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.


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*:


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.

  • $\begingroup$ 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

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