We construct an invariant measure $μ$ for the Surface Quasi-Geostrophic (SQG) equation and show that almost all functions in the support of $μ$ are initial conditions of global, unique solutions of SQG, that depend continuously on the initial data. In addition, we show that the support of $μ$ is infinite dimensional, meaning that it is not locally a subset of any compact set with finite Hausdorff dimension. Also, there are global solutions that have arbitrarily large initial condition. The measures $μ$ is obtained via fluctuation-dissipation method, that is, as a limit of invariant measures for stochastic SQG with a carefully chosen dissipation and random forcing.
We prove local well-posedness for the inviscid surface quasigeostrophic (SQG) equation in bounded domains of $\mathbb{R}^2$. When fractional Dirichlet Laplacian dissipation is added, global existence of strong solutions is obtained for small data for critical and supercritical cases. Global existence of strong solutions with arbitrary data is obtained in the subcritical cases.
We consider the family known as modified or generalized surface quasi-geostrophic equations (mSQG) consisting of the classical inviscid surface quasi-geostrophic (SQG) equation together with a family of regularized active scalars given by introducing a smoothing operator of nonzero but possibly arbitrarily small degree. This family naturally interpolates between the 2D Euler equation and the SQG equation. For this family of equations we construct an invariant measure on a rough $L^2$-based Sobolev space and establish the existence of solutions of arbitrarily large lifespan for initial data in …
We prove the existence of global $L^2$ weak solutions for a family of generalized inviscid surface-quasi geostrophic (SQG) equations in bounded domains of the plane. In these equations, the active scalar is transported by a velocity field which is determined by the scalar through a more singular nonlocal operator compared to the SQG equation. The result is obtained by establishing appropriate commutator representations for the weak formulation together with good bounds for them in bounded domains.
In this paper, we study the existence of global classical solutions to the generalized surface quasi-geostrophic equation. By using the variational method, we provide some new families of global classical solutions for to the generalized surface quasi-geostrophic equation. These solutions mainly consist of rotating solutions and travelling-wave solutions.
We consider the inviscid generalized surface quasi-geostrophic equation (gSQG) in a patch setting, where the parameter $α\in (1,2)$. The cases $α= 0$ and $α= 1$ correspond to 2d Euler and SQG respectively, and our choice of the parameter $α$ results in a velocity more singular than in the SQG case. Our main result concerns the global stability of the half-plane patch stationary solution, under small and localized perturbations. Our theorem appears to be the first construction of stable global solutions for the gSQG-patch equations. The only other nontrivial global solutions known so far in the…
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 consider the subcritical SQG equation in its natural scale invariant Sobolev space and prove the existence of a global attractor of optimal regularity. The proof is based on a new energy estimate in Sobolev spaces to bootstrap the regularity to the optimal level, derived by means of nonlinear lower bounds on the fractional laplacian. This estimate appears to be new in the literature, and allows a sharp use of the subcritical nature of the $L^\infty$ bounds for this problem. As a byproduct, we obtain attractors for weak solutions as well. Moreover, we study the critical limit of the attracto…
By studying the linearization of contour dynamics equation and using implicit function theorem, we prove the existence of co-rotating and travelling global solutions for the gSQG equation, which extends the result of Hmidi and Mateu \cite{HM} to $α\in[1,2)$. Moreover, we prove the $C^\infty$ regularity of vortices boundary, and show the convexity of each vortices component.
We consider the critical dissipative SQG equation in bounded domains, with the square root of the Dirichlet Laplacian dissipation. We prove global a priori interior $C^α$ and Lipschitz bounds for large data.
In this paper, we study the existence of rotating and traveling-wave solutions for the generalized surface quasi-geostrophic (gSQG) equation. The solutions are obtained by maximization of the energy over the set of rearrangements of a fixed function. The rotating solutions take the form of co-rotating vortices with $N$-fold symmetry. The traveling-wave solutions take the form of translating vortex pairs. Moreover, these solutions constitute the desingularization of co-rotating $N$ point vortices and counter-rotating pairs. Some other quantitative properties are also established.
We obtain estimates near the boundary for the critical dissipative SQG equation in bounded domains, with the square root of the Dirichlet Laplacian dissipation. We prove that global regularity up to the boundary holds if and only if a certain quantitative vanishing of the scalar at the boundary is maintained.
