Small $\dot B^{-1}_{\infty,\infty}$ implies regularity

Φ^*[I] 的正则性与非退化性蕴含固定边界 ∂Ω 的正则性

Regularity and non-degeneracy of $Φ^*[I]$ implies regularity of the fixed boundary $\partialΩ$

研究满足特定非退化条件的微分同胚 Φ，证明其推前 Φ^*[I] 的局部正则性可推出固定边界 ∂Ω 的局部正则性，补充了非散射非均匀介质的正则性结果。

Ill-posedness for one-dimensional wave maps at the critical regularity

We show that the wave map equation in $\R^{1+1}$ is in general ill-posed in the critical space $\dot H^{1/2}$, and the Besov space $\dot B^{1/2,1}_2$. The problem is attributed to the bad behaviour of the one-dimensional bilinear expression $D^{-1}(f Dg)$ in these spaces.

Optimal Time Decay of Navier-Stokes Equations With Low Regularity Initial Data

In this paper, we study the optimal time decay rate of isentropic Navier-Stokes equations under the low regularity assumptions about initial data. In the previous works about optimal time decay rate, the initial data need to be small in $H^{[N/2]+2}(\mathbb{R}^{N})$. Our work combined negative Besov space estimates and the conventional energy estimates in Besov space framework which is developed by R. Danchin. Though our methods, we can get optimal time decay rate with initial data just small in $\dot{B}^{N/2-1, N/2+1} \cap \dot{B}^{N/2-1, N/2}$ and belong to some negative Besov space(need not…

临界空间中Navier-Stokes方程与SQG方程的新正则性准则

New Regularity Criteria for Navier-Stokes and SQG Equations in Critical Spaces

本文针对三维不可压缩Navier-Stokes方程与二维超临界SQG方程，在Besov空间、Lorentz空间及临界空间和等多种临界空间框架下推导了新的正则性准则，拓展了两类方程正则性条件的适用范围。

Minimal regularity solutions of semilinear generalized Tricomi equations

We prove the local existence and uniqueness of minimal regularity solutions $u$ of the semilinear generalized Tricomi equation $\partial_t^2 u-t^m Δu =F(u)$ with initial data $(u(0,\cdot), \partial_t u(0,\cdot)) \in \dot{H^γ}(\mathbb R^n) \times \dot{H}^{γ-\frac2{m+2}}(\mathbb R^n)$ under the assumption that $|F(u)|\lesssim |u|^κ$ and $|F'(u)| \lesssim |u|^{κ-1}$ for some $κ>1$. Our results improve previous results of M. Beals [2] and of ourselves [15-17]. We establish Strichartz-type estimates for the linear generalized Tricomi operator $\partial_t^2 -t^m Δ$ from which the semilinear results …

Global regularity for solutions of the three dimensional Navier-Stokes equation with almost two dimensional initial data

In this paper, we will prove a new result that guarantees the global existence of solutions to the Navier--Stokes equation in three dimensions when the initial data is sufficiently close to being two dimensional. This result interpolates between the global existence of smooth solutions for the two dimensional Navier--Stokes equation with arbitrarily large initial data, and the global existence of smooth solutions for the Navier--Stokes equation in three dimensions with small initial data in $\dot{H}^\frac{1}{2}$. This result states that the closer the initial data is to being two dimensional, …

关于 $3-D$ Navier-Stokes 系统在 $L^p_T(\dot{B}^{\frac 1 2+\frac 2 p}_{2,\infty})$ 空间中的临界单分量正则性

On the Critical One Components Regularity for the $3-D$ Navier-Stokes System in $L^p_T(\dot{B}^{\frac 1 2+\frac 2 p}_{2,\infty})$ spaces

针对三维不可压缩Navier-Stokes方程的温和解，证明若解在有限时间爆破，则任意方向上的速度分量的某个范数积分必然爆破，改进了已有正则性准则。

Global Regularity for the Yang-Mills Equations on High Dimensional Minkowski Space

We prove that smallness of the critical Sobolev norm implies regularity for the Yang-Mills equations on (6+1) and higher dimensional Minkowski space.

