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.
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 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.
This paper was withdrawn by arXiv admin because it plagiarizes "Jiahong Wu The two-dimensional quasi-geostrophic equation with critical or supercritical dissipation 2005 Nonlinearity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."
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 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 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 show the global existence for critical dissipative quasi-geostrophic equations if $\|\widehat{θ^0}\|_{L^1}$ is small enough; among others we prove the analyticity of such a solution. If in addition the initial condition verifies $|D|^{-δ}θ^0\in L^1(\mathbb R^2)$ with $0<δ<1$, then the solution remains regular and $\lim_{t\rightarrow\infty}t^δ\|\widehatθ(t)\|_{L^1}=0$. Fourier analysis and standard techniques are used.
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.
The 2D quasi-geostrophic (QG) equation is a two dimensional model of the 3D incompressible Euler equations. When dissipation is included in the model then solutions always exist if the dissipation's wave number dependence is super-linear. Below this critical power the dissipation appears to be insufficient. For instance, it is not known if the critical dissipative QG equation has global smooth solutions for arbitrary large initial data. In this paper we prove existence and uniqueness of global classical solutions of the critical dissipative QG equation for initial data that have small $L^\inft…
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.
In this paper, we consider the following modified quasi-geostrophic equations $\partial_tθ+Λ^αθ+u\vec\nablaθ=0$, $u=Λ^{α-1}\mathcal{R}^\perp(θ)$ where $α\in ]0,1[$ is a fixed parameter. This equation was recently introduced by P. Constantin, G. Iyer and J. Wu in \cite{CIW} as a modification of the classical quasi-geostrophic equation. In this paper, we prove that for any initial data $θ_\ast$ in the Sobolev space $H^1(\mathbb{R}^2),$ the equation (MQG) has a global and smooth solution $θ$ in $C(\mathbb{R}^{+},H^1(\mathbb{R}^2)) .$
Recently, Silvestre proved that certain weak solutions of the slightly supercritical surface quasi-geostrophic equation eventually become smooth. To prove this, he employed a De Giorgi type argument originated in the work of Caffarelli and Vasseur. Kiselev and Nazarov proved a variation of the result of Caffarelli and Vasseur by introducing a class of test functions. Motivated by the results of Silvestre, we will modify the class of test functions from the work of Kiselev and Nazarov and use this modified class to show that a solution to the supercritical SQG that is smooth up to a certain tim…
In this article, we study the critical dissipative surface quasi-geostrophic equation (SQG) in $ \mathbb{R}^2$. Motivated by the study of the homogeneous statistical solutions of this equation, we show that for any large initial data $θ_{0}$ liying in the space $Λ^{s} (\dot H^{s}_{uloc}(\mathbb{R}^2)) \cap L^\infty(\mathbb{R}^2)$ the critical (SQG) has a global weak solution in time for all $1/2< s<1$. Our proof is based on an energy inequality verified by the truncated $(SQG)_{R,\ep}$ equation. By classical compactness arguments, we show that we are able to pass to the limit ($R \rightarrow \…
In this paper, we consider the two-dimensional surface quasi-geostrophic equation with fractional horizontal dissipation and fractional vertical thermal diffusion. Global existence of classical solutions is established when the dissipation powers are restricted to a suitable range. Due to the nonlocality of these 1D fractional operators, some of the standard energy estimate techniques no longer apply, to overcome this difficulty, we establish several anisotropic embedding and interpolation inequalities involving fractional derivatives. In addition, in order to bypass the unavailability of the …
This paper is devoted to the global (in time) regularity problem for a family of active scalar equations with fractional dissipation. Each component of the velocity field $u$ is determined by the active scalar $θ$ through $\mathcal{R} Λ^{-1} P(Λ) θ$ where $\mathcal{R}$ denotes a Riesz transform, $Λ=(-Δ)^{1/2}$ and $P(Λ)$ represents a family of Fourier multiplier operators. The 2D Navier-Stokes vorticity equations correspond to the special case $P(Λ)=I$ while the surface quasi-geostrophic (SQG) equation to $P(Λ) =Λ$. We obtain the global regularity for a class of equations for which $P(Λ)$ and …
This paper is devoted to the study of the dynamical behavior of the critically dissipative quasi-geostrophic equation in $\textbf{T}^2$. We prove that this system possesses time-dependent periodic solutions, bifurcating from a smooth steady solution, i.e. a Hopf-Bifurcation theorem.
