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Well-posed problem

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In mathematics, a system of partial differential equations is well-posed (or a well-posed problem) if it has a uniquely determined solution that depends continuously on its data.

The term was first used by Jacques Hadamard to describe systems of equations whose solutions behave as it is (heuristically) expected from a physical system: They are deterministic and they make no "jumps".

A system of equations that is not well-posed is called ill-posed.

Physical systems

According to classical mechanics a physical system behaves deterministic, i.e., the development of the system is completely determined — both in the future and in the past — by its state at a single point of time. In other words, if all laws of physics needed to describe the system and all relevant data at a certain point of time are known (and if any influence from outside the system is excluded) then, in principle, its state at any other point of time can be computed. Moreover, since physical systems do not change instantly, all changes in time or resulting from variations of the data occur without jumps.

Mathematical description

The laws of mechanics are described by partial differential equations and, more generally, by integral and functional equations. Thus a physical system is described by a system of partial differential equations and its state is defined by a set of initial values and additional constraints in the form of boundary conditions. Since the solutions of the system describe the behaviour of the system in time, they have to exhibit the properties of the physical system they are meant to represent, i.e., the solution has to be unique, and (being free of jumps) it and its derivative have to be continuous, and the solution has to change continuously when the initial conditions are varied. Such a system is called well-posed because it may correspond to a physical system. A system that fails to exhibit these properties is called ill-posed because it cannot be a correct and complete description of a physical system. (Of course, to be well-posed is only a necessary condition for a correct description of physical reality, but it is not sufficient.)

Well-posed

A system of partial differential equations is well-posed if

  • it has a solution and
  • the solution is unique, and
  • the solution depends continuously on the initial values.

In many cases it is difficult to decide whether a systen is well-posed or not. Often the answer is not known.