We use a nonlocal maximum principle to prove the global existence of smooth solutions for a slightly supercritical surface quasi-geostrophic equation. By this we mean that the velocity field $u$ is obtained from the active scalar $θ$ by a Fourier multiplier with symbol $i k^\perp |k|^{-1} m(k|)$, where $m$ is a smooth increasing function that grows slower than $\log \log |k|$ as $|k|\rightarrow \infty$.
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.
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 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.
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.
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 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 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…
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 the note, a new regularity condition for axisymmetric solutions to the non-stationary 3D Navier-Stokes equations is proven. It is slightly supercritical.
We show the existence and multiplicity of concentrating solutions to a pure Neumann slightly supercritical problem in a ball. This is the first existence result for this kind of problems in the supercritical regime. Since the solutions must satisfy a compatibility condition of zero average, all of them have to change sign. Our proofs are based on a Lyapunov-Schmidt reduction argument which incorporates the zero-average condition using suitable symmetries. Our approach also guarantees the existence and multiplicity of solutions to subcritical Neumann problems in annuli. More general symmetric d…
The Slightly Supercritical Euler Equations: Smooth Solutions and Vortex Patches
本文研究（略）超临界二维Euler方程。论文由两部分组成。在第一部分中，我们证明了所有$s>0.$在$C^s$空间中的适定性我们还给出了$0< s \leq 1.$的涡度的$C^s$范数的增长估计在第二部分中，我们证明了超临界区域中涡斑问题的整体正则性，本文推广了Chae，Constantin和Wu的结果，他们证明了所谓的LogLog-Euler方程的适定性。我们还将Chemin和Bertozzi-Constantin关于涡斑问题的经典结果推广到了微超临界情形。超临界涡补丁问题引入了几个额外的困难，这些困难可以通过利用涡补丁的额外切向规则性的精确估计来克服。我们研究的两个问题都是在整个空间的背景下进行的。
We construct radially symmetric self-similar blow-up profiles for the mass supercritical nonlinear Schrödinger equation $i\partial_t u + Δu + |u|^{p-1}u=0$ on $\mathbf{R}^d$, close to the mass critical case and for any space dimension $d\ge 1$. These profiles bifurcate from the ground state solitary wave. The argument relies on the classical matched asymptotics method suggested in [Sulem, C.; Sulem, P.-L., The nonlinear Schrödinger equation. Self-focusing and wave collapse. Applied Mathematical Sciences, 139. Springer-Verlag, New York, 1999] which needs to be applied in a degenerate case due t…
Global Existence, Hamiltonian Conservation and Vanishing Viscosity for the Surface Quasi-Geostrophic Equation
Gran Sasso Science Institute（格兰萨索科学研究所）；Laboratoire de Mathématiques et de Modélisation d’Évry (LaMME), Université d’Évry（埃夫里大学数学与建模实验室（LaMME））；Yonsei university（延世大学）
We continue our study of the dynamics of a nearly inviscid periodic surface quasi-geostrophic equation. Here we consider a slightly diffusive stochastic SQG equation of the form \begin{equation*} \begin{cases} dθ_t + |D|^{2δ}θ_t\,dx + (u_t \cdot \nabla)θ_t\,dx + |D|^δdW_t = 0 \\ u_t = \nabla^\perp|D|^{-1}θ_t. \end{cases} \end{equation*} We construct global energy solutions as introduced by P. Goncalves and M. Jara (2014) for any $δ> 0$, so that any small amount of diffusion permits us to construct solutions. We show moreover that pathwise uniqueness of these energy solutions holds in the prese…
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…
We are concerned with a nonlocal transport 1D-model with supercritical dissipation $γ\in(0,1)$ in which the velocity is coupled via the Hilbert transform, namely the so-called CCF model. This model arises as a lower dimensional model for the famous 2D dissipative quasi-geostrophic equation and in connection with vortex-sheet problems. It is known that its solutions can blow up in finite time when $γ\in(0,1/2)$. On the other hand, as stated by Kiselev (2010), in the supercritical subrange $γ\in\lbrack1/2,1)$ it is an open problem to know whether its solutions are globally regular. We show globa…
In this paper we consider the slightly $L^2$-supercritical gKdV equations $\partial_t u+(u_{xx}+u|u|^{p-1})_x=0$, with the nonlinearity $5<p<5+\varepsilon$ and $0<\varepsilon\ll 1$ . In the previous work of the author we know that there exists an stable self-similar blow-up dynamics for slightly $L^2$-supercritical gKdV equations. Such solution can be viewed as solutions with single blow-up point. In this paper we will prove the existence of solutions with multiple blow-up points, and give a description of the formation of the singularity near the blow-up time.
In this short paper we prove the global regularity of solutions to the Navier-Stokes equations under the assumption that slightly supercritical quantities are bounded. As a consequence, we prove that if a solution $u$ to the Navier-Stokes equations blows-up, then certain slightly supercritical Orlicz norms must become unbounded. This partially answers a conjecture recently made by Terence Tao. The proof relies on quantitative regularity estimates at the critical level and transfer of subcritical information on the initial data to arbitrarily large times. This method is inspired by a recent pap…
An intrinsic property of almost any physical measuring device is that it makes observations which are slightly blurred in time. We consider a nudging-based approach for data assimilation that constructs an approximate solution based on a feedback control mechanism that is designed to account for observations that have been blurred by a moving time average. Analysis of this nudging model in the context of the subcritical surface quasi-geostrophic equation shows, provided the time-averaging window is sufficiently small and the resolution of the observations sufficiently fine, that the approximat…
We consider the 1D nonlinear Schrödinger equation (NLS) with focusing \emph{point nonlinearity}, $$i\partial_tψ+ \partial_x^2ψ+ δ|ψ|^{p-1}ψ= 0$$ where $δ=δ(x)$ is the delta function supported at the origin. In the $L^2$ supercritical setting $p>3$, we construct self-similar blow-up solutions belonging to the energy space $L_x^\infty \cap \dot H_x^1$. This is reduced to finding outgoing solutions of a certain stationary profile equation. All outgoing solutions to the profile equation are obtained by using parabolic Weber functions and solving the jump condition at $x=0$ imposed by the $δ$ term.…
