Wellposed

Wellposed, in mathematical usage usually written as well-posed (also seen as wellposed in some texts), refers to problems that satisfy Hadamard's criteria for a well-behaved solution. A problem is well-posed if a solution exists, the solution is unique, and the solution depends continuously on the input data. The input data can include initial conditions, boundary data, and forcing terms.

The concept originated with Jacques Hadamard in the early 20th century as a standard for assessing the stability and reliability of solutions to differential equations and related problems. It provides a framework for distinguishing problems whose solutions are robust under small perturbations from those that exhibit instability or non-uniqueness.

In the context of ordinary and partial differential equations, well-posedness ensures stability: small changes in the input data produce only small changes in the solution. For linear problems, this is often demonstrated through a priori estimates or semigroup theory, and for nonlinear problems, through appropriate continuity and compactness arguments within function spaces.

Examples of well-posed problems include many initial value problems for ordinary differential equations with Lipschitz right-hand sides, and the heat equation with suitable initial and boundary data, where solutions exist, are unique, and depend continuously on the initial distribution. The wave equation with proper boundary conditions is another well-posed case under standard energy estimates. By contrast, ill-posed problems fail one or more Hadamard conditions, such as the backward heat equation, which is highly unstable with respect to data.

Well-posedness is central to the numerical analysis of differential equations, ensuring that numerical schemes converge to the true solution as discretization improves. Techniques to establish well-posedness include energy methods, semigroup theory, and Fourier analysis in Sobolev spaces. See also ill-posed problems and regularization.