Ngô Quốc Anh

February 7, 2010

An illustration of an ill-posed hyperbolic PDE

Filed under: PDEs — Ngô Quốc Anh @ 0:01

A dramatic example of an ill-posed, second order hyperbolic PDE problem is given by the following BVP for the one dimensional wave equation. $\displaystyle \begin{gathered} {u_{tt}} - {u_{xx}} = 0\quad 0 < t < T,0 < x < 1, \hfill \\ u(x,0) = 0\quad 0 < x < 1, \hfill \\ u(x,T) = 0\quad 0 < x < 1, \hfill \\ u(0,t) = 0\quad 0 < t < T, \hfill \\ u(1,t) = 0 \quad 0 < t < T. \hfill \\ \end{gathered}$

It can be shown that if $T$ is irrational, then the only solution of this BVP for the wave equation is $u$ identically zero; whereas if $T$ is rational, the problem has infinitely many nontrivial solution. Thus the solution fails to depend continuously on the data – namely on the size of the region on which the problem is stated.

Following in another example. Let us consider the solutions of the vibrating string equation $\displaystyle u_{tt}-u_{xx}=0$

in the rectangle $\displaystyle Q = \left\{ {(x,t) \in {\mathbb{R}^2}:0 \leqslant x \leqslant \pi ,0 \leqslant t \leqslant 2\pi } \right\}$

satisfying the conditions $\begin{gathered} u(0,t) = 0 \quad 0 < t < 2\pi , \hfill \\ u(1,t) = 0\quad 0 < t < 2\pi , \hfill \\ u(\alpha \pi ,t) = f(t)\quad 0 < t < 2\pi , \hfill \\ \end{gathered}$

where $\alpha$ is a given constant in $(0, 1)$ and $f$ is a given smooth function. We notice that, if $\alpha$ is an irrational number, then there exists one and only one smooth solution $u$ of the problem under suitable assumptions on $f$.

This is because a function $u\in C^2(Q)$ satisfying the problem can be represented by a uniformly convergent series $\displaystyle u(x,t) = \sum\limits_{n = 1}^\infty {\sin (nx)\left( {{a_n}\cos (nt) + {b_n}\sin (nt)} \right)}$

and we have $\displaystyle f(t) = \sum\limits_{n = 1}^\infty {\sin (n\alpha \pi )\left( {{a_n}\cos (nt) + {b_n}\sin (nt)} \right)}$.

Since $f \equiv 0$, then $a_n=b_n=0$ which claims $u \equiv 0$.

If $\alpha$ is rational, the uniqueness does not hold. For example, if $q$ is an integer, called $\frac{1}{q}$ then the function $u(x, t)= \sin(qx) \cos(qt)$ satisfies the first two boundary conditions and $\displaystyle u\left( {\frac{\pi }{q},t} \right) = 0, \quad t \in \left[ {0.2\pi } \right]$.

We also observe that the problem is not well posed. In fact, small changes of datum $f$ in $C^s$-norm ( $s$ any integer) lead to arbitrarily large changes of solution $u$ in $L^2$-norm.

The stability of the problem can be restored by an additional condition, for example by an a priori bound on the energy, i.e. $\displaystyle \int_0^\pi {\int_0^{2\pi } {u_x^2(x,t)dtdx} \leqslant {E^2}}$

where $E$ is a given constant.