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I have the following PDE with initial values as stated. $$u_t+u^2u_x=0\hspace{1 cm} t>0, -\infty<x<\infty$$ $$u(x,t=0)=\begin{cases} -a & x< 0 \\ 0.5 & x\geq 0 \end{cases}$$ where $a>0$.

I know that we can solve this using method of characteristics so we have $$\frac{dX}{dT}=u^2=\begin{cases} a^2 & x< 0 \\ 0.5^2 & x\geq 0 \end{cases}$$ and from this I know that when $a>0.5$ the characteristic in the middle will be a shock wave with a certain shock speed defining the slope of that shock characteristic. For this, I found $x=\frac{A(u^+)-A(u^-)}{u^{+}-u^-}t=\frac{0.5^2-0.5a+a^2}{3}t$ to be the characteristic line for the shock. I am not sure if this is correct for the characteristic lines in the fan region?

Also, when $a<0.5$ the characteristics between the two sides will be like a fan and we can find the solution for $u(x,t)$ in this fan portion, but I have no idea how I can find a $u(x,t)$ that satisfies this PDE inside of this fan portion of the characteristics. Can someone please help me find this?

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mathim1881
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1 Answers1

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The problem at hand is a Riemann problem for the conservation law $u_t+f(u)_x =0$ where the flux function $f:u\mapsto \frac13 u^3$ is non-convex. Here, we have the value $-a<0$ on the left side of the initial discontinuity, and the value $0.5>0$ on the right side. Following this post, the entropy-satisfying solution might be of the following type:

• if $a>1$, then the solution is a shock wave with speed $\frac1{3}(a -\frac12)^2$.

• if $0< a\leq 1$, then the solution is a semi-shock, with intermediate state $u^*=a/2$ satisfying $0< u^*\leq \frac12$.

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