Surface Quasi-geostrophic Equation Research Articles

Slow traveling-wave solutions for the generalized surface quasi-geostrophic equation

In this paper, we systematically study the existence, asymptotic behaviors, uniqueness, and nonlinear orbital stability of traveling-wave solutions with small propagation speeds for the generalized surface quasi-geostrophic (gSQG) equation. Firstly we obtain the existence of a new family of global solutions via the variational method. Secondly we show the uniqueness of maximizers under our variational setting. Thirdly by using the variational framework, the uniqueness of maximizers and a concentration-compactness principle we establish some stability theorems. Moreover, after a suitable transformation, these solutions constitute the desingularization of traveling point vortex pairs.

Non-existence and Strong lll-posedness in Ck,β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{k,\\beta }$$\\end{document} for the Generalized Surface Quasi-geostrophic Equation

We consider solutions to the generalized Surface Quasi-geostrophic equation (γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma $$\\end{document}-SQG) when the velocity is more singular than the active scalar function (i.e. γ∈(0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma \\in (0,1)$$\\end{document}). In this paper we establish strong ill-posedness in Ck,β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{k,\\beta }$$\\end{document} (k≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k\\ge 1$$\\end{document}, β∈(0,1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta \\in (0,1]$$\\end{document} and k+β>1+γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k+\\beta >1+\\gamma $$\\end{document}) and we also construct solutions in R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^2$$\\end{document} that initially are in Ck,β∩L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{k,\\beta }\\cap L^2$$\\end{document} but are not in Ck,β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{k,\\beta }$$\\end{document} for t>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t>0$$\\end{document}. Furthermore these solutions stay in Hk+β+1-2δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^{k+\\beta +1-2\\delta }$$\\end{document} for some small δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta $$\\end{document} and an arbitrarily long time.

Surface quasi-geostrophic equation perturbed by derivatives of space-time white noise

Surface quasi-geostrophic equation perturbed by derivatives of space-time white noise

Geometric analysis of the generalized surface quasi-geostrophic equations

AbstractWe investigate the geometry of a family of equations in two dimensions which interpolate between the Euler equations of ideal hydrodynamics and the inviscid surface quasi-geostrophic equation. This family can be realised as geodesic equations on groups of diffeomorphisms. We show precisely when the corresponding Riemannian exponential map is non-linear Fredholm of index 0. We further illustrate this by examining the distribution of conjugate points in these settings via a Morse theoretic approach

Well-posedness for the surface quasi-geostrophic front equation

We consider the well-posedness of the surface quasi-geostrophic (SQG) front equation. Hunter–Shu–Zhang (2021 Pure Appl. Anal. 3 403–72) established well-posedness under a small data condition as well as a convergence condition on an expansion of the equation’s nonlinearity. In the present article, we establish unconditional large data local well-posedness of the SQG front equation, while also improving the low regularity threshold for the initial data. In addition, we establish global well-posedness theory in the rough data regime by using the testing by wave packet approach of Ifrim–Tataru.

Optimal Gevrey Regularity for Supercritical Quasi-Geostrophic Equations

We consider the two dimensional surface quasi-geostrophic equations with super-critical dissipation. For large initial data in critical Sobolev and Besov spaces, we prove optimal Gevrey regularity endowed with the same decay exponent as the linear part. This settles several open problems in Biswas (J Differ Equ 257(6):1753–1772, 2014), Biswas et al. (J Funct Anal 269(10):3083–3119, 2015).

2D Surface Quasi Geostrophic Equations and its Regularity, A Numerical Study

In this article, we present some results on the numerical solutions of the 2-D Surface Quasi Geostrophic Equation (SQG) using the pseudospectral method along with an exponential filter. The global regularity of the solution of the inviscid SQG equation for general data remains an outstanding open problem. Our computations mainly focus on the inviscid and supercritical cases. We monitored the regions where the level curves come significantly close to one another, the L2 norm, and the growth of |∇⊥θ| throughout our computations. Our numerical findings show that there is no significant difference among the solutions of the supercritical, critical, and subcritical cases as we vary the values of the parameter α in the interval (0, 1).

