An Observation About Weak Solutions of Linear Differential Equations in Hilbert Spaces

This note addresses the well-posedness of weak solutions for a general linear evolution problem on a separable Hilbert space. For this classical problem there is a well known challenge of obtaining a priori estimates, as a constructed weak solution may not be regular enough to be utilized as a test function. This issue presents an obstacle for obtaining uniqueness and continuous dependence of solutions. Utilizing a generic weak formulation (involving the adjoint of the system’s evolution operator), the classical reference (Ball in Proceedings of the American Mathematical Society 63:370-373, 1977) provides a characterization which makes equivalent well-posedness of weak solutions and generation of a \(C_0\)-semigroup. On the other hand, the approach in (Ball in Proceedings of the American Mathematical Society 63:370-373, 1977) does not take into account any underlying energy estimate, and requires a characterization of the adjoint operator, the latter often posing a non-trivial task. We propose an alternative approach, when the problem is posed on a Hilbert space and admits an underlying “formal" energy estimate. For such a Cauchy problem, we provide a general notion of weak solution and through a straightforward observation, obtain that arbitrary weak solutions have additional time regularity and obey an a priori estimate. This yields weak well-posedness. Our result rests upon a central hypothesis asserting the existence of a “good" Galerkin basis for the construction of a weak solution. A posteriori, a \(C_0\)-semigroup may be obtained for weak solutions, and by uniqueness, weak and semigroup solutions are equivalent.