Mustapha Amara
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 $\lim\limits_{t\rightarrow +\infty}\|θ(t)\|_{H^s}=0$.
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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.
In \cite{YZ}, the author proved the global existence of the two-dimensional anisotropic quasi-geostrophic equations with condition on the parameters $α,$ $β$ in the Sobolev spaces $H^s( \R^2)$; $s\geq 2$. In this paper, we show that this equations has a global solution in the spaces $H^s(\R^2)$, where $\max\{2-2α,2-2β\}< s<2$, with additional condition over $α$ and $β$. The proof is based on the Gevrey-class regularity of the solution in neighborhood of zero.
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 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…
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 $C(\mathbb{R}^{+},H^1(\mathbb{R}^2)) .$
Stationary solutions to the critical and super-critical quasi-geostrophic equation in the scaling critical Sobolev space
本文在尺度临界 Sobolev 空间中研究临界与超临界耗散准地转方程的稳态问题，证明小外力下小解唯一存在，并阐明数据到解映射连续但不一致连续。
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…
We establish short-time existence of solutions to the surface quasi-geostrophic equation in both the Hölder spaces $C^r(\mathbb{R}^2)$ for $r>1$ and the uniformly local Sobolev spaces $H^s_{ul}(\mathbb{R}^2)$ for $s\geq 3$. Using methods similar to those for the surface quasi-geostrophic equation, we also obtain short-time existence for the three-dimensional Euler equations in uniformly local Sobolev spaces.
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…
Instantaneous gap loss of Sobolev regularity for the surface quasi-geostrophic equation
该研究针对R²上的无粘表面准地转方程，证明了其Sobolev正则性在固定间隙下的瞬时损失，构造了满足特定正则性条件的初始数据并得到唯一解，明确了正则性损失的阈值关系。
Global regularity of semi-critical case of anisotropic quasi-geostrophic equations in Sobolev spaces
本文研究一类各向异性准地转方程在半临界情形下的全局正则性，证明当初始数据属于Sobolev空间H^s(R^2)（s>1）时，方程存在全局解。
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 …
Regularity of solution maps of the generalized surface quasi-geostrophic equations
研究广义表面准地转方程族的解映射正则性，证明Lagrangian解映射为实解析，从而在Hadamard意义上局部适定，而Eulerian解映射在Sobolev拓扑下不连续。
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.
Motivated by an equation arising in magnetohydrodynamics, we address the well-posedness theroy for the non-diffusive magneto-geostrophic equation. Namely, an active scalar equation in which the divergence-free drift velocity is one derivative more singular that the active scalar. In \cite{Friedlander-Vicol_3}, the authors prove that the non-diffusive equation is ill-posed in the sense of Hadamard in Sobolev spaces, but locally well posed in spaces of analytic functions. Here, we give an example of a steady state that is nonlinearly stable for periodic perturbations with initial data localized …
Existence of Non-Decaying Solutions to the Generalized Surface Quasi-Geostrophic Equations
本文证明了广义表面准地转方程在 Hölder-Zygmund 空间和一致局部 Sobolev 空间中非衰减解的短时存在性与唯一性。
We consider the question of growth of high order Sobolev norms of solutions of the conservative surface quasi-geostrophic equation. We show that if $s>0$ is large then for every given $A$ there is exist small in $H^s$ initial data such that the corresponding solution's $H^s$ norm exceeds $A$ at some time. The idea of the construction is quasilinear. We use a small perturbation of a stable shear flow. The shear flow can be shown to create small scales in the perturbation part of the flow. The control is lost once the nonlinear effects become too large.
We consider an active scalar equation that is motivated by a model for magneto-geostrophic dynamics and the geodynamo. We prove that the non-diffusive equation is ill-posed in the sense of Hadamard in Sobolev spaces. In contrast, the critically diffusive equation is well-posed. In this case we give an example of a steady state that is nonlinearly unstable, and hence produces a dynamo effect in the sense of an exponentially growing magnetic field.
