Consider the PDE system, defined for $u>0$:
$u_x + vu_y + u^2v_y = 0$, $v_x + u_y + vv_y = 0$.
It is easy to show that $\log u \pm v$ are Riemann invariants on $\frac{dy}{dx} = v\pm u$ (I do not know if this will be needed for the question, but I am giving it just in case).
Given the boundary data $u = u_0, v = v_0$ on $y=0, x>0$ where $0<v_0 < u_0$ are
constants, give the domain of definition of the solution (i.e. the region where the solution is uniquely determined).
So if I am not mistaken, the solution must be $u = u_0$, $v=v_0$ in this domain and the characteristic directions are $y – (v_0 + u_0)x = const$, $y – (v_0 – u_0)x = const$. How to proceed from here to determine the domain?
Any help appreciated!
Best Answer
This is a quasi-linear hyperbolic system of conservation laws, which characteristic speeds are $v-u$ and $v+u$. In the present case of constant boundary data along the half-line $y=0$, $x>0$, the characteristic curves are indeed $x = x_0 + y/(v_0-u_0)$ and $x = x_0 + y/(v_0+u_0)$. The first one has negative speed $v_0-u_0$ while the second one has positive speed $v_0+u_0$. For both curves, we must have $x>0$ at $y=0$, i.e. $x_0 > 0$. That is to say, the domain of definition satisfies $$(v_0-u_0)x < y < (v_0+u_0)x , \qquad x>0 .$$