A general question about classical and weak solutions.

Question

This is a very general theoretical question about classical and weak solutions.

If I know that there exists a unique classical solution to a PDE. What can I say about the solutions to the corresponding variational (or "weak") problem?

Consider for example the common Poisson equation:

$$ \cases{- \Delta u = f \ \ \text{in } \ \ \Omega \\ u = 0 \ \ \text{on } \ \ \partial \Omega} $$

And its corresponding variational formulation: Find $u \in H^1_0(\Omega)$ such that

$$ (\nabla u, \nabla v) = (f,v) \ \ \forall v \in H^1_0(\Omega). $$

Say I know that there exists a unique classical solution. Then this is also a solution to the variational formulation. But can I deduce something about the uniqueness of a solution to the variational problem?

I know that using Lax-Milgram it is possible to deduce the both existence and uniqueness of a weak solution. But given that I already know the existence of such a solution and also the uniqueness of a classical solution, can I somehow deduce that the weak solution is also unique? Does this only hold under some conditions?

If anything is unclear, don't hesitate to ask. Thanks in advance!

Answer 1

In general: no.

When Lax-Milgram is used, you used a certain coercivity in your bilinear form (constructed from the operator) that guarantees uniqueness. In general weak solutions may be in fact too weak for classical uniqueness to persist.

Off the top of my head I don't know any examples in the elliptic context, but for evolution equations there are some well-known examples, especially connected to fluid dynamics.

• For the Boltzmann equation, local existence and uniqueness of classical solutions is known from various sources (e.g. Kaniel and Shinbrot.) For weak solutions, DiPerna and Lions should global existence. The uniqueness of these weak solutions are, to the best of my knowledge, still an open problem.

• For the Navier-Stokes equations, local existence and uniqueness of classical solutions is well-known (see, for example, the textbook of Bertozzi and Majda). Leray considered the weak solutions to the Navier-Stokes problem and arrived at global existence, but with entirely unknown uniqueness properties. For a review, see Fefferman's Clay Problem Statement.

• For the Euler equations, local existence and uniqueness of classical solutions is again well-known (see, e.g. Hou's lecture notes). For weak solutions, however, the uniqueness is known to be false, due to Scheffer and separately Shnirelman. (This is somewhat related to the famous conjecture of Onsager.)

• For a parabolic example: a result of Coron asserts that the harmonic map heat-flow, in its weak formulation, admit initial data leading to non-unique solutions. A similar result also holds in the hyperbolic case (wave maps) due to Widmayer.