Ngô Quốc Anh

February 3, 2010

An illustration of an ill-posed elliptic PDE, an example due to Hadamard

Filed under: Nghiên Cứu Khoa Học, PDEs — Ngô Quốc Anh @ 12:20

We now consider some important issues regarding the formulation and solvability of PDE problems. A solution to a PDE can be described as simply a function that reduces that PDE to an identity on some region of the independent variables. In general, a PDE alone, without any auxiliary boundary or initial conditions, will either have an infinity of solutions, or have no solution. Thus, in formulating a PDE problem there are three components

• the PDE;
• the region of space-time on which the PDE is required to be satisfied;
• the auxiliary boundary and initial conditions to be met.

For a PDE based mathematical model of a physical system to give useful results, it is generally necessary to formulate that model as what mathematicians call a well posed PDE problem. A PDE problem is said to be well-posed if

1. a solution to the problem exists
2. the solution is unique, and
3. the solution depends continuously on the problem data.

(In a PDE problem the problem data consists of the coefficients in the PDE; the functions appearing in boundary and initial conditions; and the region on which the PDE is required to hold.)

If one of these conditions is not satisfied, the PDE problem is said to be ill-posed. In practice, the question of whether a PDE problem is well-posed can be difficult to settle. Roughly speaking the following guidelines apply:

• The auxiliary conditions imposed must not be too many or a solution will not exist.
• The auxiliary conditions imposed must not be too few or the solution will not be unique.
• The kind of auxiliary conditions must be correctly matched to the type of the PDE or the solution will not depend continuously on the data.

More specific guidelines can be stated for second order linear PDE problems.

• Well-posed elliptic PDE problems usually take the form of a boundary value problem (BVP) with the PDE required to hold on the interior of some region and the solution required to satisfy a single boundary condition (BC) at each point on the boundary of the region. Typical boundary conditions are:
• Dirichlet BC – the solution value is specified on the boundary
• Neumann BC – the normal derivative of the solution is specified on the boundary
• Robin BC – a linear combination of the solution and its normal derivative is specified on the boundary.

The kind of boundary condition can vary from point to point on the boundary, but at any given point only one BC can be specified. Physically a Dirichlet BC usually corresponds to setting the value of a field variable, such as temperature; a Neumann BC usually specifies a flux condition on the boundary; and a Robin BC typically represents a radiation condition. When the region on which the PDE problem is posed is unbounded, one or more of the above boundary conditions is usually replaced by a growth condition that limits the behavior of the solution “at infinity”.

• Well-posed parabolic PDE problems usually involve one or more spatial variables and a time variable as well. Parabolic PDE models often arise in connection with evolutionary systems in which the flux of some material quantity is “down gradient” with respect to a field variable. Typically, a well posed parabolic problem requires the same boundary conditions on the spatial variables as in the case of elliptic problems. In addition an initial condition specifying the state of the system at time $t=0$ is required. Thus, a well posed second order parabolic PDE problem usually takes the form of and initial boundary value problem (IBVP).
• Wel-posed, second order, hyperbolic PDE problems also require the same boundary conditions as elliptic problems. Usually second order, hyperbolic PDE model arise in connection with physical problems involving wave motion, vibration or oscillation. In these problems, two initial conditions at time $t=0$ are required (one to describe the initial state of the system and another to describe the initial velocity).

A discussion of the well posedness of PDE problems involving systems of first order equations requires an understanding of the characteristic curves associated with such systems. Systems of first order equations are very important in the field of computational science, but are not dealt with here, since the remainder of this chapter focus on second order PDEs. To conclude this section, several examples of well posed and ill posed second order PDE problems are presented.

Example (ill-posed elliptic PDE). To illustrate that boundary value problems, not initial value problems, are the appropriate setting for elliptic PDE problems, we present the following example due to Hadamard. To view this problem as an initial value problem, one should think of $y$ as a time variable. Consider the initial value problem

$\displaystyle\begin{gathered}{u_{xx}} + {u_{yy}} = 0 \quad - \infty< x < \infty ,y > 0, \hfill \\u(x,0) = f(x)\quad- \infty< x < \infty , \hfill \\{u_x}(x,0) = g(x)\quad- \infty< x < \infty . \hfill \\ \end{gathered}$

Proof. For $f=f_1(x)=0$ and $g=g_1(x)=0$, it is clear that the corresponding solution to the above initial value problem is $u_1(x,y)=0$. For the case $f=f_2(x)=0$ and $g=g_2(x)=n^{-1}\sin (nx)$, it is easy to verify that the corresponding solution is

$\displaystyle {u_2}(x,y) = \frac{{\sinh (ny)\sin (nx)}}{{{n^2}}}$.

Observe that the functions $f_1$ and $f_2$ are identical and that

$\displaystyle\mathop {\lim }\limits_{n \to \infty } \left| {{g_1}(x) - {g_2}(x)} \right| = 0$

uniformly in $x$. Thus, we see that the data of the two problems, $(f_1,g_1)$ and $(f_2,g_2)$, can be made arbitrarily close. But, if we compare the two solutions at $x=\frac{\pi}{2}$, then we obtain

$\displaystyle\left| {{u_1}\left( {\frac{\pi }{2},y} \right) - {u_2}\left( {\frac{\pi }{2},y} \right)} \right| = \frac{1}{{{n^2}\sinh (ny)}} = \frac{{{e^{ny}} - {e^{ - ny}}}}{{2{n^2}}}$.

For $y$ positive, $e^{ny}$ approaches infinity faster than $n^2$, as $n$ goes to infinity. Therefore, we conclude that

$\displaystyle\mathop {\lim }\limits_{n \to \infty } \left| {{u_1}\left( {\frac{\pi }{2},y} \right) - {u_2}\left( {\frac{\pi }{2},y} \right)} \right| = \infty$

illustrating that as the data for the two problems becomes more alike, the solutions become increasingly different. This is what is meant by failure of the solution to depend continuously on the problem data.