A smallness regularity criterion for the 3D Navier-Stokes equations in the largest class

In this paper, we consider the three-dimensional Navier-Stokes equations, and show that if the $\dot B^{-1}_{\infty,\infty}$-norm of the velocity field is sufficiently small, then the solution is in fact classical.

Uniform Hölder regularity with small exponent in competition-fractional diffusion systems

For a class of competition-diffusion nonlinear systems involving fractional powers of the Laplacian, including as a special case the fractional Gross-Pitaevskii system, we prove that uniform boundedness implies Hölder boundedness for sufficiently small positive exponents, uniformly as the interspecific competition parameter diverges. This implies strong convergence of the family of solutions as the segregation of their support occurs.

On the regularity of weak solutions of the Boussinesq equations in Besov spaces Dedicated to Enrique Zuazua on the occasion of his sixtieth birthday

The main issue addressed in this paper concerns an extension of a result by Z. Zhang who proved, in the context of the homogeneous Besov space $\dot{B}_{\infty ,\infty }^{-1}(\mathbb{R}% ^{3})$, that, if the solution of the Boussinesq equation (\ref% {eq1.1}) below (starting with an initial data in $H^{2}$) is such that $% (\nabla u,\nabla θ)\in L^{2}\left( 0,T;\dot{B}_{\infty ,\infty }^{-1}(% \mathbb{R}^{3})\right)$, then the solution remains smooth forever after $T$. In this contribution, we prove the same result for weak solutions just by assuming the condition on the velocity $u$ and not o…

Bessel (p,s)-LaplacianDirichlet问题解的Besov正则性

Besov regularity of solutions to the Dirichlet problem for the Bessel $(p,s)$-Laplacian

本文研究了Bessel(p,s)-LaplacianDirichlet问题解的Besov正则性，通过结合Lions-Calderón空间、Besov嵌入和差商方法，建立了弱解的全局Besov正则性估计。

Well-posedness and regularity for the fractional Navier-Stokes equations

We consider the wellposedness of the fractional Navier-Stokes as a generalization of the wellposedness result in Koch-Tataru's paper. An interesting remark is that our result does not contradict to the well-known ill-posedness result for Navier-Stokes with initial data in $\dot B^{-1, \infty}_\infty$ by Bourgain-Pavlovic. In the end, we also discuss the regularity and analyticity in the space variable.

Optimal decay for the compressible MHD equations in the critical regularity framework

In this paper, we study the large time behavior of solutions to the compressible magnetohydrodynamic equations in the $L^p$-type critical Besov spaces. Precisely, we show that if the initial data in the low frequencies additionally belong to some Besov space $\dot{B}_{2,\infty}^{-σ_1}$ with $σ_1\in (1-N/2, 2N/p-N/2]$, then the $\dot{B}_{p,1}^0$ norm of the critical global solutions presents the optimal decay $t^{-\frac{N}{2}(\frac{1}{2}-\frac{1}{p})-\frac{σ_1}{2}}$ for $t\rightarrow+\infty$. The pure energy argument without the spectral analysis is performed, which allows us to remove the usua…

Global regularity for the Maxwell-Klein-Gordon equation with small critical Sobolev norm in high dimensions

We show that in dimensions $n \geq 6$ that one has global regularity for the Maxwell-Klein-Gordon equations in the Coulomb gauge provided that the critical Sobolev norm $\dot H^{n/2-1} \times \dot H^{n/2-2}$ of the initial data is sufficiently small. These results are analogous to those recently obtained for the high-dimensional wave map equation but unlike the wave map equation, the Coulomb gauge non-linearity cannot be iterated away directly. We shall use a different approach, proving Strichartz estimates for the covariant wave equation. This in turn will be achieved by use of Littlewood-Pal…

Regularity Conditions of 3D Navier-Stokes flow in terms of large spectral components

We develop Ladyzhenskaya-Prodi-Serrin type spectral regularity criteria for 3D incompressible Navier-Stokes equations in a torus. Concretely, for any $N>0$, let $w_N$ be the sum of all spectral components of the velocity fields whose all three wave numbers are greater than $N$ absolutely. Then, we show that for any $N>0$, the finiteness of the Serrin type norm of $w_N$ implies the regularity of the flow. It implies that if the flow breaks down in a finite time, the energy of the velocity fields cascades down to the arbitrarily large spectral components of $w_N$ and corresponding energy spectru…

