[Math] Lax entropy condition and uniqueness for shallow water equations

fluid dynamicshyperbolic-equationspartial differential equations

Why is the Lax entropy condition $${\lambda _i}({{\mathbf{u}}_R}) \leqslant {\sigma _i} \leqslant {\lambda _i}({{\mathbf{u}}_L}),$$ where $i = 1,2$, a sufficient condition for uniqueness of the solution of the shallow water equations
$$\begin{split}
{h_t} + {\left( {hu} \right)_x} = 0, \hfill \\
{\left( {hu} \right)_t} + {\left( {h{u^2} + \tfrac{1}{2}g{h^2}} \right)_x} = 0 \hfill \\
\end{split} $$
?

Here $\sigma_i$ is the i-th shock speed and $\lambda_i$ is the i-th characteristic speed of the shallow water equations.

Furthermore, ${{\mathbf{u}}_L}$ and ${{\mathbf{u}}_R}$ are the states to the left and right of the shocks respectively, so ${\lambda _1}({{\mathbf{u}}_L}) = {u_L} – \sqrt {g{h_L}} ,\;\;\;{\lambda _2}({{\mathbf{u}}_L}) = {u_L} + \sqrt {g{h_L}} ,\;\;\;{\lambda _1}({{\mathbf{u}}_R}) = {u_R} – \sqrt {g{h_R}} ,\;\;\;{\lambda _2}({{\mathbf{u}}_R}) = {u_R} + \sqrt {g{h_R}} $ where $u_L, u_R$ are the fluid speeds to the left and right of the shocks, and $h_L, h_R$ are the heights of the fluid columns to the left and right of the shocks. Also, ${\sigma _i} = {u_L} \pm {h_R}\sqrt {\frac{g}{2}\left( {\frac{1}{{{h_R}}} + \frac{1}{{{h_L}}}} \right)} = {u_R} \pm {h_L}\sqrt {\frac{g}{2}\left( {\frac{1}{{{h_R}}} + \frac{1}{{{h_L}}}} \right)}$ where $-$ corresponds to $i=1$ and $+$ corresponds to $i=2$ are the shock speeds.

I'm reading some lecture notes (not in English, so there's no point in uploading them here) where it just says that it's easy to show that the Lax entropy conditions are sufficient for uniqueness of solution of the shallow water equations (no proof), but I don't see why this is true.
Are there any books where I can read more about this?

Best Answer

From (1) p. 274

For the shallow water equations, the characteristic speed $\lambda_p(\tilde q(\xi))$ varies monotonically as we move along an integral curve. [...] If $\lambda_p(\tilde q(\xi))$ varies monotonically with $ξ$ along every integral curve, then we say that the $p$th field is genuinely nonlinear.

Existence and uniqueness results need specifications. Are we speaking of classical solutions or weak solutions? Are we speaking of the Cauchy problem (general IVP) or of the Riemann problem (particular IVP)? In the latter case, there is the following result ((2) p. 84):

Assume that for all $k \in 1, ... , p$, the $k$th characteristic field is either genuinely nonlinear or linearly degenerate. Then for all ${\bf u}_L \in \Omega$ there exists a neighborhood $\vartheta$ of ${\bf u}_L$ in $\Omega$ with the following property: If ${\bf u}_R$ belongs to $\vartheta$, the Riemann problem (6.1) has a weak solution that consists of at most $(p + 1)$ constant states separated by rarefaction waves, admissible shock waves, or contact discontinuities. Moreover, a weak solution of this kind is unique.

Here, "admissible" means satisfying the Lax entropy condition.

More results of this king can be found in (3).

(1) R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002 doi:10.1017/CBO9780511791253

(2) E. Godlewski, P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Springer, 1996 doi:10.1007/978-1-4612-0713-9

(3) C.M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 4th ed., Springer, 2016 doi:10.1007/978-3-662-49451-6

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