Some Conditions of Non-Blow-Up of Generalized Inviscid Surface Quasigeostrophic Equation

In this paper, we survey some non-blow-up results for the following generalized modified inviscid surface quasigeostrophic equation (GSQG) θ t + u · ∇ θ = 0 , u = ∇ ⊥ ψ , − Λ β ψ = θ , θ x , 0 = θ 0 x . . This is a generalized surface quasigeostrophic equation (GSQG) with the velocity field u related to the scalar θ by u = − ∇ ⊥ Λ − β θ , where 1 ≤ β ≤ 2 . We prove that there is no finite-time singularity if the level set of generalized quasigeostrophic equation does not have a hyperbolic saddle, and the angle of opening of the saddle can go to zero at most as an exponential decay. Moreover, we give some conditions that rule out the formation of sharp fronts for generalized inviscid surface quasigeostrophic equation, and we obtain some estimates on the formation of semiuniform fronts. These results greatly weaken the geometrical assumptions which rule out the collapse of a simple hyperbolic saddle in finite time.

Hölder regularity for collapses of point-vortices

The first part of this article studies the collapses of point-vortices for the Euler equation in the plane and for surface quasi-geostrophic equations in the general setting of α models. In these models the kernel of the Biot–Savart law is a power function of exponent . It is proved that, under a standard non-degeneracy hypothesis, the trajectories of the point-vortices have a Hölder regularity up to the time of collapse. The Hölder exponent obtained is and this exponent is proved to be optimal for all α by exhibiting an example of a three-vortex collapse. The same question is then addressed for the Euler point-vortex system in smooth bounded connected domains. It is proved that if a given point-vortex has an accumulation point in the interior of the domain as t → T, then it converges towards this point and displays the same Hölder continuity property. A partial result for point-vortices that collapse with the boundary is also established: we prove that their distance to the boundary is Hölder regular.

The evolution of surface quasi-geostrophic modons on sloping topography

This work discusses modons, or dipolar vortices, propagating along sloping topography. Two different regimes exist, which are studied separately using the surface quasi-geostrophic equations. First, when the modon propagates in the direction opposite to topographic Rossby waves, steady solutions exist and a semi-analytical method is presented for calculating these solutions. Second, when the modon propagates in the same direction as the Rossby waves, a wave wake is generated. This wake removes energy from the modon, causing it to decay slowly. Asymptotic predictions are presented for this decay and found to agree closely with numerical simulations. Over long times, decaying vortices are found to break down due to an asymmetry resulting from the generation of waves inside the vortex. A monopolar vortex moving along a wall is shown to behave in a similar way to a dipole, though the presence of the wall is found to stabilise the vortex and prevent the long-time breakdown. The problem is equivalent mathematically to a dipolar vortex moving along a density front, hence our results apply directly to this case.

Emergence of time periodic solutions for the generalized surface quasi-geostrophic equation in the disc

In this paper we address the existence of time periodic solutions for the generalized inviscid SQG equation in the unit disc with homogeneous Dirichlet boundary condition when α∈(0,1). We show the existence of a countable family of bifurcating curves from the radial patches. In contrast with the preceding studies in active scalar equations, the Green function is no longer explicit and we circumvent this issue by a suitable splitting into a singular explicit part (which coincides with the planar one) and a smooth implicit one induced by the boundary of the domain. Another problem is connected to the analysis of the linear frequencies which admit a complicated form through a discrete sum involving Bessel functions and their zeros. We overcome this difficulty by using Sneddon's formula leading to a suitable integral representation of the frequencies.

Energy Conservation for the Generalized Surface Quasi-geostrophic Equation

Energy Conservation for the Generalized Surface Quasi-geostrophic Equation

A note on the existence of stationary vortex patches for the SQG equation in bounded domain

By studying the contour dynamics equation and using the implicit function theorem, we prove the existence of stationary vortex patches with fixed vorticity and total flux for each patch for the surface quasi-geostrophic equation in a bounded domain near non-degenerate critical points of the Kirchhoff-Routh function.

Self-similar solutions to the hypoviscous Burgers and SQG equations at criticality

After reviewing the source-type solution of the Burgers equation with standard dissipativity, we study the hypoviscous counterpart of the Burgers equation. (1) We determine an equation that governs the near-identity transformation underlying its self-similar solution. (2) We develop its approximation scheme and construct the first-order approximation. (3) We obtain the source-type solution numerically by the Newton–Raphson iteration scheme and find it to agree well with the first-order approximation. Implications of the source-type solution are given, regarding the possibility of linearisation of the hypoviscous Burgers equation. Finally we address the problems of the incompressible fluid equations in two dimensions, centering on the surface quasi-geostrophic equation with standard and hypoviscous dissipativity.

Evolution Equations with Sectorial Operator on Fractional Power Scales

Originating with the famous monograph by Dan Henry, the semigroup approach to evolution problems having a positive sectorial operator in the main part allows us to settle them at various levels of the fractional power scale associated with the main linear operator. This translates into different regularity properties of local solutions to such equations. Specific applications of the abstract results to the 2D surface quasi-geostrophic equation or the fractional chemotaxis system are presented.