热方程在混合Lebesgue范数下的最大正则性和热迹估计

Maximal regularity and caloric trace estimates in mixed Lebesgue norms for the heat equation

研究半空间上非齐次Dirichlet边界数据的热方程，建立了边界数据g的混合Besov和Lizorkin-Triebel型范数与热延拓v的混合Lebesgue范数之间的最大正则性估计和逆热迹估计，覆盖端点情形。

Global regularity of wave maps I. Small critical Sobolev norm in high dimension

We show that wave maps from Minkowski space $R^{1+n}$ to a sphere are globally smooth if the initial data is smooth and has small norm in the critical Sobolev space $\dot H^{n/2}$ in the high dimensional case $n \geq 5$. A major difficulty, not present in the earlier results, is that the $\dot H^{n/2}$ norm barely fails to control $L^\infty$, potentially causing a logarithmic divergence in the nonlinearity; however, this can be overcome by using co-ordinate frames adapted to the wave map by approximate parallel transport. In the sequel of this paper we address the more interesting two-dimensio…

Critical one component anisotropic regularity for 3-D Navier-Stokes system

Let us consider an initial data $v_0$ for the classical 3D Navier-Stokes equation with vorticity belonging to $L^{\frac 32}\cap L^2$. We prove that if the solution associated with $v_0$ blows up at a finite time $T^\star$, then for any $p\in]4,\infty[,~q_1\in[1,2[,~μ>0, ~q_2\in\bigl[2,\bigl(1/p+μ\bigr)^{-1}\bigr[,~κ\in ]1,\infty[$, and any unit vector $e$, the $L^p$ estimate in time of $\bigl\|(v(t)|e)_{\mathbb{R}^3}\bigr\|_{L^{\frac{3p}{p-2}}}^p +\bigl\|(v(t)|e)_{\mathbb{R}^3}\bigr\|^p_{ \bigl(\dot{B}^{μ+\frac2p+\frac2{q_1}-1}_{q_1,κ}\bigr)_{\rm h} \bigl(\dot{B}^{\frac1{q_2}-μ}_{q_2,κ}\bigr)_…

偶数奇异积分算子在补片上的定量Hölder估计

Quantitative Hölder Estimates for Even Singular Integral Operators on Patches

本文提出构造性方法，获得偶数奇异积分算子在$C^{1+σ}$域特征函数上的定量$\u0307{C}^σ$估计，线性依赖于域正则性，并应用于粘度对比自由边界Navier-Stokes方程，得到新的全局时间正则性结果。

A regularity criterion for three-dimensional micropolar fluid equations in Besov spaces of negative regular indices

In this article, we study regularity criteria for the 3D micropolar fluid equations in terms of one partial derivative of the velocity. It is proved that if \begin{equation*} \int^{T}_{0}\|\partial_{3}u\|^{\frac{2}{1-r}}_{\dot{B}^{-r}_{\infty,\infty}} dt<\infty \quad \text{with} \quad 0< r<1, \end{equation*} then, the solutions of the micropolar fluid equations actually are smooth on $(0, T)$. This improves and extends many previous results.

Global Attractor for Weakly Damped Forced KdV Equation in Low Regularity on T

In this paper we consider the long time behavior of the weakly damped, forced Korteweg-de Vries equation in the Sololev spaces of the negative indices in the periodic case. We prove that the solutions are uniformly bounded in $\dot{H}^s(\T)$ for $s>-\dfrac{1}{2}$. Moreover, we show that the solution-map possesses a global attractor in $\dot{H}^s(\T)$ for $s>-\dfrac{1}{2}$, which is a compact set in $H^{s+3}(\T)$.

