Past exam of the differential equations and applications course of the University of Oxford / 2010 / B / 5

Write the system as $U_x+A(U)U_y=0$ with $U=(u,v{)}^T$ andIts two characteristic speeds are ${\lambda }_{\pm }=v\pm \sqrt{u}$, so it is strictly hyperbolic precisely when $u>0$. At $u=0$ they coincide and the matrix is not diagonalizable; for $u<0$ the speeds are nonreal.

For $u>0$ define $R_{\pm }=v\pm 2\sqrt{u}$ and $D_{\pm }={\partial }_x+(v\pm \sqrt{u}){\partial }_y$. Substitution from the two differential equations givesThereforeThese are the two Riemann invariants; their constancy applies within smooth regions, not across a shock.

For a smooth solution, the first equation is $u_x+(uv{)}_y=0$. Multiplying it by $v$ and adding $u$ times the second equation givesThus the appropriate shallow water equations conservation variables and flux areUse this conservative extension to define a weak solution across $y=F(x)$. A jump contributes $-F^{\prime }[W]\delta (y-F(x))$ to $W_x$ and $[H]\delta (y-F(x))$ to $H_y$. Cancellation of the singular parts gives the Rankine-Hugoniot conditionsEquivalently this follows by integrating each conservation law over a thin space-time control region crossing the discontinuity.

The conservative extension is essential here. Piecewise smoothness by itself does not imply any jump law. Moreover the smooth second equation can also be written $v_x+(u+v^2/2{)}_y=0$; imposing that different conservation law distributionally would give a different shock condition. Weak conservation laws depend on conserved variables. The requested jump law specifically selects mass $u$ and momentum $uv$, and this is the extension used below.

Test a single discontinuity connecting the two constant initial states. With state 1 below the discontinuity and state 2 above it, their jumps areThe first Rankine-Hugoniot condition gives $F^{\prime }=\sqrt{3}$, and the second independently gives $(-\sqrt{3})F^{\prime }=-3$, the same value. Since the initial jump is at the origin,Values on the shock itself do not affect the weak solution. No intermediate state or rarefaction is needed: the prescribed states already lie on the same compressive shock curve.

For admissibility, the upper-family characteristic speed is $\sqrt{3}/2+\sqrt{2}$ behind the shock and $1$ ahead of it. They satisfyBoth lower-family speeds, $-1$ and $\sqrt{3}/2-\sqrt{2}$, are less than the shock speed. These are the Lax shock admissibility conditions for a family-plus shock. This is the entropy-admissible solution for the mass/momentum conservative extension specified by the preceding jump law; imposing an additional velocity conservation law across the same jump would describe a different problem.