Blow-up of strong solutions of the Thermal Quasi-Geostrophic equation
Dan Crisan, Prince Romeo Mensah
备注：
12 pages. 'Blow-up' in title was previously 'Breakdown'. Submitted to the proceedings: 2021 STUOD - 2nd Stochastic Transport in Upper Ocean Dynamics Annual Workshop
The Thermal Quasi-Geostrophic (TQG) equation is a coupled system of equations that governs the evolution of the buoyancy and the potential vorticity of a fluid. It has a local in time solution as proved in [4]. In this paper, we give a criterion for the blow-up of solutions to the Thermal Quasi-Geostrophic equation, in the spirit of the classical Beale-Kato-Majda blow-up criterion (cf. [3]) for the solution of the Euler equation.
In this article we apply the technique proposed in Deng-Hou-Yu (Comm. PDE, 2005) to study the level set dynamics of the 2D quasi-geostrophic equation. Under certain assumptions on the local geometric regularity of the level sets of $θ$, we obtain global regularity results with improved growth estimate on $| \nabla^{\bot} θ|$. We further perform numerical simulations to study the local geometric properties of the level sets near the region of maximum $| \nabla^{\bot} θ|$. The numerical results indicate that the assumptions on the local geometric regularity of the level sets of $θ$ in our theore…
The problem we are concerned with is whether singularities form in finite time in incompressible fluid flows. It is well known that the answer is ``no'' in the case of Euler and Navier-Stokes equations in dimension two. In dimension three it is still an open problem for these equations. In this paper we focus on a two-dimensional active scalar model for the 3D Euler vorticity equation. Constantin, Majda and Tabak suggested, by studying rigorous theorems and detailed numerical experiments, a general principle: ``If the level set topology in the temperature field for the 2D quasi-geostrophic act…
We derive regularized contour dynamics equations for the motion of infinite sharp fronts in the two-dimensional incompressible Euler, surface quasi-geostrophic (SQG), and generalized surface quasi-geostrophic (gSQG) equations. We derive a cubic approximation of the contour dynamics equation and prove the short-time well-posedness of the approximate equations for generalized surface quasi-geostrophic fronts and weak well-posedness for surface quasi-geostrophic fronts.
We prove new \emph{a priori} estimates for the 3D Euler, the 3D Navier-Stokes and the 2D quasi-geostrophic equations by the method of similarity transforms.
We prove the global well-posedness of the critical dissipative quasi-geostrophic equation for large initial data belonging to the critical Besov space $\dot B^0_{\infty,1}(\RR^2).$
This paper was withdrawn by arXiv admin because it plagiarizes "Jiahong Wu The two-dimensional quasi-geostrophic equation with critical or supercritical dissipation 2005 Nonlinearity 18 139-154."
Inspired by recent developments in Berdina-like models for turbulence, we propose an inviscid regularization for the surface quasi-geostrophic (SQG) equations. 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 $α$ 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 invi…
In this paper, we study the super-critical Quasi-Geostrophic equation in Gevrey-Sobolev space. We prove the local existence of $(QG)$ for any large initial data and we give an exponential type of Blow-up to the solution. Moreover, we establish the existence global for a small initial data and we show that $\|θ\|_{H^s_{a, α^{-1}}}$ decays to zero as time goes to infinity. Fourier analysis and standard techniques are used.
We prove that weak solutions of a slightly supercritical quasi-geostrophic equation become smooth for large time. We prove it using a De Giorgi type argument using ideas from a recent paper by Caffarelli and Vasseur.
We consider the 2D quasi-geostrophic model and its two different regularizations. Global regularity results are established for the regularized models with subcritical or critical indices. The proof of Onsager's conjecture concerning weak solutions of the 3D Euler equations and the notion of dissipative solutions of Duchon and Robert are extended to weak solutions of the quasi-geostrophic equation.
In this paper, by using Fourier splitting method and the expanded properties of decay character $r^*$, we establish the algebraic decay rate of higher order derivative of solutions to 2D dissipative quasi-geostrophic flows.
We revisit a model for three-dimensional, inviscid quasi-geostrophic flow on bounded, cylindrical domains introduced by the authors in \cite{nv18}. We prove the local-in-time existence of classical solutions.
In a recent article by Gravejat and Smets, it is built smooth solutions to the inviscid surface quasi-geostrophic equation that have the form of a traveling wave. In this article we work back on their construction to provide solution to a more general class of quasi-geostrophic equation where the half-laplacian is replaced by any fractional laplacian.
