Proving a weak solution to PDE

[dochalladay32]

Since the hurricane has killed school for awhile, I was working on some stuff that I knew we would not cover in class any more, but could end up on a test and I got to this question. It seems like it should be simple, but I have been stumped all day. Don't know if anyone here will have the expertise to explain what is happening on the discontinuities. Luckily the question is an ebook; much easier to post as an image than try to type. It is Salsa's PDE's in Action, by the way, if anyone wants to see more.

So it is obvious that in each region, it is a weak (actually a strong) solution since they are constants. The problem is what happens when you pass over the discontinuities? As it is a Riemann problem with u_L < u_R, the rarefaction fan is the entropy solution we are looking for and the lines for those discontinuities have nothing to do with a RH condition. If you try to use that formula, where f(u) = 1/2*u^2, they don't match up. We are fine at 0, but when you try to show it is a weak solution across the other two lines, I can't get them to go to 0.

Anybody have some advice on how to look at this problem? It doesn't really seem to follow from what I found in the text, but maybe I missed something.

 

[gtoguy97]

I think the key here is to realize that the solution is a piecewise constant function, so you don’t have to worry about how the values change across the discontinuities. Instead, focus on the fact that the solution is continuous, and satisfies the boundary conditions. You can then use the Rankine-Hugoniot condition to check that the jump in the derivative of the solution across the discontinuities is consistent with the jump in the right hand side of the equation.