wellposedness

Wellposedness is the property of a mathematical problem that satisfies three conditions according to Hadamard: existence of a solution, uniqueness of that solution, and continuous dependence of the solution on the input data. If any of these conditions fails, the problem is called ill-posed. The term is used especially for differential equations, including ordinary and partial differential equations, as well as integral equations. For evolution problems, wellposedness is often considered locally in time (a solution exists for a short time) or globally (for all times).

To prove wellposedness, one typically shows existence, uniqueness, and stability (continuous dependence) using a priori bounds, energy estimates, fixed-point theorems, or semigroup theory. Methods from functional analysis and operator theory, such as contraction mappings or the theory of C0-semigroups, are common when dealing with evolution equations. The choice of function spaces, such as Sobolev spaces, greatly influences the statement and proof of wellposedness.

Examples illustrate the concept. The linear ordinary differential equation y' = Ay + f(t) with appropriate Lipschitz conditions is well-posed by standard existence-uniqueness results. The heat equation u_t = Δu with suitable boundary and initial data is well-posed in L2 or Sobolev spaces. By contrast, the backward heat equation u_t = -Δu is ill-posed: small perturbations in data can lead to large changes in the solution, and continuous dependence fails.

Wellposedness can be local or global, and solutions may be strong, weak, or mild, depending on the regularity assumed. It is a central concept in the analysis of nonlinear PDEs, fluid dynamics, and control theory, and it underpins the reliability of numerical approximations.