We consider the initial value problem for the 2D quasi-geostrophic equation with weak dissipation term $κ(-Δ)^{α/2}θ (0<α\leqslant 1)$ and dispersive forcing term $Au_2$. We establish a unique global solution for a given initial data $θ_0$ which belongs to the scaling subcritical Sobolev space $H^s(\mathbb{R}^2)\ (s>2-α)$ if the size of dispersion parameter is sufficiently large. This phenomenon is so-called the global regularity. We also obtain the relationship between the initial data and the dispersion parameter, which ensures the existence of the global solution. Moreover, we show the glob…
We prove the global well-posedness of the critical dissipative quasi-geostrophic equation for large initial data belonging to the critical Besov space $\dot B^0_{\infty,1}(\RR^2).$
We address the decay of the norm of weak solutions to the 2D dissipative quasi-geostrophic equation. When the initial data is in $L^2$ only, we prove that the $L^2$ norm tends to zero but with no uniform rate, that is, there are solutions with arbitrarily slow decay. For the initial data in $L^p \cap L^2$, with $1 \leq p < 2$, we are able to obtain a uniform decay rate in $L^2$. We also prove that when the $L^{\frac{2}{2 α-1}}$ norm of the initial data is small enough, the $L^q$ norms, for $q > \frac{2}{2 α-1}$ have uniform decay rates. This result allows us to prove decay for the $L^q$ norms,…
This is a remark that by using an adaptation of the technique invented by A. Kiselev, F. Nazarov, and A. Voldberg, with a modified scaling argument, we can prove global regularity of the critical 2-D dissipative quasi-geostrophic equation with smooth periodic force, under the assumption that the initial data is smooth and periodic, and the force is $α$-Hölder continuous in space, $α> 0$.
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.
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".
We study the critical dissipative quasi-geostrophic equations in $\bR^2$ with arbitrary $H^1$ initial data. After showing certain decay estimate, a global well-posedness result is proved by adapting the method in [11] with a suitable modification. A decay in time estimate for higher order homogeneous Sobolev norms of solutions is also discussed.
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.
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 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.
We consider the 2D quasi-geostrophic equation with supercritical dissipation and dispersive forcing in the whole space. When the dispersive amplitude parameter is large enough, we prove the global well-posedness of strong solution to the equation with large initial data. We also show the strong convergence result as the amplitude parameter goes to $\infty$. Both results rely on the Strichartz-type estimates for the corresponding linear equation.
In this paper, we prove that if the initial data $θ_0$ and its Riesz transforms ($\mathcal{R}_1(θ_0)$ and $\mathcal{R}_2(θ_0)$) belong to the space $(\overline{S(\mathbb{R}^2))}^{B_{\infty}^{1-2α,\infty}}$, where $α\in ]1/2,1[$, then the 2D Quasi-Geostrophic equation with dissipation $α$ has a unique global in time solution $θ$. Moreover, we show that if in addition $θ_0 \in X$ for some functional space $X$ such as Lebesgue, Sobolev and Besov's spaces then the solution $θ$ belongs to the space $C([0,+\infty [,X).$
In this paper, we study transport equations with nonlocal velocity fields with rough initial data. We address the global existence of weak solutions of an one dimensional model of the surface quasi-geostrophic equation and the incompressible porous media equation, and one dimensional and $n$ dimensional models of the dissipative quasi-geostrophic equations and the dissipative incompressible porous media equation.
We consider the $2$D dissipative quasi-geostrophic equation with the time periodic external force and prove the existence of a unique time periodic solution in the case of the supercritical dissipation. In this case, the smoothing effect of the semigroup generated by the dissipation term is too weak to control the nonlinearity in the Duhamel term of the correponding integral equation. In this paper, we give a new approach which does not depend on the contraction mapping principle for the integral equation.
