A mathematical problem is well-posed in the sense of Hadamard if, for every admissible set of data, a solution exists, is unique within the specified solution class, and depends continuously on the data in the chosen topologies. The choices of data space, solution space, and topology are therefore part of the statement (Hadamard 1902). A problem that violates any of these requirements is ill-posed.
Well-posedness concerns the existence, uniqueness, and stability of solutions, whereas well-defined concerns whether a definition or construction determines a unique value or meaning.
