Surface quasi-geostrophic equation
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The surface quasi-geostrophic (SQG) equation consists of an evolution equation for a scalar function $\theta: \R^+ \times \R^2 \to \R$. In the inviscid case the equation is $$ \theta_t + u \cdot \nabla \theta = 0,$$ where $u = R^\perp \theta$ and $R$ stands for the Riesz transform.
Fractional diffusion is often added to the equation $$ \theta_t + u \cdot \nabla \theta + (-\Delta)^s \theta = 0.$$
The equation is used as a toy model for the 3D Euler equation and Navier-Stokes. The main question is to determine whether the Cauchy problem is well posed in the classical sense. In the inviscid case, it is a major open problem as well as in the supercritical diffusive case when $s<1/2$. It is believed that inviscid SQG equation presents a similar difficulty as 3D Euler equation in spite of being a scalar model in two dimensions ^{[1]}. The same comparison can be made between the supercritical SQG equation and Navier-Stokes.
The key feature of the model is that the drift $u$ is a divergence free vector field related to the solution $\theta$ by a zeroth order singular integral operator.
For the diffusive case, the well posedness of the equation follows from perturbative techniques in the subcritical case ($s>1/2$). In the critical case the proof is more delicate and can be shown using three essentially different methods. In the sueprcritical regime ($s<1/2$) only partial results are known.
Global weak solutions, as well as classical solutions locally in time, are known to exist globally for the full range of $s \in [0,1]$ ^{[2]}.
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Conserved quantities
The following simple a priori estimates are satisfied by solutions (in order from strongest -locally- to weakest).
- Maximum principle
The supremum of $\theta$ occurs at time zero: $||\theta(t,.)||_{L^\infty} \leq ||\theta(0,.)||_{L^\infty}$.
- Conservation of energy.
A classical solution $u$ satisfies the energy equality $$ \int_{\R^2} \theta(0,x)^2 \ dx = \int_{\R^2} \theta(t,x)^2 \ dx + \int_0^t \int_{\R^2} |(-\Delta)^{s/2}\theta(r,x)|^2 \ dx \ dr.$$
In the case of weak solutions, only the energy inequality is available $$ \int_{\R^2} \theta(0,x)^2 \ dx \geq \int_{\R^2} \theta(t,x)^2 \ dx + \int_0^t \int_{\R^2} |(-\Delta)^{s/2}\theta(r,x)|^2 \ dx \ dr.$$
- $H^{-1/2}$ estimate
The $H^{-1/2}$ norm of $\theta$ does not increase in time.
$$ \int_{\R^2} |(-\Delta)^{-1/4} \theta(0,x)|^2 \ dx = \int_{\R^2} |(-\Delta)^{-1/4}\theta(t,x)|^2 \ dx + \int_0^t \int_{\R^2} |(-\Delta)^{s/2-1/4}\theta(r,x)|^2 \ dx \ dr.$$
Scaling and criticality
If $\theta$ solves the equation, so does the rescaled solution $\theta_r(t,x) = r^{2s-1} \theta(r^{2s} t,rx)$.
The $L^\infty$ norm is invariant by the scaling of the equation if $s=1/2$. This observation makes $s=1/2$ the critical exponent for the strongest a priori estimate available. For larger values of $s$, the diffusion dominates the drift in small scales and the equation is well posed. For larger values of $s$, the drift might be dominant at small scales.
Well posedness results
Sub-critical case: $s>1/2$
The equation is well posed globally. The proof can be done with perturbation methods using only soft functional analysis or Fourier analysis.
Critical case: $s=1/2$
The equation is well posed globally. There are four known proofs.
- Evolution of a modulus of continuity ^{[3]}: An explicit modulus of continuity which is comparable to Lipschitz in small scales but growth logarithmically in large scales is shown to be preserved by the flow. The method is vaguely comparable to Ishii-Lions.
- De Giorgi approach ^{[4]}: From the $L^\infty$ modulus of continuity, it is concluded that $u$ stays bounded in $BMO$. A variation to the parabolic De Giorgi-Nash-Moser can be carried out to obtain Holder continuity of $\theta$. The result does not use the relations $u = R^\perp \theta$, but only that $u$ is a divergence-free vector field in BMO. Therefore, it is actually a regularity result for arbitrary drift-diffusion equations.
