On the global regularity for anisotropic dissipative surface quasi-geostrophic equation
Zhuan Ye
备注：
To appear in Nonlinearity. We have added some more details. The new title is "On the global regularity for anisotropic dissipative surface quasi-geostrophic equation"
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 classical Gronwall inequality, we establish a new logarithmic type Gronwall inequality, which may be of independent interest and potential applications.
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This paper was withdrawn by arXiv admin because it plagiarizes "Jiahong Wu The two-dimensional quasi-geostrophic equation with critical or supercritical dissipation 2005 Nonlinearity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 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 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…
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.
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 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 $C(\mathbb{R}^{+},H^1(\mathbb{R}^2)) .$
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 $α\in (0,2)$.
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 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 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 \…
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 …
Regularity of solution maps of the generalized surface quasi-geostrophic equations
University of Notre Dame（圣母大学）；Institute for Mathematical Sciences, Stony Brook University（石溪大学数学科学研究所）；University of Illinois at Chicago（伊利诺伊大学芝加哥分校）；Hitotsubashi University（一桥大学）
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,…
In this article we consider the following generalized quasi-geostrophic equation \partial_tθ+ u\cdot\nabla θ+ νΛ^βθ=0, \quad u= Λ^α\mathcal{R}^\botθ, \quad x\in\mathbb{R}^2, where $ν>0$, $Λ:=\sqrt{-Δ}$, $α\in ]0,1[$ and $β\in ]0,2[$. We first show a general criterion yielding the nonlocal maximum principles for the whole space active scalars, then mainly by applying the general criterion, for the case $α\in]0,1[$ and $β\in ]α+1,2]$ we obtain the global well-posedness of the system with smooth initial data; and for the case $α\in ]0,1[$ and $β\in ]2α,α+1]$ we prove the local smoothness and the …
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.
It is well known, by now, that the three-dimensional non-viscous planetary geostrophic model, with vertical hydrostatic balance and horizontal Rayleigh friction, coupled to the heat diffusion and transport, is mathematically ill-posed. This is because the no-normal flow physical boundary condition implicitly produces an additional boundary condition for the temperature at the literal boundary. This additional boundary condition is different, because of the Coriolis forcing term, than the no heat flux physical boundary condition. Consequently, the second order parabolic heat equation is over de…
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.
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…
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.
We study the three dimensional active scalar equation called the magneto-geostropic equation which was proposed by Moffatt and Loper as a model for the geodynamo processes in the Earth's fluid core. When the viscosity of the fluid is positive, the constitutive law that relates the drift velocity $u(x,t)$ and the scalar temperature $θ(x,t)$ produces two orders of smoothing. We study the implications of this property. For example, we prove that in the case of the non-diffusive ($\varepsilon_κ=0$) active scalar equation, initial data $θ_0\in L^3$ implies the existence of unique, global weak solut…
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.
Instantaneous gap loss of Sobolev regularity for the surface quasi-geostrophic equation
Institute of Applied Physics and Computational Mathematics（应用物理与计算数学研究所）；School of Mathematics and Physics, University of Science and Technology Beijing（北京科技大学数理学院）
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 $Q_α^{β,-1}(\mathbb{R}^{2}).$
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.
Any classical solution of the 2D incompressible Euler equation is global in time. However, it remains an outstanding open problem whether classical solutions of the surface quasi-geostrophic (SQG) equation preserve their regularity for all time. This paper studies solutions of a family of active scalar equations in which each component $u_j$ of the velocity field $u$ is determined by the scalar $θ$ through $u_j =\mathcal{R} Λ^{-1} P(Λ) θ$ where $\mathcal{R}$ is a Riesz transform and $Λ=(-Δ)^{1/2}$. The 2D Euler vorticity equation corresponds to the special case $P(Λ)=I$ while the SQG 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.
We show a global existence result of weak solutions for a class of generalized Surface Quasi-Geostrophic equation in the inviscid case. We also prove the global regularity of such solutions for the equation with slightly supercritical dissipation, which turns out to correspond to a logarithmically supercritical diffusion due to the singular nature of the velocity. Our last result is the eventual regularity in the supercritical cases for such weak solutions. The main idea in the proof of the existence part is based on suitable commutator estimates along with a careful cutting into low/high freq…