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…
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 construct solutions in $\mathbb{R}^2$ with finite energy of the surface quasi-geostrophic equations (SQG) that initially are in $C^k$ ($k\geq 2$) but that are not in $C^{k}$ for $t>0$. We prove a similar result also for $H^{s}$ in the range $s\in(\frac32,2)$. Moreover, we prove strong ill-posedness in the critical space $H^{2}$.
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 establish two types of characterizations for high order anisotropic Sobolev spaces. In particular, we prove high order anisotropic versions of Bourgain-Brezis- Mironescu's formula and Nguyen's formula.
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.
Strong illposedness for SQG in critical Sobolev spaces
本文证明了无粘SQG方程在临界Sobolev空间中的强不适定性，揭示了存在无解的初值及光滑数据的强临界范数膨胀，并将该鲁棒证明推广至修正SQG方程族。
We are concerned with the non-stationary Stokes system with non-homogeneous external force and non-zero initial data in ${\mathbb R}^n_+ \times (0,T)$. We obtain new estimates of solutions including pressure in terms of mixed anisotropic Sobolev spaces. As an application, some anisotropic Sobolev estimates are presented for weak solutions of the Navier-Stokes equations in a half-space in dimension three.
This paper aims to study time periodic solutions for 3D inviscid quasi-geostrophic model. We show the existence of non trivial rotating patches by suitable perturbation of stationary solutions given by generic revolution shapes around the vertical axis. The construction of those special solutions are done through bifurcation theory. In general, the spectral problem is very delicate and strongly depends on the shape of the initial stationary solutions. More specifically, the spectral study can be related to an eigenvalue problem of a self-adjoint compact operator. We are able to implement the b…
Optimal Gevrey regularity for supercritical quasi-geostrophic equations
针对带超临界耗散的二维地表类地转方程，本文证明临界 Sobolev 与 Besov 空间中大初值具有和线性部分同衰减指数的最优 Gevrey 正则性，解决了相关公开问题。
Eigenvalue problems in fractional anisotropic Orlicz-Sobolev spaces
本文引入分数阶各向异性Orlicz-Sobolev空间，利用变分方法研究含椭圆型非局部积分微分算子的分数阶各向异性问题，揭示了各向异性系数增长率竞争对特征值集合存在性与非存在性的关键影响。
Peridynamics and Anisotropic Fractional Sobolev Spaces with Variable Exponents
该数学分析领域研究构建近场动力学分数阶Sobolev空间及对应新型BBM型结果，探究带变指数的各向异性分数阶Sobolev空间的BBM型结果，还研究了各向同性情形的特征值问题。
On Fractional Anisotropic Musielak-Sobolev Spaces with Applications to Nonlocal Eigenvalue Problems
本文引入并研究分数阶各向异性Musielak-Sobolev空间，建立其基本性质与嵌入结果，将其应用于非局部各向异性特征值问题并证明特征值存在性，扩展并统一了相关现有模型。
In this paper we prove Bourgain-Brezis-Mironescu's type results (cf. \cite{BBM2001}) (BBM for short) for an energy functional which is strongly related to the fractional anisotropic p-Laplacian. We also provide with the analogous of Maz'ya-Shaposhnikova (see \cite{MS}) type results for these energies and finally we apply these results to analyze the stability of solutions to anisotropic fractional $p-$laplacian equations.
In this paper we establish the local and global well-posedness of the real valued fifth order Kadomstev-Petviashvili I equation in the anisotropic Sobolev spaces with nonnegative indices. In particular, our local well-posedness improves Saut-Tzvetkov's one and our global well-posedness gives an affirmative answer to Saut-Tzvetkov's $L^2$-data conjecture.
We prove higher differentiability of bounded local minimizers to some widely degenerate functionals, verifying superquadratic anisotropic growth conditions. In the two dimensional case, we prove that local minimizers to a model functional are locally Lipschitz continuous functions, without any restriction on the anisotropy.
