Justification of the Point Vortex Approximation for Modified Surface Quasi-Geostrophic Equations

This paper studies the finite difference method for the surface quasi-geostrophic (SQG) equations with the fractional Laplacian and Riesz potential. A numerical discretization for the Riesz potential in the SQG equations is proposed. The second-order convergence of the method in l ∞ norm has been verified through both theoretical analysis and numerical experiments. Based on the upwind scheme, an implicit-explicit finite difference (IMEX-FD) scheme for the SQG equations is established, in which the convection term is discretized explicitly and implicitly, while the discretization of other terms employs an implicit structure. The stability of this scheme is proven when the order of fractional Laplacian is one, and the first-order convergence in time and space is also verified through numerical experiments. Finally, the evolutions of single and double vortices problems in SQG equations are simulated in this paper to verify the effectiveness of the IMEX-FD scheme.

Whether or not classical solutions to the surface quasi-geostrophic (SQG) equation can develop finite time singularities remains an outstanding open problem. This paper constructs a class of large global-in-time classical solutions to the SQG equation with supercritical dissipation. The construction process presented here implies that any solution of the supercritical SQG equation must be globally regular if its initial data is sufficiently close to a function (measured in a Sobolev norm) whose Fourier transform is supported in a suitable region away from the origin.

Local well-posedness for the generalized surface quasi-geostrophic equation with singular velocity in optimal space

Generalized singular integral with rough kernel and approximation of surface quasi-geostrophic equation

This paper examines the global regularity problem on several 2D incompressible fluid models with partial dissipation. They are the surface quasi-geostrophic (SQG) equation, the 2D Euler equation and the 2D Boussinesq equations. These are well-known models in fluid mechanics and geophysics. The fundamental issue of whether or not they are globally well-posed has attracted enormous attention. The corresponding models with partial dissipation may arise in physical circumstances when the dissipation varies in different directions. We show that the SQG equation with either horizontal or vertical dissipation always has global solutions. This is in sharp contrast with the inviscid SQG equation for which the global regularity problem remains outstandingly open. Although the 2D Euler is globally well-posed for sufficiently smooth data, the associated equations with partial dissipation no longer conserve the vorticity and the global regularity is not trivial. We are able to prove the global regularity for two partially dissipated Euler equations. Several global bounds are also obtained for a partially dissipated Boussinesq system.

This paper establishes several existence and uniqueness results for two families of active scalar equations with velocity fields determined by the scalars through very singular integrals. The first family is a generalized surface quasigeostrophic (SQG) equation with the velocity fieldurelated to the scalarθby$u=\nabla^\perp\Lambda^{\beta-2}\theta$, where$1<\beta\le 2$and$\Lambda=(-\Delta)^{1/2}$is the Zygmund operator. The borderline caseβ= 1 corresponds to the SQG equation and the situation is more singular forβ> 1. We obtain the local existence and uniqueness of classical solutions, the global existence of weak solutions, and the local existence of patch‐type solutions. The second family is a dissipative active scalar equation with$u=\nabla^\perp (\log(I-\Delta))^\mu\theta\ {\rm for}\ \mu>0$, which is at least logarithmically more singular than the velocity in the first family. We prove that this family with any fractional dissipation possesses a unique local smooth solution for any given smooth data. This result for the second family constitutes a first step towards resolving the global regularity issue recently proposed by K. Ohkitani. © 2012 Wiley Periodicals, Inc.

This article is devoted to the study of the critical dissipative surface quasi-geostrophic (SQG) equation in $${\mathbb{R}^2}$$ . For any initial data $${\theta_{0}}$$ belonging to the space $${\Lambda^{s} ( H^{s}_{uloc}(\mathbb{R}^2)) \cap L^\infty(\mathbb{R}^2)}$$ , we show that the critical (SQG) equation has at least one global weak solution in time for all 1/4 ≤ s ≤ 1/2 and at least one local weak solution in time for all 0 < s < 1/4. The proof for the global existence is based on a new energy inequality which improves the one obtain in Lazar (Commun Math Phys 322:73–93, 2013) whereas the local existence uses more refined energy estimates based on Besov space techniques.

In this paper, we consider weighted jump and variational inequalities of truncated singular integral operator with rough kernel [Formula: see text] where the kernel [Formula: see text] satisfies the cancellation condition and the homogeneous condition of degree [Formula: see text] and [Formula: see text]. This kind of singular integral appears in the approximation of the surface quasi-geostrophic (SQG) equation from the generalized SQG equation. We establish the [Formula: see text] estimate of the jump and variational inequalities of the families [Formula: see text] for [Formula: see text]. Moreover, one can get the weighted [Formula: see text] boundedness of the Calderón–Zygmund operator by letting [Formula: see text].

The focus of the course is on small scale formation in solutions of the incompressible Euler equation of fluid dynamics and associated models. We first review the regularity results and examples of small scale growth in two dimensions. Then we discuss a specific singular scenario for the three-dimensional Euler equation discovered by Hou and Luo, and analyze some associated models. Finally, we will also talk about the surface quasi-geostrophic (SQG) equation, and construct an example of singularity formation in the modified SQG patch solutions as well as examples of unbounded growth of derivatives for the smooth solutions.

