The SMGT equation from the boundary: regularity and stabilization

Abstract

We consider the third-order (in time) linear equation known as SMGT-equation, as defined on a multidimensional bounded domain. Part A gives optimal interior and boundary regularity results from $L^2(0,T;L^2(\mathrm{\Gamma }\left)\right)$ – Dirichlet or Neumann boundary terms. Explicit representation formulas are given that can be taken to define the notion of solution in the canonical case $(\gamma =0)$, while the same regularity results hold for $\gamma \in L^{\mathrm{\infty }}\left(\mathrm{\Omega }\right).$ Part B considers the SMGT equation under Neumann dissipative boundary conditions and critical parameter $\gamma \in L^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$ and $\gamma \left(x\right)⩾0$ a.e. in Ω. We provide two results: (i) uniform stabilization under minimal checkable geometric conditions, and (ii) strong stabilization in the absence of geometrical conditions.

Keywords:

• SMGT equation
• boundary regularity
• boundary stabilization

2020 Mathematics Subject Classification:

• 35Lxx

Acknowledgements

The authors wish to thank the referees for their much appreciated comments. The research of I. L. and R. T. was partially supported by the National Science Foundation under Grant DMS-1713506. The work was partially carried out while I. L. and R. T. were members of the MSRI program “Mathematical Problems in Fluid Dynamics” during the Spring 2021 semester (NSF DMS-1928930); while M. B. was a member of the Weierstrass Institute for Applied Analysis and Stochastics. The authors thank their host organizations.

Disclosure statement

No potential conflict of interest was reported by the author(s).