \rho_t +q_x=0\\ q_t+\left( \frac{q^2-1}{\rho} \right)_x=0 \end{eqnarray}$$ admits delta shock solution.

Why is this so? What is the difference between these two equations?.

Why the procedure to get solution of the Riemann problem as a combination of shock and rarefaction for strictly hyperbolic equation as explained in "Numeric approximations of hyperbolic systems of conservation laws" by Godlewsky and Raviart, does not work for this equation?

## 2 Answers

I have no idea if your claims are true, and I am not a specialist of the two systems above. Nevertheless, one can make the following observations.

For the first system ("shallow water"), the eigenvalues of the flux's Jacobian matrix are $$\lambda_\pm = q \pm \sqrt{\rho + q^2}$$. Therefore the system is strictly hyperbolic if $$\rho + q^2 > 0$$. The gradient $$\nabla \lambda_\pm = \pm\left(\tfrac12, \lambda_\pm \right)^\top / \sqrt{\rho + q^2}$$ of the eigenvalue $$\lambda_\pm$$ is never orthogonal to the corresponding right eigenvector $$(-\lambda_\mp/\rho, 1)^\top$$ over the domain of hyperbolicity. Therefore, both characteristic fields are genuinely nonlinear, and the Riemann solution is a combination of shocks and rarefaction waves.

For the second system ("Chaplign gas equation"), the eigenvalues are $$(q\pm 1)/\rho$$. Therefore, the system is strictly hyperbolic if $$0 \neq |\rho| < +\infty$$. The gradient $$\nabla \lambda_\pm = \left(-\lambda_\pm, 1 \right)^\top /\rho$$ of the eigenvalue $$\lambda_\pm$$ is always orthogonal to the corresponding right eigenvector $$\left(1/\lambda_\pm, 1 \right)^\top$$ over the domain of hyperbolicity. Therefore, both characteristic fields are linearly degenerate. You may find the article (1) interesting, which may present a similar problem.

(1) H. Cheng, "Delta Shock Waves for a Linearly Degenerate Hyperbolic System of Conservation Laws of Keyfitz-Kranzer Type", Adv. Math. Phys. (2013), 958120 doi:10.1155/2013/958120

The procedure explained in "Numeric approximations of hyperbolic systems of conservation laws" by Godlewsky and Raviart is applicable for strictly hyperbolic systems provided $$U_L$$ and $$U_R$$ are sufficiently close. The method in general is not applicable for any $$U_L$$ and $$U_R.$$

Strict hyperbolicity helps in solving Riemann problem for the data whose TV is sufficiently small. For general data Riemann problems do not always admit bounded solutions.

However there are certain strictly hyperbolic systems like shallow water which admits bounded solutions for all the Riemann data. Chaplign gas equation does not come under this category. For this system Riemann problems admit bounded solutions for data with small total variation. But not for all Riemann data.

Note: However if we remove strict hyperbolicity assumptions there are systems which do not admit bounded solutions even for the Riemann data whose total variation is small. For example Pressureless gas equation.