Well-posedness

What is well-posedness?

By well-posedness in $H^{s}$ we generally mean that there exists a unique solution u for some time T for each set of initial data in $H^{s}$, which stays in $H^{s}$ and depends continuously on the initial data as a map from $H^{s}$ to $H^{s}$. However, there are a couple subtleties involved here.

• Existence. For classical (smooth) solutions it is clear what it means for a solution to exist; for rough solutions one usually asks (as a bare minimum) for a solution to exist in the sense of distributions. (One may sometimes have to write the equation in conservation form before one can make sense of a distribution). It is possible for negative regularity solutions to exist if there is a sufficient amount of local smoothing available.
• Uniqueness. There are many different notions of uniqueness. One common one is uniqueness in the class of limits of smooth solutions. Another is uniqueness assuming certain spacetime regularity assumptions on the solution. A stronger form of uniqueness is in the class of all $H^{s}$ functions. Stronger still is uniqueness in the class of all distributions for which the equation makes sense.
• Time of existence. In subcritical situations the time of existence typically depends only on the $H^{s}$ norm of the initial data, or at a bare minimum one should get a fixed non-zero time of existence for data of sufficiently small norm. When combined with a conservation law this can often be extended to global existence. In critical situations one typically obtains global existence for data of small norm, and local existence for data of large norm but with a time of existence depending on the profile of the data (in particular, the frequencies where the norm is largest) and not just on the norm itself.
• Continuity. There are many different ways the solution map can be continuous from $H^{s}$ to $H^{s}$. One of the strongest is real analyticity (which is what is commonly obtained by iteration methods). Weaker than this are various types of $C^{k}$ continuity ( $C^{1}$, $C^{2}$ , $C^{3}$, etc.). If the solution map is C^k, then this implies that the k^th derivative at the origin is in $H^{s}$, which roughly corresponds to some iterate (often the k^th iterate) lying in $H^{s}$. Weaker than this is Lipschitz continuity, and weaker than that is uniform continuity. Finally, there is just plain old continuity. Interestingly, several examples have emerged recently in which one form of continuity holds but not another; in particular we now have several examples (critical wave maps, low-regularity periodic KdV and mKdV, Benjamin-Ono, quasilinear wave equations, ...) where the solution map is continuous but not uniformly continuous.

For a survey of LWP and GWP issues, see Ta2002.