We consider the initial value problem for the fractionally dissipative quasi-geostrophic equation \[ \partial_t θ+ \mathcal{R}^\perp θ\cdot \nabla θ+ Λ^γθ= 0, \qquad θ(\cdot,0) =θ_0 \] on $\mathbb{T}^2 = [0,1]^2$, with $γ\in (0,1)$. The coefficient in front of the dissipative term $Λ^γ= (-Δ)^{γ/2}$ is normalized to $1$. We show that given a smooth initial datum with $\|θ_0\|_{L^2}^{γ/2} \|θ_0\|_{\dot{H}^2}^{1-γ/2}\leq R$, where {\em $R$ is arbitrarily large}, there exists $γ_1 = γ_1(R) \in (0,1)$ such that for $γ\geq γ_1$, the solution of the supercritical SQG equation with dissipation $Λ^γ$ d…
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 focus on the two-dimensional surface quasi-geostrophic equation with fractional horizontal dissipation and fractional vertical thermal diffusion. On the one hand, when the dissipation powers are restricted to a suitable range, the global regularity of the surface quasi-geostrophic equation is obtained by some anisotropic embedding and interpolation inequalities involving fractional derivatives. One the one hand, we obtain the optimal large time decay estimates for global weak solutions by an anisotropic interpolation inequality. Moreover, based on the argument adopted in esta…
We consider the dissipative SQG equation in bounded domains, first introduced by Constantin and Ignatova in 2016. We show global Holder regularity up to the boundary of the solution, with a method based on the De Giorgi techniques. The boundary introduces several difficulties. In particular, the Dirichlet Laplacian is not translation invariant near the boundary, which leads to complications involving the Riesz transform.
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 \…
It is well known that the incompressible Euler equations in two dimensions have globally regular solutions. The inviscid surface quasi-geostrophic (SQG) equation has a Biot-Savart law which is one derivative less regular than in the Euler case, and the question of global regularity for its solutions is still open. We study here the patch dynamics in the half-plane for a family of active scalars which interpolates between these two equations, via a parameter $α\in[0,\frac 12]$ appearing in the kernels of their Biot-Savart laws. The values $α=0$ and $α=\frac 12$ correspond to the 2D Euler and SQ…
We consider the global attractor of the critical SQG semigroup $S(t)$ on the scale-invariant space $H^1(\mathbb{T}^2)$. It was shown in~\cite{CTV13} that this attractor is finite dimensional, and that it attracts uniformly bounded sets in $H^{1+δ}(\mathbb{T}^2)$ for any $δ>0$, leaving open the question of uniform attraction in $H^1(\mathbb{T}^2)$. In this paper we prove the uniform attraction in $H^1(\mathbb{T}^2)$, by combining ideas from DeGiorgi iteration and nonlinear maximum principles.
We prove the existence of a compact global attractor for the dynamics of the forced critical surface quasi-geostrophic equation (SQG) and prove that it has finite fractal (box-counting) dimension. In order to do so we give a new proof of global regularity for critical SQG. The main ingredient is the nonlinear maximum principle in the form of a nonlinear lower bound on the fractional Laplacian, which is used to bootstrap the regularity directly from $L^\infty$ to $C^α$, without the use of De Giorgi techniques. We prove that for large time, the norm of the solution measured in a sufficiently str…
We consider the evolution of weak vanishing viscosity solutions to the critically dissipative surface quasi-geostrophic equation. Due to the possible non-uniqueness of solutions, we rephrase the problem as a set-valued dynamical system and prove the existence of a global attractor of optimal Sobolev regularity. To achieve this, we derive a new Sobolev estimate involving Hölder norms, which complement the existing estimates based on commutator analysis.
Aggregation equations, such as the parabolic-elliptic Patlak-Keller-Segel model, are known to have an optimal threshold for global existence vs. finite-time blow-up. In particular, if the diffusion is absent, then all smooth solutions with finite second moment can exist only locally in time. Nevertheless, one can ask whether global existence can be restored by adding a suitable noise to the equation, so that the dynamics are now stochastic. Inspired by the work of Buckmaster et al. arXiv:1806.03734 showing that, with high probability, the inviscid SQG equation with random diffusion has global …
We construct a determining form for the surface quasi-geostrophic (SQG) equation with subcritical dissipation. In particular, we show that the global attractor for this equation can be embedded in the long-time dynamics of an ordinary differential equation (ODE) called a determining form. Indeed, there is a one-to-one correspondence between the trajectories in the global attractor of the SQG equation and the steady state solutions of the determining form. The determining form is a true ODE in the sense that its vector field is Lipschitz. This is shown by combining De Giorgi techniques and elem…
We consider the well-posedness of the surface quasi-geostrophic (SQG) front equation. Hunter-Shu-Zhang [9] 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.
We prove the global existence of solutions with small and smooth initial data of a nonlinear dispersive equation for the motion of generalized surface quasi-geostrophic (GSQG) fronts in a parameter regime $1<α<2$, where $α=1$ corresponds to the SQG equation and $α=2$ corresponds to the incompressible Euler equations. This result completes previous global well-posedness results for $0<α\le 1$. We also use contour dynamics to derive the GSQG front equations for $1<α<2$.
The paper is devoted to the study of slightly supercritical active scalars with nonlocal diffusion. We prove global regularity for the surface quasi-geostrophic (SQG) and Burgers equations, when the diffusion term is supercritical by a symbol with roughly logarithmic behavior at infinity. We show that the result is sharp for the Burgers equation. We also prove global regularity for a slightly supercritical two-dimensional Euler equation. Our main tool is a nonlocal maximum principle which controls a certain modulus of continuity of the solutions.
