Veli Shakhmurov
The maximal B_{p,q}^{s}-regularity properties of a nonlocal fractional elliptic equation is studied. Particularly, it is proven that the operator generated by this nonlocal ellıpıtıc equation in B_{p,q}^{s} is sectorial and also is a generator of an analytic semigroup. Moreover, well-posedeness of nonlocal fractional parabolic equation in Besov spaces are established.
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In this paper, the multipoint Cauchy problem for nonlocal nonlinear wave type equatıons are studied.The equation involves a convolution integral operator with a general kernel function whose Fourier transform is nonnegative. We establish local and global existence and uniqueness of solutions assuming enough smoothness on the initial data together with some growth conditions on the nonlinear term
In this paper we consider the pointwise convergence to the initial data for the Schrödinger-Dirac equation $i\tfrac{\partial u}{\partial t}=D^βu$ with $u(x,0)=u^0$ in a dyadic Besov space. Here $D^β$ denotes the fractional derivative of order $β$ associated to the dyadic distance $δ$ on $\mathbb{R}^+$. The main tools are a sumability formula for the kernel of $D^β$ and pointwise estimates of the corresponding maximal operator in terms of the dyadic Hardy-Littlewood function and the Calderón sharp maximal operator.
The well-posedness of nonlocal elliptic equation with singular drift is investigated in Besov-Hölder spaces. As an application, we show the existence and uniqueness for corresponding martingale problem. Moreover, we prove that the one dimensional distribution of the martingale solution has a density in some Besov space.
On Limit Formulas for Besov Seminorms and Nonlocal Perimeters in the Dunkl Setting
本文在 Dunkl 框架下推广了 Besov 半范数极限公式与非局部周长极限公式，提出了无维数依赖的 Maz'ya--Shaposhnikova 公式，并弱化了边界正则性条件，结果在经典拉普拉斯情形下亦为新发现。
In this paper, we consider the fractional Camassa-Holm equation modelling the propagation of small-but-finite amplitude long unidirectional waves in a nonlocally and nonlinearly elastic medium. First, we establish the local well-posedness in Besov space $B^{s_0}_{2,1}$ with $s_0=2ν-\frac 1 2$ for $ν>\frac 3 2 $ and $s_0=\frac 5 2$ for $1<ν\leq \frac 3 2 $. Then, with a given analytic initial data, we establish the analyticity of the solutions in both variables, globally in space and locally in time.
On the decay estimates of a nonlocal convection-diffusion Hamer system
本文研究多维Hamer模型的双曲-椭圆耦合形式，在混合Besov空间中建立小初值全局适定性，放宽了已有正则性假设，还得到临界Besov空间下的最优时间衰减估计，并在零质量条件下推导了改进衰减速率。
In this note we consider the pointwise convergence to the initial data for the solutions of some nonlocal dyadic Schrödinger equations on spaces of homogeneous type. We prove the a.e. convergence when the initial data belongs to a dyadic version of an $L^2$ based Besov space.
In this note we study the behaviour as $s\to 0^+$ of some semigroup based Besov seminorms associated with a non-symmetric and hypoelliptic diffusion with a drift. Our results generalise a previous one of Maz'ya and Shaposhnikova for the classical fractional Sobolev spaces $W^{s,p}$, and they also underscore a new phenomenon caused by the presence of the drift.
Geometric topics related to Besov type spaces on the Grushin setting
研究 Grushin 空间上基于 Grushin 半群和分数阶半群的两类 Besov 型空间，结合半群理论与非局部微积分建立 Sobolev 型不等式及等周不等式，并导出 Besov 型空间的嵌入定理。
We establish the global well-posedness of the 3D fractional Boussinesq-Coriolis system with stratification in a framework of Fourier type, namely spaces of Fourier-Besov type with underlying space being Morrey spaces (FBM-spaces, for short). Under suitable conditions and rescaled density fluctuation, the result is uniform with respect to the Coriolis and stratification parameters. We cover the critical case of the dissipation, namely half-Laplacian, in which the nonlocal dissipation has the same differential order as the nonlinearity and balances critically the scaling of the quadratic nonline…
In the past years, the phenomenon of fractional regularity has been addressed for a large class of linear and/or quasilinear differential operators, mostly, in terms of certain Besov spaces. As it turned out, for equations governed by the $p$-Laplacian, in general, the regularity of solutions appears in terms of functional spaces with nonlinear order of smoothness. Moreover, despite its own interest, fractional regularity methods may be used as a tool for the investigation of some Partial Differential Equations which are not usually addressed in this manner. Thus, the purpose of the present pa…
Quantitative bounds for bounded solutions to the Navier-Stokes equations in endpoint critical Besov spaces
本文基于Tao的定量正则性理论，在端点临界Besov空间中推导三维Navier-Stokes方程有界解的导数定量界，得到耦合临界量的混合爆破准则，解决了低正则性与二进可和性缺失的问题。
Schauder-type Estimates and Well-posedness for Nonlocal Quasilinear Evolution Equations in Fluid Dynamics
本文通过Schauder型估计研究非局部拟线性演化方程的适定性，提出核适应冻结系数法，建立临界空间中的适定性框架，应用于Muskat方程和Peskin问题。
We prove a mean value formula for weak solutions of $div(|y|^{a}\grad u)=0$ in $\mathbb{R}^{n+1}=\{(x,y): x\in\mathbb{R}^{n}, y\in\mathbb{R}\}$, $-1<a<1$ and balls centered at points of the form $(x,0)$. We obtain an explicit nonlocal kernel for the mean value formula for solutions of $(-\triangle)^{s}f=0$ on a domain $D$ of $\mathbb{R}^{n}$. When $D$ is Lipschitz we prove a Besov type regularity improvement for the solutions of $(-\triangle)^{s}f=0$.