- Dual flow method ^{[5]}: Also from the information that $u$ is $BMO$ and divergence free, it can be shown that the solution $\theta$ becomes Holder continuous by studying the dual flow and characterizing Holder functions in terms of how they integrate against simple test functions. This is a regularity result for general drift-diffusion equations as well.
- Nonlinear maximum principle ^{[6]}: By studying the evolution of $|\nabla \theta|^2$ and using a nonlinear lower bound on the fractional Laplacian when evaluated at extrema, one may prove that a solution of SQG which has only (sufficiently) small jumps, then it is in fact smooth. By looking at the evolution of finite differences $\delta_h \theta(x) = \theta(x+h) - \theta(x)$, one may measure the evolution of these small jumps, and prove that if the initial data has only small jumps, then so does the corresponding solution of SQG. Combined, the above statements imply global regularity.
Supercritical case: $s<1/2$
The global well posedness of the equation is an open problem. Some partial results are known:
- Existence of solutions locally in time.
- Existence of global weak solutions. ^{[2]}
- Global smooth solution if the initial data is sufficiently small. ^{[7]}
- Smoothness of weak solutions for sufficiently large time. ^{[8]} ^{[9]} ^{[10]}
Inviscid case
The global well posedness of the equation is an open problem. Some partial results are known:
- Existence of smooth solutions locally in time^{[citation needed]}
- Existence of weak solutions globally in time^{[citation needed]}
References
- ↑ Constantin, Peter; Majda, Andrew J.; Tabak, Esteban (1994), "Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar", Nonlinearity 7 (6): 1495–1533, ISSN 0951-7715, http://stacks.iop.org/0951-7715/7/1495
- ↑ ^{2.0} ^{2.1} Resnick, Serge G. (1995), Dynamical problems in non-linear advective partial differential equations, ProQuest LLC, Ann Arbor, MI, http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:9542767
- ↑ Kiselev, A.; Nazarov, F.; Volberg, A. (2007), "Global well-posedness for the critical 2D dissipative quasi-geostrophic equation", Inventiones Mathematicae 167 (3): 445–453, doi:10.1007/s00222-006-0020-3, ISSN 0020-9910, http://dx.doi.org/10.1007/s00222-006-0020-3
- ↑ Caffarelli, Luis A.; Vasseur, Alexis (2010), "Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation", Annals of Mathematics. Second Series 171 (3): 1903–1930, doi:10.4007/annals.2010.171.1903, ISSN 0003-486X, http://dx.doi.org/10.4007/annals.2010.171.1903
- ↑ Kiselev, A.; Nazarov, F. (2009), "A variation on a theme of Caffarelli and Vasseur", Rossiĭskaya Akademiya Nauk. Sankt-Peterburgskoe Otdelenie. Matematicheski\u\i Institut im. V. A. Steklova. Zapiski Nauchnykh Seminarov (POMI) 370: 58–72, ISSN 0373-2703
- ↑ Constantin, Peter; Vicol, Vlad (2012), "Nonlinear maximum principles for dissipative linear nonlocal operators and applications", Geometric and Functional Analysis,to appear, http://arxiv.org/abs/1110.0179
- ↑ Yu, Xinwei (2008), "Remarks on the global regularity for the super-critical 2D dissipative quasi-geostrophic equation", Journal of Mathematical Analysis and Applications 339 (1): 359–371, doi:10.1016/j.jmaa.2007.06.064, ISSN 0022-247X, http://dx.doi.org/10.1016/j.jmaa.2007.06.064
- ↑ Silvestre, Luis (2010), "Eventual regularization for the slightly supercritical quasi-geostrophic equation", Annales de l'Institut Henri Poincaré. Analyse Non Linéaire 27 (2): 693–704, doi:10.1016/j.anihpc.2009.11.006, ISSN 0294-1449, http://dx.doi.org/10.1016/j.anihpc.2009.11.006
- ↑ Dabkowski, M. (2011), "Eventual Regularity of the Solutions to the Supercritical Dissipative Quasi-Geostrophic Equation", Geometric and Functional Analysis (Berlin, New York: Springer-Verlag) 21 (1): 1–13, ISSN 1016-443X
- ↑ Kiselev, A. (2010), "Regularity and blow up for active scalars", Mathematical Modelling of Natural Phenomena 5 (4): 225–255, doi:10.1051/mmnp/20105410, ISSN 0973-5348, http://dx.doi.org/10.1051/mmnp/20105410