A regularity criterion to the 3d Boussinesq equations

The paper deals with the regularity criterion for the weak solutions to the 3D Boussinesq equations in terms of the partial derivatives in Besov spaces. It is proved that the weak solution $(u,θ)$ becomes regular provided that $(\nabla_{h}u,\nabla_{h}θ)\in L^{\frac{8}{3}}(0,T;\dot{B}_{\infty ,\infty}^{-1}(\mathbb{R}^{3}))$ Our results improve and extend the well-known results by Fang-Qian for the Navier-Stokes equations.

Small data scattering and soliton stability in $\dot{H}^{-\frac16}$ for the quartic KdV Equation

In this note we prove scattering for perturbations of solitons in the scaling space appropriate for the quartic nonlinearity, namely $\dot{H}^{-\frac16}$. The article relies strongly on refined estimates for a KdV equation linearized at the soliton. In contrast to the work of Tao (2006), we are able to work purely in the scaling space without additional regularity assumptions, allowing us to prove some results on the existence of inverse wave operators.

Uniform Rectifiability and harmonic measure IV: Ahlfors regularity plus Poisson kernels in $L^p$ implies uniform rectifiability

Let $E\subset \mathbb{R}^{n+1}$, $n\ge 2$, be an Ahlfors-David regular set of dimension $n$. We show that the weak-$A_\infty$ property of harmonic measure, for the open set $Ω:= \mathbb{R}^{n+1}\setminus E$, implies uniform rectifiability of $E$.

三次非线性薛定谔方程在 $\dot{H}^{\frac12}(\Bbb R^3)$ 中的无条件唯一性

Unconditional uniqueness for the cubic nonlinear Schrödinger equation in $\dot{H}^{\frac12}(\Bbb R^3)$

本文通过引入临界 Besov 空间并采用 Born 型双重 Duhamel 迭代，解决了三次 NLS 在临界 Sobolev 空间中的无条件唯一性难题。

Maximal Regularity for Non-Autonomous Second Order Cauchy Problems

We consider non-autonomous wave equations \[ \left\{ \begin{aligned} \&\ddot u(t) + \B(t)\dot u(t) + \A(t)u(t) = f(t) \quad t\text{-a.e.}\\ \&u(0)=u_0,\, \dot u(0) = u_1. \end{aligned} \right. \] where the operators $\A(t)$ and $\B(t)$ are associated with time-dependent sesquilinear forms $\fra(t,.,.)$ and $\frb$ defined on a Hilbert space $H$ with the same domain $V$. The initial values satisfy $ u_0 \in V$ and $u_1 \in H$. We prove well-posedness and maximal regularity for the solution both in the spaces $V'$ and $H$. We apply the results to non-autonomous Robin-boundary conditions and also …

三维Navier–Stokes方程的一个临界单分量正则性准则

A Critical one-component regularity criteria for the three-dimensional Navier--Stokes equations

本文为三维Navier–Stokes方程建立两个临界单分量正则性准则，通过垂直涡量估计、各向异性能量估计及热流分解等方法，放宽了正则性条件并去除了对初始涡量的额外可积性假设。

Regularity problem for the nematic LCD system with Q-tensor in $\mathbb R^3$

We study the regularity problem of a nematic liquid crystal model with local configuration represented by Q-tensor in three dimensions. It was an open question whether the classical Prodi-Serrin condition implies regularity for this model. Applying a wavenumber splitting method, we show that a solution does not blow-up under certain extended Beale-Kato-Majda condition solely imposed on velocity. This regularity criterion automatically implies that the classical Prodi-Serrin or Beale-Kato-Majda condition prevents blow-up of solutions.

临界正则性框架下可压缩无压Navier-Stokes系统的一致稳定性和最优时间衰减率

Uniform stability and optimal time decay rates of the compressible pressureless Navier-Stokes system in the critical regularity framework

本文研究了可压缩无压Navier-Stokes系统的Cauchy问题，在临界Besov空间中证明了强解的整体适定性和一致稳定性，并在低频条件下克服导数损失建立了速度的最优衰减估计，揭示了密度随时间保持一致有界的新渐近行为。

Sharp one component regularity for Navier-Stokes

We consider the conditional regularity of mild solution $v$ to the incompressible Navier-Stokes equations in three dimensions. Let $e \in \mathbb{S}^2$ and $0 < T^\ast < \infty$. J. Chemin and P. Zhang \cite{CP} proved the regularity of $v$ on $(0,T^\ast]$ if there exists $p \in (4, 6)$ such that $$\int_0^{T^\ast}\|v\cdot e\|^p_{\dot{H}^{\frac{1}{2}+\frac{2}{p}}}dt < \infty.$$ J. Chemin, P. Zhang and Z. F. Zhang \cite{CPZ} extended the range of $p$ to $(4, \infty)$. In this article we settle the case $p \in [2, 4]$. Our proof also works for the case $p \in (4,\infty)$.

