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.

See also: Ill-posed elliptic PDE, Ill-posed parabolic PDE.


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