Help me for nonlinear hyperbolic first order system of pde

mr.mu_vn · #1 $Old$ June 19th, 2008, 05:56 AM

Hi,

i have a problem with the solving of nonlinear hyperbolic first oder system of pde. This look like following

Can anyone help me to solve it?

Thank you in advance.
Best regards
Minh

Rebesques · #2 $Old$ July 20th, 2008, 09:21 PM

Well... that's just great

This system is not trivial to solve -- at least in general. Start by scaling and then define $\underbar{u}=(p, q)$ , to obtain $(P): \ \frac{\partial}{\partial t}\underbar{u}=A\frac{\partial}{\partial x}\underbar{u}$ , where $A=\left[ \begin{matrix} -a_2/a_1 & -a_3/a_1 \\ -b_2/b_1 & -b_3/b_1\end{matrix} \right]$ . By saying this is hyperbolic, I presume you mean the (constant!) matrix has no real eigenvalues. Then, prescribing differentiable Cauchy data $(C): \ u(0,x)=u_0(x)$ might not mean there is a solution at all! In fact, $u_0$ has to be at least analytic, and I have no idea on obtaining an actual solution.

For the sake of argument, suppose...

The matrix of the system has real and distinct eigenvalues. Then the Cauchy problem (P)&(C) has always a solution of the form

$\underbar{u}(t,x)=(p(t,x),q(t,x))=\left( \int\exp(c_1ts+c_2xs)u_0(s)ds,\int\exp(d_1ts+d_2xs)u_0(s)ds\right)$

where we have to pick the constants involved so that the integrals converge and $\underbar{u}$ is truly a solution, the latter part being algebraic manipulations.
The matrix of the system is self adjoint. Then, for the Cauchy data (C) being a tempered distribution, there is a unique solution for (P)&(C). In fact, a stronger result on regularity is true: This solution must necessarily be $C^{\infty}!$

Hope this helps.