zbMATH — the first resource for mathematics

An inviscid regularization for the surface quasi-geostrophic equation. (English) Zbl 1149.35018
Summary: Inspired by recent developments in Berdina-like models for turbulence, we propose an inviscid regularization for the surface quasi-geostrophic (SQG) equations with periodic boundary conditions on a basic periodic square \(\Omega= [0,1]^2\subset\mathbb R^2\),
\[ \frac{\partial\theta}{\partial t}+\text{div}({\mathbf v}\theta)=0, \qquad (-\Delta)^{1/2}\psi=\theta, \]
\[ \nabla^\perp\psi={\mathbf v}, \qquad \theta({\mathbf x},0)= \theta_0({\mathbf x})\quad\text{in }\Omega, \]
\[ \int_\Omega \theta\,d{\mathbf x}=0, \qquad \int_\Omega\psi\,d{\mathbf x}=0, \qquad \int_\Omega{\mathbf v}\,d{\mathbf x}=0. \]
Here \(\Delta\) is the horizontal Laplacian operator and \((-\Delta)^{1/2}\) is the pseudodifferential operator defined in the Fourier space \(\widehat{(-\Delta)^{1/2}u({\mathbf k})}= |{\mathbf k}|\widehat{u}({\mathbf k})\).
We are particularly interested in the celebrated question of blowup in finite time of the solution gradient of the SQG equations. The new regularization yields a necessary and sufficient condition, satisfied by the regularized solution, when a regularization parameter tends to zero for the solution of the original SQG equations to develop a singularity in finite time. As opposed to the commonly used viscous regularization, the inviscid equations derived here conserve a modified energy. Therefore, the new regularization provides an attractive numerical procedure for finite-time blowup testing. In particular, we prove that, if the initial condition is smooth, then the regularized solution remains as smooth as the initial data for all times.

35B65 Smoothness and regularity of solutions to PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
86A05 Hydrology, hydrography, oceanography
Full Text: DOI arXiv
[1] Adams, Sobolev spaces 65 (1975)
[2] Cao, Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models, Commun Math Sci 4 (4) pp 823– (2006) · Zbl 1127.35034
[3] Constantin, On the critical dissipative quasi-geostrophic equation, Indiana Univ Math J 50 (1) pp 97– (2001) · Zbl 0989.86004
[4] Constantin, Navier-Stokes equations (1988)
[5] Constantin, Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar, Nonlinearity 7 (6) pp 1495– (1994) · Zbl 0809.35057
[6] Constantin, Nonsingular surface quasi-geostrophic flow, Phys Lett A 241 (3) pp 168– (1998) · Zbl 0974.76512
[7] Constantin, Front formation in an active scalar equation, Phys Rev E (3) 60 (3) pp 2858– (1999)
[8] Cordoba, Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation, Ann of Math (2) 148 (3) pp 1135– (1998) · Zbl 0920.35109
[9] Ladyzhenskaya, The mathematical theory of viscous incompressible flow 2 (1969) · Zbl 0184.52603
[10] Ladyzhenskaya, The boundary value problems of mathematical physics 49 (1985) · Zbl 0588.35003
[11] Majda, Vorticity and incompressible flow 27 (2002)
[12] Majda, A two-dimensional model for quasigeostrophic flow: comparison with the two-dimensional Euler flow. Nonlinear phenomena in ocean dynamics (Los Alamos, NM, 1995), Phys D 98 (2) pp 515– (1996)
[13] Oskolkov, The uniqueness and solvability in the large of boundary value problems for the equations of motion of aqueous solutions of polymers. Boundary value problems of mathematical physics and related questions in the theory of functions, 7, Zap Naučn Sem Leningrad Otdel Mat Inst Steklov (LOMI) 38 (2) pp 98– (1973)
[14] Pedlosky, Geophysical fluid dynamics (1987)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.