For $(t,r)\in[0,\infty)\times[0,1]$ let be the following PDE's system
$$\dot{\vartheta}= w' +w\vartheta' $$ $$\dot{w}= \vartheta'+ww' $$ along with initial conditions (i.c.) $$w(0,r)=w_i(r)\qquad \vartheta(0,r)=\vartheta_i(r) $$ and the boundary condition (b.c) $$\boxed{w(t,1)=0}\;\Rightarrow\;w(t,1)=\dot{w}(t,1)=0\;\Rightarrow\;\boxed{\vartheta'(t,1)=0} $$
My question is:
- What is the minimum extra b.c. needed for the problem to be well posed?
- Is there any specific analytical solution, if not a generic one?
- How can one modify the equations so that no shock waves occur? (I mean some modification that does not change the type and number of b.c. needed.)
Remark: If range $[0,1]$ specified for $r$ is interpreted as a sphere of radious $R=1$ then $r=0$ is just an interior point. So the fields $w$ and $\vartheta$ at this point should not be restricted by any b.c. just as it happens for any other interior point.
Above all: What would be a good source (lecture notes, book, webpage, etc.) for studying this kind of system without having to revise all of PDE's theory?
I would appreciate any help.
PS: I use "dot" for $\partial_t$ and "prime" for $\partial_r$.
This system can be written in the quasi-linear form $u_t + A(u)\, u_r = 0$ where $u = (\vartheta,w)^\top$ and $$ A(u) = -\begin{pmatrix}w & 1 \\ 1 & w\end{pmatrix} $$ is a symmetric matrix. The above PDE system is a first-order symmetric hyperbolic quasi-linear system. Thus, the matrix $A(u)$ can be diagonalized with real eigenvalues $\lambda \in \lbrace-1-w, 1-w\rbrace$. Here, the two characteristic fields associated with the eigenvalues are both genuinely nonlinear, i.e. the gradient $\lambda_u$ of the eigenvalue $\lambda$ is not orthogonal to the corresponding eigenvector (1). Since this system is of first order, the boundary value problem is locally well-posed for small data if we know the function $\vartheta_1$ such that $\vartheta(t,1) = \vartheta_1(t)$ along the boundary $r=1$. Indeed, we can verify that there is no zero eigenvalue on the boundary $r=1$ (which is the case here), so that information will propagate from the boundary.
Concerning further results such as analytical solutions, it would be advantageous to write this system in conservation form $u_t + f(u)_r = 0$, but I doubt that this is possible. The present type of system produces discontinuous solutions in finite time even if the boundary data is smooth. To prevent this, a dissipative right-hand side of the form $\epsilon u_{rr}$ or $-\epsilon u$ can be added.
(1) P.D. Lax (1973): Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM. doi:10.1137/1.9781611970562