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.
We study Sobolev spaces with weights in the half-space $\mathbb{R}^{N+1}_+=\{(x,y): x \in \mathbb{R}^N, y>0\}$, adapted to the singular elliptic operators \begin{equation*} \mathcal L =y^{α_1}Δ_{x} +y^{α_2}\left(D_{yy}+\frac{c}{y}D_y -\frac{b}{y^2}\right). \end{equation*}
We consider the well-posedness of the initial value problem associated to the k-generalized Zakharov-Kuznetsov equation in fractional weighted Sobolev spaces. Our method of proof is based on the contraction mapping principle and it mainly relies on the well-posedness results recently obtained for this equation in the Sobolev spaces H^s(\R^2) and a new pointwise commutator type formula involving the group induced by the linear part of the equation and the fractional anisotropic weights to be considered
Local well-posedness for a moving rigid region in Surface Quasi-Geostrophic equations
针对具有移动刚性障碍物的临界表面准地转方程，通过适应坐标变换、分数阶椭圆算子的积分表示和非线性不动点论证，建立了Sobolev空间中的局部适定性、唯一性、局部稳定性和爆破准则。
Anisotropic fractional Sobolev spaces with variable exponent and application to nonlocal problems
本文引入具有变指数的分数阶各向异性Sobolev空间，建立其完备性、可分性、自反性及嵌入性质，证明各向异性分数阶$\vec{p}(.,.)$-Laplace算子的泛函性质，并利用山路定理和Ekeland变分原理证明非局部各向异性问题弱解的存在性。
The existence of multiple nonnegative solutions to the anisotropic critical problem - \sum_{i=1}^{N} \frac{\partial}{\partial x_i} (| \frac{\partial u}{\partial x_i} |^{p_i-2} \frac{\partial u}{\partial x_i}) = |u|^{p^*-2} u {in} \mathbb{R}^N is proved in suitable anisotropic Sobolev spaces. The solutions correspond to extremal functions of a certain best Sobolev constant. The main tool in our study is an adaptation of the well-known concentration-compactness lemma of P.-L. Lions to anisotropic operators. Futhermore, we show that the set of nontrival solutions $\calS$ is included in $L^\infty(…
Existence of weak solutions to the generalized SQG equations in Sobolev spaces
该研究针对二维耗散型广义表面准地转方程，在初始数据属于Ḣ^{β/2 -1}(ℝ^2)的条件下证明了弱解的整体存在性，将Marchand的解框架推广到更大的主动标量方程族。
Well-posedness for Ohkitani model and long-time existence for surface quasi-geostrophic equations
本文研究Ohkitani对数奇异SQG方程，在随时间递减的Sobolev空间中证明光滑解局部适定性，并将其用于小参数$δ$-SQG方程的长时间存在性及对数耗散情形的全局适定性。
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 the initial-value problem associated with the dispersion generalized-Benjamin-Ono-Zakharov-Kuznetsov equation, $$ u_{t}+D^{a+1}_x \partial_{x}u+u_{xyy}+uu_{x}=0, \qquad a\in(0,1). $$ More specifically, we study the persistence property of the solution in the weighted anisotropic Sobolev spaces $$ H^{(1+a)s,2s}(\R^{2})\cap L^{2}((x^{2r_1} +y^{2r_2})dxdy), $$ for appropriate $s$, $r_1$ and $r_2$. By establishing unique continuation properties we also show that our results are sharp with respect to the decay in the $x$-direction.
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 …
We consider a quasilinear parabolic Cauchy problem with spatial anisotropy of orthotropic type and study the spatial localization of solutions. Assuming the initial datum is localized with respect to a coordinate having slow diffusion rate, we bound the corresponding directional velocity of the support along the flow. The expansion rate is shown to be optimal for large times.