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 a nonlinear, spatially-nonlocal initial value problem in one space dimension on $\mathbb{R}$ that describes the motion of surface quasi-geostrophic (SQG) fronts. We prove that the initial value problem has a unique local smooth solution under a convergence condition on the multilinear expansion of the nonlinear term in the equation, and, for sufficiently smooth and small initial data, we prove that the solution is global.
In this paper, we prove the global regularity of smooth solutions to 2D surface quasi-geostrophic (SQG) equations with super-critical dissipation for a class of large initial data, where the velocity and temperature can be arbitrarily large in spaces $L^\infty(\mathbb{R}^2)$ and $H^3(\mathbb{R}^2)$. This result could be seen as an improvement work of Liu-Pan-Wu \cite{Liu}, for it's without any smallness hypothesis of the $L^\infty(\mathbb{R}^2)$ norm of the initial data.
We obtain a family of nonlinear maximum principles for linear dissipative nonlocal operators, that are general, robust, and versatile. We use these nonlinear bounds to provide transparent proofs of global regularity for critical SQG and critical d-dimensional Burgers equations. In addition we give applications of the nonlinear maximum principle to the global regularity of a slightly dissipative anti-symmetric perturbation of 2d incompressible Euler equations and generalized fractional dissipative 2d Boussinesq equations.
We prove the existence of the V-states for the generalized inviscid SQG equations with $α\in ]0,1[.$ These structures are special rotating simply connected patches with $m-$ fold symmetry bifurcating from the trivial solution at some explicit values of the angular velocity. This produces, inter alia, an infinite family of non stationary global solutions with uniqueness.
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 …
In this paper we construct an invariant weighted Wiener measure associated to the periodic derivative nonlinear Schrödinger equation in one dimension and establish global well-posedness for data living in its support. In particular almost surely for data in a Fourier-Lebesgue space ${\mathcal F}L^{s,r}(\T)$ with $s \ge \frac{1}{2}$, $2 < r < 4$, $(s-1)r <-1$ and scaling like $H^{\frac{1}{2}-ε}(\T),$ for small $ε>0$. We also show the invariance of this measure.
We consider the asymptotic behavior of the surface quasi-geostrophic equation, subject to a small external force. Under suitable assumptions on the forcing, we first construct the steady states and we provide a number of useful a posteriori estimates for them. Importantly, to do so, we only impose minimal cancellation conditions on the forcing function. Our main result is that all $L^1\cap L^\infty$ localized initial data produces global solutions of the forced SQG, which converge to the steady states in $L^p(\mathbf R^2), 1<p\leq 2$ as time goes to infinity. This establishes that the steady s…
In this paper, we construct invariant measures and global-in-time solutions for a fractional Schr\" odinger equation with a Moser-Trudinger type nonlinearity $$ i\partial_t u= (-Δ)^αu+ 2βu e^{β|u|^2},\qquad\mbox{for}\qquad(x,t)\in \ M\times \mathbb{R} $$ on a compact Riemannian manifold $M$ without boundary of dimension $d\geq 2$. To do so, we use the so-called Inviscid-Infinite-dimensional limits introduced by Sy ('19) and Sy and Yu ('21). More precisely, we show that if $s>d/2$ or if $s\leq d/2$ and $s\leq 1+α$, there exists an invariant measure $μ^{s}$ and a set $Σ^s \subset H^s$ containing…
We are concerned with a family of dissipative active scalar equation with velocity fields coupled via multiplier operators that can be of high-order. We consider sub-critical values for the fractional diffusion and prove global well-posedness of solutions with small initial data belonging to a framework based on Fourier transform, namely Fourier-Besov-Morrey spaces. Since the smallness condition is with respect to the weak norm of this space, some initial data with large $L^{2}$-norm can be considered. Self-similar solutions are obtained depending on the homogeneity of the initial data and cou…
In this note, by analyzing the behavior at infinity of the matrix symbol of an invariant operator $P$ with respect to a fixed elliptic operator, we obtain a necessary and sufficient condition to guarantee that $P$ is globally hypoelliptic. We also investigate relations between the global hypoellipticity of $P$ and global subelliptic estimates.
The non-existence of global solutions for semirelativistic equations with non-gauge invariant power type nonlinearity is revisited by a relatively direct way with a pointwise estimate of fractional derivative of some test functions.
The purpose of this paper is the construction of invariant regions in which we establish the global existence of solutions for m-component reaction-diffusion systems with a tridiagonal symmetric toeplitz matrix of diffusion coefficients and with nonhomogeneous boundary conditions. The proposed technique is based on invariant regions and Lyapunov functional methods. The nonlinear reaction term has been supposed to be of polynomial growth.
