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We present a study on asymptotically compatible Galerkin discretizations for a class of parametrized nonlinear variational problems. The abstract analytical framework is based on variational convergence, or Gamma-convergence. We demonstrate the broad applicability of the theoretical framework by developing asymptotically compatible finite element discretizations of some representative nonlinear nonlocal variational problems on a bounded domain. These include nonlocal nonlinear problems with classically-defined, local boundary constraints through heterogeneous localization at the boundary, as well as nonlocal problems posed on parameter-dependent domains.
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This content will become publicly available on January 1, 2026
Global regularity for critical SQG in bounded domains
Abstract We prove the existence and uniqueness of global smooth solutions of the critical dissipative SQG equation in bounded domains in . We introduce a new methodology of transforming the single nonlocal nonlinear evolution equation in a bounded domain into an interacting system of extended nonlocal nonlinear evolution equations in the whole space. The proof then uses the method of the nonlinear maximum principle for nonlocal operators in the extended system.
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- Award ID(s):
- 2204614
- PAR ID:
- 10591327
- Publisher / Repository:
- Wiley
- Date Published:
- Journal Name:
- Communications on Pure and Applied Mathematics
- Volume:
- 78
- Issue:
- 1
- ISSN:
- 0010-3640
- Page Range / eLocation ID:
- 3 to 59
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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