Regularity criterion for 3D Navier-Stokes Equations in Besov spaces

Several regularity criterions of Leray-Hopf weak solutions $u$ to the 3D Navier-Stokes equations are obtained. The results show that a weak solution $u$ becomes regular if the gradient of velocity component $\nabla_{h}{u}$ (or $ \nabla{u_3}$) satisfies the additional conditions in the class of $L^{q}(0,T; \dot{B}_{p,r}^{s}(\mathbb{R}^{3}))$, where $\nabla_{h}=(\partial_{x_{1}},\partial_{x_{2}})$ is the horizontal gradient operator. Besides, we also consider the anisotropic regularity criterion for the weak solution of Navier-Stokes equations in $\mathbb{R}^3$. Finally, we also get a further re…

齐次临界正则性框架下的Boltzmann方程

The Boltzmann equation in the homogeneous critical regularity framework

本文在空间临界齐次Besov空间中构造了3D Boltzmann方程的唯一全局解，并利用无需常规$L^2$估计的新亚强制性框架和时间加权Lyapunov能量方法，证明了解的大时间最优衰减。

小质量范围内Hartree-Ohta-Kawasaki形状优化问题的存在性与正则性

Existence and Regularity in the Small-Mass Regime for a Hartree--Ohta-Kawasaki Shape Optimization Problem

本文研究了混合能函数的小质量范围内形状优化问题，证明了体积约束下的极小值解存在且为球体的C^{2,α}扰动，结合手术技术、Γ收敛、椭圆PDE理论和一相自由边界正则性。

Remarks on local regularity of axisymmetric solutions to the 3D Navier--Stokes equations

In this note, a new local regularity criteria for the axisymmetric solutions to the 3D Navier--Stokes equations is investigated. It is slightly supercritical and implies an upper bound for the oscillation of $Γ=r u^θ$: for any $0< τ<1$, there exists a constant $c>0$, $$ |Γ(r,x_{3},t)|\leq N e^{-c\, |\ln r|^τ},\ 0<r\leq \frac{1}{4}. $$

Flatness implies smoothness for solutions of the porous medium equation

One of the major problems in the theory of the porous medium equation is the regularity of the solutions and the free boundaries. Here we assume flatness of the solution in space time cylinder and derive smoothness of the interface after a small time, as well as smoothness of the solution in the positivity set and up to the free boundary for some time interval. We use these facts to prove the following eventual regularity result: solutions with compactly supported initial data are smooth after a finite time T that depends on mass and the size of the initial support. This result eliminates the …

抛物型正则性与Neumann问题的扰动理论

Perturbation theory for the parabolic Regularity and Neumann problem

该研究针对抛物型正则性边值问题与Neumann问题，在Lipschitz区域对应的抛物柱体域内，针对算子$L=\u2202_t-div(A\u2207\u00b7)$，推导了不同边界数据空间下的Carleson扰动结果。

Besov regularity for the Dirichlet integral fractional Laplacian in Lipschitz domains

We prove Besov regularity estimates for the solution of the Dirichlet problem involving the integral fractional Laplacian of order $s$ in bounded Lipschitz domains $Ω$: \[ \begin{aligned} \|u\|_{\dot{B}^{s+r}_{2,\infty}(Ω)} \le C \|f\|_{L^2(Ω)}, & \quad r = \min\{s,1/2\}, & \quad \mbox{if } s \neq 1/2, \\ \|u\|_{\dot{B}^{1-ε}_{2,\infty}(Ω)} \le C \|f\|_{L^2(Ω)}, & \quad ε\in (0,1), & \quad \mbox{if } s = 1/2, \end{aligned} \] with explicit dependence of $C$ on $s$ and $ε$. These estimates are consistent with the regularity on smooth domains and show that there is no loss of regularity due to L…