We prove Hoelder continuity for n/2-harmonic maps from subsets of Rn into a sphere. This extends a recent one-dimensional result by F. Da Lio and T. Riviere to arbitrary dimensions. The proof relies on compensation effects which we quantify adapting an approach for Wente's inequality by L. Tartar, instead of Besov-space arguments which were used in the one-dimensional case. Moreover, fractional analogues of Hodge decomposition and higher order Poincare inequalities as well as several localization effects for nonlocal operators similar to the fractional laplacian are developed and applied.
Regularity of sets of finite fractional perimeter and nonlocal minimal surfaces in metric measure spaces
本文在倍增度量测度空间中研究有限s-周长集合的正则性，给出周长有限性的充分条件并证明测度论边界的Hausdorff测度为零，同时推广非局部周长泛函并证明极小化子的存在性、一致密度性与多孔性。
Large time behavior for solutions to the anisotropic Navier-Stokes equations in a 3D half-space
本研究针对三维半空间各向异性Navier-Stokes方程解的大时间行为，通过分析线性化特性得到非线性解最优衰减估计，揭示速度场第三分量增强耗散机制，引入Besov型空间结合切空间Littlewood–Paley分析解决了非局部算子带来的L¹估计难题。
We present an $L_q(L_{p})$-theory for the equation $$ \partial_{t}^αu=ϕ(Δ) u +f, \quad t>0,\, x\in \mathbb{R}^d \quad\, ;\, u(0,\cdot)=u_0. $$ Here $p,q>1$, $α\in (0,1)$, $\partial_{t}^α$ is the Caputo fractional derivative of order $α$, and $ϕ$ is a Bernstein function satisfying the following: $\exists δ_0\in (0,1]$ and $c>0$ such that \begin{equation} \label{eqn 8.17.1} c \left(\frac{R}{r}\right)^{δ_0}\leq \frac{ϕ(R)}{ϕ(r)}, \qquad 0<r<R<\infty. \end{equation} We prove uniqueness and existence results in Sobolev spaces, and obtain maximal regularity results of the solution. In particular, we…
Muckenhoupt-weighted $L_q(L_p)$ boundedness for time-space fractional nonlocal operators
本文针对分数阶演化方程，在Muckenhoupt加权混合范数空间中证明了$L_q(L_p)$估计，扩展了先前结果，为时空非局部方程提供了统一分析框架。
Positive-regularity norm inflation for the 3D hypodissipative Navier--Stokes equations
研究具有特定耗散项的三维次耗散纳维 - 斯托克斯方程，通过将各向异性涡环混合机制用于分数耗散，证明存在初始数据在特定空间中任意小，但对应局部光滑解在短时间内于同一空间中任意大的情况。
Let $Z=(Z_t)_{t\geq0}$ be an additive process with a bounded triplet $(0,0,Λ_t)_{t\geq0}$. Suppose that for any Schwartz function $φ$ on $\mathbb{R}^d$ whose Fourier transform is in $C_c^{\infty}(B_{c_s} \setminus B_{c_s^{-1}} )$, there exist positive constants $N_0$, $N_1$, and $N_2$ such that \begin{equation*} \int_{\mathbb{R}^d}|\mathbb{E}[φ(x+r^{-1}Z_t)]|dx\leq N_0 e^{- \frac{N_1 t}{s(r)}},\quad \forall (r,t)\in(0,1)\times[0,T], \end{equation*} and $$ \|ψ^μ(r^{-1}D)φ\|_{L_1(\mathbb{R}^d)}\leq \frac{N_2}{s(r)},\quad \forall r\in(0,1). $$ where $s$ is a scaling function (Definition 2.4), $c_…
Global Solutions to the 3D Half-Wave Maps Equation with Angular Regularity
针对3D半波映射方程因缺失$L^2_tL^\infty_x$估计导致全局适定性证明受阻的问题，本文通过引入角正则性与导数加权衰减条件，结合改进的Strichartz估计与交换向量场构建三线性估计，成功证明了该方程在加权空间下的弱全局适定性。
A Priori Estimates for Singular Fractional Stochastic Burgers Equations
针对奇异分数阶随机伯格斯方程，在特定参数条件下结合响应分解与保守 Zvonkin 变换建立余项估计，得到补偿型非局部扩散算子，为该类方程的先验分析提供了关键方法。
This work contributes in two areas, with sharp results, to the current investigation of regularity of solutions of heat equations (*) $Pu+\partial_tu=f$ on $Ω\times I $, where $P$ is a nonlocal operator, and $Ω\subset R^n$, $I\subset R$. 1) For $P$ a strongly elliptic pseudodifferential operator ($ψ$do) on $R^n$ of order $d\in R_+$, a symbol calculus on $R^{n+1}$ is introduced, that allows showing optimal regularity of solutions in the scale of anisotropic Bessel-potential spaces $H^{(s,s/d)}$, globally over $R^n\times R$, and locally over $Ω\times I$, for $s\in R$, $1<p<\infty $. Similar resu…
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