# An initial- and boundary-value problem for Burgers' equation with no solution

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?

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.

• 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? – dmtri Dec 5 '18 at 17:27
• 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 $(0,-1)$. – dmtri Dec 5 '18 at 18:06
• 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? – dmtri Dec 13 '18 at 10:13
• 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. – dmtri Dec 20 '18 at 12:53
• @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) – Harry49 Dec 20 '18 at 13:27