A class of supercritical/critical singular stochastic PDEs: Existence, non-uniqueness, non-Gaussianity, non-unique ergodicity

We study the surface quasi-geostrophic equation with an irregular spatial perturbation∂tθ+u⋅∇θ=−ν(−Δ)γ/2θ+ζ,u=∇⊥(−Δ)−1θ, on [0,∞)×T2, with ν⩾0, γ∈[0,3/2) and ζ∈B∞,∞−2+κ(T2) for some κ>0. This covers the case of ζ=(−Δ)α/2ξ for α<1 and ξ a spatial white noise on T2. Depending on the relation between γ and α, our setting is subcritical, critical or supercritical in the language of Hairer's regularity structures [38]. Based on purely analytical tools from convex integration and without the need of any probabilistic arguments including renormalization, we prove existence of infinitely many analytically weak solutions in Llocp(0,∞;B∞,1−1/2)∩Cb([0,∞);B∞,1−1/2−δ)∩Cb1([0,∞);B∞,1−3/2−δ) for all p∈[1,∞) and δ>0. We are able to prescribe an initial as well as a terminal condition at a finite time T>0, and to construct steady state, i.e. time independent, solutions. In all cases, the solutions are non-Gaussian, but we may as well prescribe Gaussianity at some given times. Moreover, a coming down from infinity with respect to the perturbation and the initial condition holds. Finally, we show that our solutions generate statistically stationary solutions as limits of ergodic averages, and we obtain existence of infinitely many non-Gaussian time dependent ergodic stationary solutions. We also extend our results to a more general class of singular SPDEs.

Existence of solutions to fluid equations in Hölder and uniformly local Sobolev spaces

We establish short-time existence of solutions to the surface quasi-geostrophic (SQG) equation in the Hölder spaces Cr(R2) for r>1; to avoid an integrability assumption (such as membership of the data in an Lq space) we introduce a generalization of the SQG constitutive law. As an application of the Hölder theory, we use these solutions when forming an approximation sequence in the proof of existence of solutions of SQG in another class of non-decaying function spaces, the uniformly local Sobolev spaces Huls(R2) for s≥3. Using methods similar to those for the surface quasi-geostrophic equation, we also obtain short-time existence for the three-dimensional Euler equations in uniformly local Sobolev spaces.

Global well-posedness of slightly supercritical SQG equations and gradient estimate

We prove the global regularity of smooth solutions for a dissipative surface quasi-geostrophic (SQG) equation with both velocity and dissipation logarithmically supercritical compared to the critical equation. By this, we mean that a symbol defined as a power of logarithm is added to both velocity and dissipation terms to penalize the equation’s criticality. Our primary tool is the nonlinear maximum principle which provides transparent proofs of global regularity for nonlinear dissipative equations. Combining the nonlinear maximum principle with a modulus of continuity, we prove an uniform-in-time gradient estimate for the critical and slightly supercritical SQG equation. It improves the previous double exponential bound by Kiselev–Nazarov–Volberg to the single exponential. In addition, we prove the eventual exponential decay of the solutions.

On the fractional Laplacian of some positive definite kernels with applications in numerically solving the surface quasi-geostrophic equation as a prominent fractional calculus model

This paper provides the Riesz potential and fractional Laplacian (−Δ)s, s∈R of the famous radial kernels, including the Gaussian, multiquadric, Sobolev spline, and mainly focuses on Wendland kernels. We show that (−Δ)s maps these kernels into the kernels constructed by the generalized hypergeometric functions or the Meijer G–functions. An essential application of the results applies to developing recent meshless methods for solving equations involving fractional Laplacian. Compared to the recently proposed Gaussian meshless method, our numerical results show that working with other kernels to solve fractional equations can bring better results. As a real word application, we consider the numerical study of the surface quasi-geostrophic equations.

Oceanic dipoles in a surface quasi-geostrophic model

Analysis of satellite altimetry and Argo float data leads Ni et al. (J. Geophys. Res., 125, 2020, e2020JC016479) to argue that mesoscale dipoles are widespread features of the global ocean having a relatively uniform three-structure that can lead to strong vertical exchanges. Almost all the features of the composite dipole they construct can be derived from a model for multipoles in the surface quasi-geostrophic equations for which we present a straightforward novel solution in terms of an explicit linear algebraic eigenvalue problem, allowing simple evaluation of the higher radial modes that appear to be present in the observations and suggesting that mass conservation may explain the observed frontogenetic velocities.