We give an elementary proof of the global well-posedness for the critical 2D dissipative quasi-geostrophic equation. The argument is based on a non-local maximum principle involving appropriate moduli of continuity.
Motivated by the critical dissipative quasi-geostrophic equation, we prove that drift-diffusion equations with L^2 initial data and minimal assumptions on the drift are locally Holder continuous. As an application we show that solutions of the quasi-geostrophic equation with initial L^2 data and critical diffusion (-Δ)^{1/2}, are locally smooth for any space dimension.
We study the existence of doubly connected rotating patches for the inviscid surface quasi- geostrophic equation left open in \cite{HHH}. By using the approach proposed by \cite{CCGS} we also prove that close to the annulus the boundaries are actually analytic curves.
In this article, the authors prove the existence of global weak solutions to the inviscid three-dimensional quasi-geostrophic equation. This equation models the evolution of the temperature on the surface of the earth. It is widely used in geophysics and meteorology.
We show that the the 2 D quasi-geostrophic equation has global and unique strong solution, when the (large) data belongs in the critical, scale invariant space $\dot{B}^{2-2\al}_{2, \infty}\cap L^{2/(2\al-1)}$.
The initial value problem for the two dimensional dissipative quasi-geostrophic equation derived from geophisical fluid dynamics is studied. The dissipation of this equation is given by the fractional Laplacian. It is known that the half Laplacian is a critical dissipation for the quasi-geostrophic equation. In this paper, far field asymptotics of solutions are given in the critical and the supercritical cases.
In this paper, we show that the global solution of the surface anisotropic two-dimensional quasi-geostrophic equation with fractional horizontal dissipation and vertical thermal diffusion established by the author in [2] is bounded in Sobolev spaces uniformly with respect to time.
Consider the surface quasi-geostrophic equation with random diffusion, white in time. We show global existence and uniqueness in high probability for the associated Cauchy problem satisfying a Gevrey type bound. This article is inspired by recent work of Glatt-Holtz and Vicol.
We consider surface quasi-geostrophic equation with dispersive forcing and critical dissipation. We prove global existence of smooth solutions given sufficiently smooth initial data. This is done using a maximum principle for the solutions involving conservation of a certain family of moduli of continuity.
In this paper, we prove the existence of relative equilibria with holes for quasi-geostrophic shallow-water equations. More precisely, using bifurcation techniques, we establish for any m large enough the existence of two branches of m-fold doubly-connected V-states bifurcating from any annulus of arbitrary size.
This paper was removed by arXiv admin because it plagiarizes the electronic article HYKE 2005-010 (http://www.hyke.org/preprint/index.php), by Jose-A. Carrillo and Lucas C.f. Ferreira, entitled "Self-similar solutions and large time asymptotics for the dissipative quasi-geostrophic equations".
Motivated by the De Giorgi type argument used in a recent paper by Caffarelli and Vasseur, we prove Hölder-regularity for weak solutions of the supercritical quasi-geostrophic equation with minimal assumptions on the initial datum.
In the present study, we find that the surface quasi-geostrophic equation admits exact solutions, which evolve with time in quasi-stationary states. The solutions presented are available for any dissipation effect $κ(-Δ)^α$ ($κ>0$, $0\le α<1$), involved in the equation. When the equation is supercritical ($0\le α<\frac12$), the problem on the existence of large global regular solutions remains open. This study, however, provides explicit sample solutions for the understanding of the uncertain problem.
We compute the sharp time decay rates of the solutions of the IVP for quasi-geostrophic equation and the Boussinesq model, subject to fractional dissipation. Moreover, we explicitly identify the asymptotic profiles, the kernel of the $α$ stable processes, which are analogues of the Oseen vortices.
We provide two-sided pointwise estimates and uniform asymptotics of the solutions to the subcritical quasi-geostrophic equation with initial data in $L^{2/(α-1)}(\mathbb{R}^2)$. Furthermore, we give upper bound of similar type for any derivative of the solutions. Initial data in $L^{p}(\mathbb{R}^2)$, $p>2/(α-1)$, are also discussed.
The inviscid barotropic quasi-geostrophic equation with a free surface is considered. The free surface mandates a non-standard boundary condition. The global existence existence and uniqueness of a weak solution is established, thanks to the uniform in time bounds on the potential vorticity. The solution is also shown to satisfy the initial and boundary conditions in the classical sense.