For a given Finsler-Minkowski norm $\mathcal{F}$ in $\mathbb{R}^N$ and a bounded smooth domain $Ω\subset\mathbb{R}^N$ $\big(N\geq 2\big)$, we establish the following weighted anisotropic Sobolev inequality $$ S\left(\int_Ω|u|^q f\,dx\right)^\frac{1}{q}\leq\left(\int_Ω\mathcal{F}(\nabla u)^p w\,dx\right)^\frac{1}{p},\quad\forall\,u\in W_0^{1,p}(Ω,w)\leqno{\mathcal{(P)}} $$ where $W_0^{1,p}(Ω,w)$ is the weighted Sobolev space under a class of $p$-admissible weights $w$, where $f$ is some nonnegative integrable function in $Ω$. We discuss the case $0<q<1$ and observe that $$ μ(Ω):=\inf_{u\in W_{0…
We obtain well-posedness results in $L_p$-based weighted Sobolev spaces for a transmission problem for anisotropic Stokes and Navier-Stokes systems with $L_{\infty}$ strongly elliptic coefficient tensor, in complementary Lipschitz domains of ${\mathbb R}^n$, $n\ge 3$. The strong ellipticity allows to explore the associated pseudostress setting. First, we use a variational approach that reduces two linear transmission problems for the anisotropic Stokes system to equivalent mixed variational formulations with data in $L_p$-based weighted Sobolev and Besov spaces. We show that such a mixed varia…
We prove local well-posedness in the Sobolev spaces $\dot H^s(\mathbb{T})$, with $s>7/2$, for an initial value problem for a nonlocal, cubically nonlinear, dispersive equation that provides an approximate description of the evolution of surface quasi-geostrophic (SQG) fronts with small slopes.
In this paper, we consider the Cauchy problem for the generalized KP-II equation \begin{eqnarray*} u_{t}-|D_{x}|^αu_{x}+\partial_{x}^{-1}\partial_{y}^{2}u+\frac{1}{2}\partial_{x}(u^{2})=0,α\geq4. \end{eqnarray*} The goal of this paper is two-fold. Firstly, we prove that the problem is locally well-posed in anisotropic Sobolev spaces H^{s_{1},\>s_{2}}(\R^{2}) with s_{1}>\frac{1}{4}-\frac{3}{8}α, s_{2}\geq 0 and α\geq4. Secondly, we prove that the problem is globally well-posed in anisotropic Sobolev spaces H^{s_{1},\>0}(\R^{2}) with -\frac{(3α-4)^{2}}{28α}<s_{1}\leq0. and α\geq4. Thus, our glob…
We address the well-posedness theory for the magento-geostrophic equation, namely an active scalar equation in which the divergence-free drift velocity is one derivative more singular than the active scalar. In the presence of supercritical fractional diffusion given by (-Δ)^γ, where 0<γ<1, we discover that for γ>1/2 the equations are locally well-posed, while for γ<1/2 they are ill-posed, in the sense that there is no Lipschitz solution map. The main reason for the striking loss of regularity when γgoes below 1/2 is that the constitutive law used to obtain the velocity from the active scalar …
On approximation theorems for solutions to strongly parabolic systems in anisotropic Sobolev spaces
本文针对特殊非柱形区域上常系数强一致抛物方程组的Sobolev解，证明了边界光滑且无平行于$t=0$部分的两区域构成Runge对的充要条件。
In this paper, we consider the Cauchy problem for the fifth-order KP-I equation \begin{align*} u_t + \partial_x^5u+\partial_x^{-1}\partial_y^2u + \frac{1}{2}\partial_x(u^2)=0. \end{align*} Firstly, we establish the local well-posedness of the problem in the anisotropic Sobolev spaces $H^{s_1, s_2}(\mathbb{R}^2)$ with $s_1>-\frac{9}{8}$ and $s_2\geq 0$. Secondly, we establish the global well-posedness of the problem in $H^{s_1,0}(\mathbb{R}^2)$ with $s_1>-\frac{4}{7}$. Our result improves considerably the results of Saut and Tzvetkov (J. Math.\ Pures Appl.\ 79(2000), 307--338.) and Li and Xiao …
We consider an inhomogeneous initial-boundary value problem for a Petrovskii parabolic system of second order PDEs. We prove that this problem induces isomorphisms between appropriate anisotropic generalized Sobolev spaces. The regularity of these spaces are given by a pair of real numbers and by a function parameter. The latter allows us to characterize the regularity of solutions to the problem more finely as compared with anisotropic Sobolev spaces.