In this paper we study a three dimensional thermocline planetary geostrophic ``horizontal" hyper--diffusion model of the gyre-scale midlatitude ocean. We show the global existence and uniqueness of the weak and strong solutions to this model. Moreover, we establish the existence of a finite dimensional global attractor to this dissipative evolution system. Preliminary computational tests indicate that our hyper--diffusion model does not exhibit any of the nonphysical instabilities near the literal boundary which are observed numerically in other models.
The $β$-generalized quasi-geostrophic equation is studied in the range of $α\in (0, 1), β\in (1/2, 1), 1/2 < α+ β< 3/2$. When $α\in (1/2, 1), β\in (1/2, 1)$ such that $1 \leq α+ β< 3/2$, using the method introduced in [12] and [9], we prove global regularity of the unique and analytic solution and when $α\in (0, 1/2), β\in (1/2, 1)$ such that $1/2 < α+ β< 1$, that there exists a constant such that $\lVert \nablaθ_{0}\rVert_{L^{\infty}}^{2-2α-2β}\lVertθ_{0}\rVert_{L^{\infty}}^{2α+ 2β- 1} \leq c_{α,β}$ implies global regularity.
In this paper we consider the following modified quasi-geostrophic equation \partial_{t}θ+u\cdot\nablaθ+ν|D|^αθ=0, \quad u=|D|^{α-1}\mathcal{R}^{\bot}θ,\quad x\in\mathbb{R}^2 with $ν>0$ and $α\in ]0,1[\,\cup \,]1,2[$. When $α\in]0,1[$, the equation was firstly introduced by Constantin, Iyer and Wu in \cite{ref ConstanIW}. Here, by using the modulus of continuity method, we prove the global well-posedness of the system with the smooth initial data. As a byproduct, we also show that for every $α\in ]0,2[$, the Lipschitz norm of the solution has a uniform exponential bound.
In this article, we prove that data assimilation by feedback nudging can be achieved for the three-dimensional quasi-geostrophic equation in a simplified scenario using only large spatial scale observables on the dynamical boundary. On this boundary, a scalar unknown (buoyancy or surface temperature of the fluid) satisfies the surface quasi-geostrophic equation. The feedback nudging is done on this two-dimensional model, yet ultimately synchronizes the streamfunction of the three-dimensional flow. The main analytical difficulties are due to the presence of a nonlocal dissipative operator in th…
The present article is devoted to the 3D dissipative quasi-geostrophic system (QG). This system can be obtained as limit model of the Primitive Equations in the asymptotics of strong rotation and stratification, and involves a non-radial, non-local, homogeneous pseudo-differential operator of order 2 denoted by $Γ$ (and whose semigroup kernel reaches negative values). After a refined study of the non-local part of $Γ$, we prove apriori estimates (in the general L^p setting) for the 3D QG-model. The main difficulty of this article is to study the commutator of $Γ$ with a Lagrangian change of va…
In this paper, we are concerned with the Cauchy problem of the generalized surface quasi-geostrophic (SQG) equation in which the velocity field is expressed as $u=K\astω$, where $ω=ω(x,t)$ is an unknown function and $K(x)=\frac{x^\perp}{|x|^{2+2α}}, 0\leα\le \frac12.$ When $α=0$, it is the two-dimensional Euler equations. When $α=\frac 12$, it corresponds to the inviscid SQG. We will prove that if the existence interval of the smooth solution to the generalized SQG for some $0<α_0\le\frac12$ is $[0,T]$, then under the same initial data, the existence interval of the generalized SQG with $α$ wh…
In this paper, we give a rigorous justification of the point vortex approximation to the family of modified surface quasi-geostrophic (mSQG) equations globally in time in both the inviscid and vanishing dissipative cases. This result completes the justification for the remaining range of the mSQG family unaddressed by Geldhauser and Romito in the article arXiv:1812.05166, in the case of identically signed vortices.