Motivated by the recent work of Hassainia and Hmidi [Z. Hassainia, T. Hmidi - On the {V}-states for the generalized quasi-geostrophic equations,arXiv preprint arXiv:1405.0858], we close the question of the existence of convex global rotating solutions for the generalized surface quasi-geostrophic equation for $α\in [1,2)$. We also show $C^{\infty}$ regularity of their boundary for all $α\in (0,2)$.
We consider the stationary Quasi-Geostrophic equation in the whole space $\mathbb R^2$ driven by a force $f$. Under certain smallness assumptions of $f$, we establish the existence of solutions with finite $L^2$ norm. This solution is unique among all solutions with finite energy. The unique solution $Θ$ is also shown to be stable in the sense: any solution of the evolutionary Quasi-Geostrophic equation driven by $f$ and starting with finite energy, will return to $Θ$.
We construct a linear approximation of the solution to the Surface Quasi-Geostrophic Equation in $\mathbb{R}^2$, and obtain a convergence rate in $L^p$ between the solution and this approximation with respect to time. We also demonstrate that the nonlinear term of the solution is bounded sharply in $L^p$ by the same function of time.
The one-dimensional quasi-geostrophic equation is the one-dimensional Fourier-space analogue of the famous Navier-Stokes equations. In their work Li and Sinai have proposed a renormalization approach to the problem of existence of finite-time blow up solutions of this equation. In this setting, existence of finite time blow ups is a consequence of existence of a fixed point for a certain renormalization operator on an appropriate functional space. They have provided a proof of existence of complex-valued finite time blow up solutions of the quasi-geostrophic equation. In this paper we revisit …
We study the critical and super-critical dissipative quasi-geostrophic equations in $\bR^2$ or $\bT^2$. Higher regularity of mild solutions with arbitrary initial data in $H^{2-γ}$ is proved. As a corollary, we obtain a global existence result for the critical 2D quasi-geostrophic equations with periodic $\dot H^1$ data. Some decay in time estimates are also provided.
In this paper, we prove the boundedness of Riesz transforms $\partial_{j}(-Δ)^{-1/2}$ ($j=1,2,...,n$) on the Q-type spaces $Q_α^β(\mathbb{R}^{n})$. As an application, we get the well-posedness and regularity of the quasi-geostrophic equation with initial data in $Q_α^{β,-1}(\mathbb{R}^{2}).$
We present a regularity result for weak solutions of the 2D quasi-geostrophic equation with supercritical ($α< 1/2$) dissipation $(-Δ)^α$ : If a Leray-Hopf weak solution is Hölder continuous $θ\in C^δ({\mathbb R}^2)$ with $δ>1-2α$ on the time interval $[t_0, t]$, then it is actually a classical solution on $(t_0,t]$.
In this paper we study the super-critical 2D dissipative quasi-geostrophic equation. We obtain some regularization effects allowing us to prove global well-posedness result for small initial data lying in critical Besov spaces constructed over Lebesgue spaces $L^p,$ \mbox{with $p\in[1,\infty].$ Local results for arbitrary initial data are also given.
In this paper we prove, if $θ\in C([0,\infty),H^{2-2α}(\mathbb R^2))$ is a global solution of supercritical surface Quasi-Geostrophic equation with small initial data, then $\|θ(t)\|_{H^{2-2α}}$ decays to zero as time goes to infinity. Fourier analysis and standard techniques are used.
We construct families of smooth travelling-wave solutions to the inviscid surface quasi-geostrophic equation (SQG). These solutions can be viewed as the equivalents for this equation of the vortex anti-vortex pairs in the context of the incompressible Euler equation. Our argument relies on the stream function formulation and eventually amounts to solving a fractional nonlinear elliptic equation by variational methods.
In this article we apply the method used in the recent elegant proof by Kiselev, Nazarov and Volberg of the well-posedness of critically dissipative 2D quasi-geostrophic equation to the super-critical case. We prove that if the initial value is smooth and periodic, and $\left\| \nabla θ_0 \right\|_{L^{\infty}}^{1 - 2 s} \left\| θ_0 \right\|_{L^{\infty}}^{2 s}$ is small, where $s$ is the power of the fractional Laplacian, then no finite time singularity will occur for the super-critically dissipative 2D quasi-geostrophic equation.
The purpose of this note is to study the weak solutions to the inviscid quasi-geostrophic system for initial data belonging to Lebesgue spaces. We give a global existence result as well as detail the connections between several different notions of weak solutions. In addition, we give a condition under which the energy of the system is conserved.
We study the Cauchy problem for the quasi-geostrophic equations with the critical dissipation in the two dimensional half space under the homogeneous Dirichlet boundary condition. We show the global existence, the uniqueness and the analyticity of solutions, and the real analyticity up to the boundary is obtained. We will show one of natural ways to estimate the nonlinear term for functions satisfying the Dirichlet boundary condition.