The goal of this paper is three-fold. Firstly, we prove that the Cauchy problem for generalized KP-I equation \begin{eqnarray*} u_{t}+|D_{x}|^α\partial_{x}u+\partial_{x}^{-1}\partial_{y}^{2}u+\frac{1}{2}\partial_{x}(u^{2})=0,α\geq4 \end{eqnarray*} is locally well-posed in the anisotropic Sobolev spaces$ H^{s_{1},\>s_{2}}(\R^{2})$ with $s_{1}>-\frac{α-1}{4}$ and $s_{2}\geq 0$. Secondly, we prove that the problem is globally well-posed in $H^{s_{1},\>0}(\R^{2})$ with $s_{1}>-\frac{(α-1)(3α-4)}{4(5α+3)}$ if $4\leq α\leq5$. Finally, we prove that the problem is globally well-posed in $H^{s_{1},\>0…
Bourgain, Brezis & Mironescu showed that (with suitable scaling) the fractional Sobolev $s$-seminorm of a function $f\in W^{1,p}(\mathbb{R}^n)$ converges to the Sobolev seminorm of $f$ as $s\rightarrow1^-$. Ludwig introduced the anisotropic fractional Sobolev $s$-seminorms of $f$ defined by a norm on $\mathbb{R}^n$ with unit ball $K$, and showed that they converge to the anisotropic Sobolev seminorm of $f$ defined by the norm whose unit ball is the polar $L_p$ moment body of $K$, as $s\rightarrow1^-$. The asymmetric anisotropic $s$-seminorms are shown to converge to the anisotropic Sobolev sem…
Embeddings of anisotropic Sobolev spaces into spaces of anisotropic Hölder-continuous functions
本文提出在矩形区域中将各向异性变指数Sobolev空间嵌入各向异性变指数Hölder连续函数空间的新框架，建立了变指数各向异性Hölder连续性的基础方法，深化了对函数方向性正则性的理解，并为数学与应用科学提供了新工具。
We study an eigenvalue problem involving a fully anisotropic elliptic differential operator in arbitrary Orlicz-Sobolev spaces. The relevant equations are associated with constrained minimization problems for integral functionals depending on the gradient of competing functions through general anisotropic $N$-functions. In particular, the latter need neither be radial, nor have a polynomial growth, and are not even assumed to satisfy the so called $Δ_2$-condition. The resulting analysis requires the development of some new aspects of the theory of anisotropic Orlicz-Sobolev spaces.
Bourgain, Brezis & Mironescu showed that (with suitable scaling) the fractional Sobolev $s$-seminorm of a function $f\in W^{1,p}(\rn)$ converges to the Sobolev seminorm of $f$ as $s\to 1^-$. The anisotropic $s$-seminorms of $f$ defined by a norm on $\rn$ with unit ball $K$ are shown to converge to the anisotropic Sobolev seminorm of $f$ defined by the norm with unit ball $\,\ompd K$, the polar $L_p$ moment body of $K$. The limiting behavior for $s\to 0^+$ is also determined (extending results by Maz$'$ya & Shaposhnikova).
This paper studies the dissipative generalized surface quasi-geostrophic equations in a supercritical regime where the order of the dissipation is small relative to order of the velocity, and the velocities are less regular than the advected scalar by up to one order of derivative. We also consider a non-degenerate modification of the endpoint case in which the velocity is less smooth than the advected scalar by slightly more than one order. The existence and uniqueness theory of these equations in the borderline Sobolev spaces is addressed, as well as the instantaneous smoothing effect of the…
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