Ngô Quốc Anh

April 20, 2010

Entropy conditions for conservation laws

Filed under: PDEs — Ngô Quốc Anh @ 0:46

This entry deals with the conservation law $u_t+F(u)_x=0$

and the problem of determining which weak solutions of the above equation are to be considered acceptable. We shall use the idea of vanishing viscosity whereby one considers instead the equation $u_t^\varepsilon+F(u^\varepsilon)_x=\varepsilon u_{xx}^\varepsilon$

in the limit as $\varepsilon$ decreases to zero.

We consider a jump discontinuity in the solution $u$ with left state $u_l$ and right state $u_r$ and shock speed $s'$ satisfying the Rankine-Hugoniot condition $\displaystyle s'=\frac{F(u_l)-F(u_r)}{u_l-u_r}$

and the problem is to determine further conditions on the shock that will hopefully lead to uniqueness as well of existence of weak solutions to the initial value problem for the problem.

Traveling waves. One way to get an answer to our problem is to look for traveling wave solutions to our problem, i.e., solutions on the form $\displaystyle u^\varepsilon(x,t)=w\left( \frac{x-s' t}{\varepsilon}\right)$.

Substituting this into the equation gives $\displaystyle -s' w'+F(w)'=w''$.

We can integrate this once to get $\displaystyle w'(\xi) = F(w(\xi))-s' w(\xi)-C$.

Now we get $\displaystyle\mathop {\lim }\limits_{\varepsilon \searrow 0} {u^\varepsilon }(x,t) = \left\{ \begin{gathered} \mathop {\lim }\limits_{\xi \to - \infty } w(\xi ), \quad x < s't, \hfill \\ \mathop {\lim }\limits_{\xi \to + \infty } w(\xi ), \quad x > s't, \hfill \\ \end{gathered} \right.$

and therefore we shall insist that $\displaystyle\mathop {\lim }\limits_{\xi \to - \infty } w(\xi ) = {u_l}, \quad \mathop {\lim }\limits_{\xi \to + \infty } w(\xi ) = u_r$.

In particular, the right hand side of the above ODE has a limit as $\xi \to \pm\infty$ and so $w'(\xi)$ has a limit as well. If this limit is nonzero then of course $w$ itself cannot have a limit so in fact the limit must be zero. Thus we get $\displaystyle F({u_l}) - s'{u_l} - C = 0 = F({u_r}) - s'{u_r} - C$.

Clearly, $\displaystyle\int {\frac{{w'(\xi )}}{{F(w(\xi )) - s'w(\xi ) - C}}d\xi } = \int {d\xi } = \xi$.

Besides, the above identity states that $z= s'w +C$ is the equation of the straight line joining the two points $(u_l, F(u_l))$ and $(u_r,F(u_r))$ on the graph of $F$, that is the secant between those two points. So we have discovered that the graph $z=F(w)$ must lie to one side of that secant for $w$ between $u_l$ and $u_r$.

Theorem. The given equation admits a traveling wave solution of the form above satisfying $\displaystyle\mathop {\lim }\limits_{\xi \to - \infty } w(\xi ) = {u_l}, \quad \mathop {\lim }\limits_{\xi \to + \infty } w(\xi ) = u_r$

if and only if $\displaystyle (u_r-u_l)(F(w)-s'w-C)>0$

for all $w$ between $u_l$ and $u_r$ where $s'$ and $C$ are chosen so that the above expression is zero when $w=u_l$ or $w=u_r$.

O. Oleinik entropy condition. The entropy condition given by the above theorem is equivalent to the inequalities $\displaystyle\frac{{F(w) - F({u_l})}}{{w - {u_l}}} > s' > \frac{{F(w) - F({u_r})}}{{w - {u_r}}}$.

This entropy condition was suggested by O. Oleinik but since she gave another entropy condition more commonly associated with her name this condition is often referred to as the viscous profile entropy condition.

Lax entropy condition. The Lax entropy condition is the pair of inequalities $\displaystyle F'(u_l) \geqslant s' \geqslant F'(u_r)$

in the case when $F$ is either strictly convex or strictly concave. Of course the Lax condition has the advantage that it is very easy to check.