# PDE system and domain of definition

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 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!

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 $$(v_0-u_0)x < y < (v_0+u_0)x , \qquad x>0 .$$