3
$\begingroup$

Prove that there is no solution to the following Cauchy problem: $$\begin{align} u_t+uu_x& =0&\quad x&\in(-1,1), t\gt0 \label{1}\tag{1}\\ u(x,0)&=x&\quad x&\in[-1,1] \label{2}\tag{2}\\ u(-1,t)&=-1& \quad t&\geq0 \label{3}\tag{3}\\ u(1,t)&=1&\quad t&\geq0 \label{4}\tag{4} \end{align}$$

My attempt: Using the method of characteristics, I found the classical solution $$ u=\frac{x}{1+t} \quad t\gt{-1} $$ which satisfies conditions \eqref{1} and \eqref{2}, but not \eqref{3} and \eqref{4}.

But how can I show that there is also no weak solution to this problem?

Thanks in advance.

$\endgroup$
1
$\begingroup$

In facts, it is impossible for a classical solution to solve the initial- and boundary-value problem. A plot of the characteristic curves in the $x$-$t$ plane deduced from the initial data is given below*:

characteristics

The possibility of admissible discontinuities near the boundaries (weak solutions) should be examined. A look at the characteristics in the $x$-$t$ plane shows that such solutions are not admissible in the sense of Lax** (characteristics do not intersect appropriately). For example, at $x\simeq 1$, we have $u_R = 1$ and $u_L = 1/(1+t)<1$, so that $u_L < u_R$. No shock wave is admissible. Alternatively, we could have investigated the boundary $x\simeq -1$, where $u_L = -1$ and $u_R = -1/(1+t) > -1$. Here too, no shock wave is admissible, since $u_L < u_R$.


*The lines drawn outside $]-1,1[$ are the characteristics starting at the boundaries $x=\pm 1$, which have been represented outside the interior domain to keep the figure readable.

**The Lax entropy condition reads $u_L>s>u_R$ where $u_L$ is the value on the left of the discontinuity, $u_R$ is the value on the right of the discontinuity, and $s$ is the speed of shock given by Rankine-Hugoniot.

$\endgroup$
  • $\begingroup$ Thanks for your answer, but the only criterion I know about weak solutions is the Runkine - Hugionot theorem, which I cannot apply here as I can not figure one possible curve...should it be characteristic? $\endgroup$ – dmtri Dec 5 '18 at 17:27
  • $\begingroup$ one more question please, what are the lines outside of the area $x\ge1$ and $-1\ge{x}$, all the characteristics should pass by the point $(0,-1)$. $\endgroup$ – dmtri Dec 5 '18 at 18:06
  • $\begingroup$ I would like to ask something more, as I think I do not get clearly what is really asked in an initial and boundary problem (IBP). In the IBP, I posted, are we looking for a classical solution of equation $(1)$ valid in the interior (open set) of the area given and this solution should take at boundary of this set the values described by equations $(2), (3), (4)$ ? If this is the case, we can easily conclude that $u=\frac{x}{t+1}$ is the only solution satisfying $(1) , (2)$ but not $(3)$. Why do we need $(4)$ then to show that there is no solution? $\endgroup$ – dmtri Dec 13 '18 at 10:13
  • $\begingroup$ I would be gradefull, If you could provide me a refference, as simple as possible, for this admissible shock wave....My level at this area of maths is really low. $\endgroup$ – dmtri Dec 20 '18 at 12:53
  • 1
    $\begingroup$ @dmtri Sure: I'd suggest LeVeque's book Numerical Methods for Conservation Laws (Part I), which is very easy. Alternatively, Evans' book Partial Differential Equations (Chap. 3) $\endgroup$ – Harry49 Dec 20 '18 at 13:27

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.