We study the behavior at infinity in time of the global solution of the anisotropic quasi-geostrophic equation $θ\in C_b(\mathbb{R}^+,H^s( \mathbb{R}^2))$. We prove that this solution decays to zero as time goes to infinity in $L^p(\mathbb{R}^2)$, $p\in [2,+\infty],$ moreover, we prove also that $\lim\limits_{t\rightarrow +\infty}\|θ(t)\|_{H^s}=0$.
In this article I study Hölder regularity for solutions of a transport equation based in the dissipative quasi-geostrophic equation. Following a recent idea of A. Kiselev and F. Nazarov, I will use the molecular characterization of local Hardy spaces in order to obtain information on Hölder regularity of such solutions. This will be done by following the evolution of molecules in a backward equation.
We show a new Bernstein's inequality which generalizes the results of Cannone-Planchon, Danchin and Lemarié-Rieusset. As an application of this inequality, we prove the global well-posedness of the 2D quasi-geostrophic equation with the critical and super-critical dissipation for the small initial data in the critical Besov space, and local well-posedness for the large initial data.
We introduce a determining wavenumber for the surface quasi-geostrophic (SQG) equation defined for each individual trajectory and then study its dependence on the force. While in the subcritical and critical cases this wavenumber has a uniform upper bound, it may blow up when the equation is supercritical. A bound on the determining wavenumber provides determining modes, which in some sense measure the number of degrees of freedom of the flow, or resolution needed to describe a solution to the SQG equation.
In this paper, we prove the nonlinear orbital stability of vortex dipoles for the quasi-geostrophic shallow-water (QGSW) equations. The vortex dipoles are explicit travelling wave solutions to the QGSW equations, which are analogues of the classical circular vortex of Lamb and Chaplygin for the steady planar Euler equations. We establish a variational characterization of these vortex poles, which provides a basis for the stability result.
In this paper, we focus on the two-dimensional surface quasi-geostrophic equation with fractional horizontal dissipation and vertical thermal diffusion which represents a general case of the classical surface quasi-geostrophic equation. On the one hand, we will show the local existence and uniqueness of the solution in Sobolev space $H^{2-2α}(\mathbb{R}^2)\cap H^{2-2β}(\mathbb{R}^2)$, which is the critical space in the classical case. Furthermore, we will demonstrate that the solution is global even when the initial data is very small. Finally, we will study the asymptotic representation of ou…
We consider classical solutions of the inviscid Surface Quasi-geostrophic equation that are a small perturbation $ε$ from a radial stationary solution $θ=|x|$. We use a modified energy method to prove the existence time of classical solutions from $\frac{1}ε$ to a time scale of $\frac{1}{ε^4}$. Moreover, by perturbing in a suitable direction we construct global smooth solutions, via bifurcation, that rotate uniformly in time and space.
This paper studies the regularity and energy conservation problems for the 2D supercritical quasi-geostrophic (SQG) equation. We apply an approach of splitting the dissipation wavenumber to obtain a new regularity condition which is weaker than all the Prodi-Serrin type regularity conditions. Moreover, we prove that any weak solution of the supercritical SQG in $L^2(0,T; B^{1/2}_{2,c(\mathbb N)})$ satisfies energy equality.
In this paper, following the techniques of Foias and Temam, we establish suitable Gevrey class regularity of solutions to the supercritical quasi-geostrophic equations in the whole space, with initial data in "critical" Sobolev spaces. Moreover, the Gevrey class that we obtain is "near optimal" and as a corollary, we obtain temporal decay rates of higher order Sobolev norms of the solutions. Unlike the Navier-Stokes or the subcritical quasi-geostrophic equations, the low dissipation poses a difficulty in establishing Gevrey regularity. A new commutator estimate in Gevrey classes, involving the…
In this paper, we consider the modified quasi-geostrophic equation \begin{gather*} \del_t θ+ (u \cdot \grad) θ+ κΛ^αθ= 0 u = Λ^{α- 1} R^{\perp}θ. \end{gather*} with $κ> 0$, $α\in (0,1]$ and $θ_0 \in \lp{2}(\R^2)$. We remark that the extra $Λ^{α- 1}$ is introduced in order to make the scaling invariance of this system similar to the scaling invariance of the critical quasi-geostrophic equations. In this paper, we use Besov space techniques to prove global existence and regularity of strong solutions to this system.