We prove the global regularity of smooth solutions for a dissipative surface quasi-geostrophic 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 a uniform-in-time gr…
In this paper, we establish Gevrey class regularity of solutions to a class of dissipative equations with an analytic nonlinearity in the whole space. This generalizes the results of Ferrari and Titi in the periodic space case with initial data in $L^2-$based Sobolev spaces to the $L^p$ setting and in the whole space. Our generalization also includes considering rougher initial data, in negative Sobolev spaces in some cases including the Navier-Stokes and the subcritical quasi-geostrophic equations, and allowing the dissipation operator to be a fractional Laplacian. Moreover, we derive global …
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 quasi-geostrophic (SQG) equation with the velocity field $u$ related to the scalar $θ$ by $u=\nabla^\perpΛ^{β-2}θ$, where $1<β\le 2$ and $Λ=(-Δ)^{1/2}$ is the Zygmund operator. The borderline case $β=1$ corresponds to the SQG equation and the situation is more singular for $β>1$. We obtain the local existence and uniqueness of classical solutions, the global existence…
In this paper, we first generalize a new energy approach, developed by Y. Guo and Y. Wang \cite{GW12}, in the framework of homogeneous Besov spaces for proving the optimal temporal decay rates of solutions to the fractional power dissipative equation, then we apply this approach to the supercritical and critical quasi-geostrophic equation and the critical Keller-Segel system. We show that the negative Besov norm of solutions is preserved along time evolution, and obtain the optimal temporal decay rates of the spatial derivatives of solutions by the Fourier splitting approach and the interpolat…
In this paper we study the Cauchy problem for one multidimensional compressible nonlocal model of the dissipative quasi-geostrophic equations. First, we obtain the local existence and uniqueness of the smooth non-negative solution or the strong solution in time. Secondly, for the sub-critical and critical case $1\leα\leq 2$, we obtain the global existence and uniqueness results of the nonnegative smooth solution. Then, we prove the global existence of the weak solution for $0\le α\le 2$ and $ν\ge 0$. Finally, for the sub-critical case, we establish the global $H^1$ and $L^p, p>2,$ decay rate o…
We prove nonlinear lower bounds and commutator estimates for the Dirichlet fractional Laplacian in bounded domains. The applications include bounds for linear drift-diffusion equations with nonlocal dissipation and global existence of weak solutions of critical surface quasi-geostrophic equations.
This article is devoted to the study of the critical dissipative surface quasi-geostrophic $(SQG)$ equation in $\mathbb{R}^2$. For any initial data $θ_{0}$ belonging to the space $Λ^{s} ( H^{s}_{uloc}(\mathbb{R}^2)) \cap L^\infty(\mathbb{R}^2)$, we show that the critical (SQG) equation has at least one global weak solution in time for all $1/4\leq s \leq 1/2$ 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 \cite{Laz} whereas the local existence uses more refined energy estima…
We present remarkably simple proofs of Burkholder-Davis-Gundy inequalities for stochastic integrals and maximal inequalities for stochastic convolutions in Banach spaces driven by Lévy-type processes. Exponential estimates for stochastic convolutions are obtained and two versions of Itô's formula in Banach spaces are also derived. Based on the obtained maximal inequality, the existence and uniqueness of mild solutions of stochastic quasi-geostrophic equation with Lévy noise is established.
We examine the regularity of weak solutions of quasi-geostrophic (QG) type equations with supercritical ($α<1/2$) dissipation $(-Δ)^α$. This study is motivated by a recent work of Caffarelli and Vasseur, in which they study the global regularity issue for the critical ($α= 1/2$) QG equation \cite{CV}. Their approach successively increases the regularity levels of Leray-Hopf weak solutions: from $L^2$ to $L^\infty$, from $L^\infty$ to Hölder ($C^δ$, $δ>0$), and from Hölder to classical solutions. In the supercritical case, Leray-Hopf weak solutions can still be shown to be $L^\infty$, but it do…
We consider the Cauchy problem for systems of cubic nonlinear Klein-Gordon equations in one space dimension. Under a suitable structural condition on the nonlinearity, we will show that the small amplitude solution gains an additional logarithmic decay in comparison with the free evolution in the sense of $L^p$, $2\le p \le \infty$.