Patch solutions for the surface quasigeostrophic (SQG) equation model sharp temperature fronts in atmospheric and oceanic flows. Boundedness of curvature plays an important role in the theoretical [F. Gancedo and R. M. Strain, Absence of splash singularities for surface quasi-geostrophic sharp fronts and the Muskat problem, Proc. Natl. Acad. Sci. USA 111 (2014) 635–639] and numerical [D. Córdoba, M. A. Fontelos, A. M. Mancho and J. L. Rodrigo, Evidence of singularities for a family of contour dynamics equations, Proc. Natl. Acad. Sci. USA 102 (2005) 5949–5952; R. K. Scott and D. G. Dritschel, Numerical simulation of a self-similar cascade of filament instabilities in the surface quasigeostrophic system, Phys. Rev. Lett. 112 (2014) 144505] study of singularity formation. In this paper, we establish local well-posedness for SQG sharp fronts of low Sobolev regularity, [Formula: see text] for arbitrarily small [Formula: see text]. This is the first construction for SQG sharp front solutions with unbounded curvature.

In this paper we study the time evolution of a temperature patch in $\mathbb{R}^2$ according to the modified Surface Quasi-Geostrophic (SQG) patch equation. In particular we give a temporal estimate on the growth of the support, providing a rigorous proof of the confinement of a hot patch of temperature in absence of external forcing, under the quasi-geostrophic approximation.

An intrinsic property of almost any physical measuring device is that it makes observations which are slightly blurred in time. The authors consider a nudging-based approach for data assimilation that constructs an approximate solution based on a feedback control mechanism that is designed to account for observations that have been blurred by a moving time average. Analysis of this nudging model in the context of the subcritical surface quasi-geostrophic equation shows, provided the time-averaging window is sufficiently small and the resolution of the observations sufficiently fine, that the approximating solution converges exponentially fast to the observed solution over time. In particular, the authors demonstrate that observational data with a small blur in time possess no significant obstructions to data assimilation provided that the nudging properly takes the time averaging into account. Two key ingredients in our analysis are additional bounded-ness properties for the relevant interpolant observation operators and a non-local Gronwall inequality.

We propose one-dimensional reduced models for the three-dimensional electron magnetohydrodynamics which involves a highly nonlinear Hall term with intricate structure. The models contain nonlocal nonlinear terms which are more singular than that of the one-dimensional models for the Euler equation and the surface quasi-geostrophic equation. Local well-posedness is obtained in certain circumstances. Moreover, for a model with nonlocal transport term, we show that singularity develops in finite time for a class of initial data.

For the generalized surface quasi-geostrophic equation { a m p ; ∂ t θ + u ⋅ ∇ θ = 0 , in R 2 × ( 0 , T ) , a m p ; u = ∇ ⊥ ψ , ψ = ( − Δ ) − s θ in R 2 × ( 0 , T ) , \begin{equation*} \left \{ \begin {aligned} &amp; \partial _t \theta +u\cdot \nabla \theta =0, \quad \text {in } \mathbb {R}^2 \times (0,T), \\ &amp; u=\nabla ^\perp \psi , \quad \psi = (-\Delta )^{-s}\theta \quad \text {in } \mathbb {R}^2 \times (0,T) , \end{aligned} \right . \end{equation*} 0 &gt; s &gt; 1 0&gt;s&gt;1 , we consider for k ≥ 1 k\ge 1 the problem of finding a family of k k -vortex solutions θ ε ( x , t ) \theta _\varepsilon (x,t) such that as ε → 0 \varepsilon \to 0 θ ε ( x , t ) ⇀ ∑ j = 1 k m j δ ( x − ξ j ( t ) ) \begin{equation*} \theta _\varepsilon (x,t) \rightharpoonup \sum _{j=1}^k m_j\delta (x-\xi _j(t)) \end{equation*} for suitable trajectories for the vortices x = ξ j ( t ) x=\xi _j(t) . We find such solutions in the special cases of vortices travelling with constant speed along one axis or rotating with same speed around the origin. In those cases the problem is reduced to a fractional elliptic equation which is treated with singular perturbation methods. A key element in our construction is a proof of the non-degeneracy of the radial ground state for the so-called fractional plasma problem ( − Δ ) s W = ( W − 1 ) + γ , in R 2 , 1 &gt; γ &gt; 1 + s 1 − s \begin{equation*} (-\Delta )^sW = (W-1)^\gamma _+ , \quad \text {in } \mathbb {R}^2, \quad 1&gt;\gamma &gt; \frac {1+s}{1-s} \end{equation*} whose existence and uniqueness have recently been proven in Chan, del Mar González, Huang, Mainini, and Volzone [Calc. Var. Partial Differential Equations 59 (2020), p. 42].

The well-posedness of the surface quasi-geostrophic equations in the Besov–Morrey spaces