DE Seminar: Justin Webster
UMBC
Location
Mathematics/Psychology : 401
Date & Time
March 10, 2025, 11:00 am – 12:00 pm
Description
Title: A Fresh Look
at Weak Solutions for Linear Evolutionary PDEs
Speaker: Justin Webster
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.
Abstract: This talk 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), a classical work of John Ball provides a characterization which makes equivalent well-posedness of weak solutions and generation of a strongly continuous-semigroup. On the other hand, this approach 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.
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