On the critical one component regularity for 3-D Navier-Stokes system: general case

Let us consider an initial data $v_0$ for the homogeneous incompressible 3D Navier-Stokes equation with vorticity belonging to $L^{\frac 32}\cap L^2$. We prove that if the solution associated with $v_0$ blows up at a finite time $T^\star$, then for any $p$ in $]4,\infty[$, and any unit vector $e$ of $\R^3$, the $L^p$ norm in time with value in $\dot{H}^{\frac 12+\frac 2 p }$ of $(v|e)_{\R^3}$ blows up at $T^\star$

Global regularity of wave maps II. Small energy in two dimensions

We show that wave maps from Minkowski space $\R^{1+n}$ to a sphere $S^{m-1}$ are globally smooth if the initial data is smooth and has small norm in the critical Sobolev space $\dot H^{n/2}$, in all dimensions $n \geq 2$. This generalizes the results in the prequel [math.AP/0010068] of this paper, which addressed the high-dimensional case $n \geq 5$. In particular, in two dimensions we have global regularity whenever the energy is small, and global regularity for large data is thus reduced to demonstrating non-concentration of energy.

On the well-posedness of the Hall-magnetohydrodynamics system in critical spaces

We investigate the existence and uniqueness issues of the 3D incompressible Hall-magnetohydrodynamic system supplemented with initial velocity $u_0$ and magnetic field $B_0$ in critical regularity spaces.In the case where $u_0,$ $B_0$ and the current $J_0:=\nabla\times B_0$ belong to the homogeneous Besov space $\dot B^{\frac 3p-1}_{p,1},$ $\:1\leq p<\infty,$ and are small enough, we establish a global result and the conservation of higher regularity.If the viscosity is equal to the magnetic resistivity, then we obtain the global well-posedness provided $u_0,$ $B_0$ and $J_0$ are small enough …

可压缩Navier-Stokes方程解的稳定性

On the stability of the solutions to compressible Navier-Stokes equations

本文证明临界Besov空间中可压缩Navier-Stokes方程解在欧拉坐标下连续依赖于初值，无正则性损失，完善了Hadamard适定性。

负指标齐次 Besov 空间中三维 Boussinesq 方程的一个正则性准则

A regularity criterion for the 3D Boussinesq equations in homogeneous Besov spaces with negative indices

本文针对三维 Boussinesq 方程，利用速度的第三个偏导数在负指标齐次 Besov 空间中的时空积分条件，给出并证明了一个保证解在给定时间区间上保持正则的新准则。

具有角正则性的3D半波映射方程的全局解

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。

与双调和波映射相关的柯西问题的全局结果

Global results for a Cauchy problem related to biharmonic wave maps

本文在尺度临界 Besov 空间中证明了一类带非泛型二次非线性项的导数双调和波方程对小初值的全局存在性，并由此得到双调和波映射方程在包括球面在内的一类目标流形上的小数据全局正则性。

Well-posedness and Ill-posedness for the Nonlinear Beam Equation

We investigate Strichartz estimates for the nonlinear beam equation with initial data $f\in\dot{H}^s, g\in\dot{H}^{s-2}$ and $f\in H^s, g\in H^{s-2}$. We extend results of H. Lindblad and C. D.Sogge [10] and T. Cazenave and F. B. Weissler [4] to nonlinear beam equations to determine the minimal regularity that is needed to prove well-posedness and scattering results with low regularity data. Finally, we also use small dispersion analysis of M. Christ, J. Colliander and T. Tao [2] to prove the nonlinear beam equation is ill-posed in defocusing case $ω=-1$ when $ 0<s<s_c=\frac{n}{2}-\frac{4}{κ-1…

Boundary Regularity Criteria for the 6D Steady Navier-Stokes and MHD Equations

It is shown in this paper that suitable weak solutions to the 6D steady incompressible Navier-Stokes and MHD equations are Hölder continuous near boundary provided that either $r^{-3}\int_{B_r^+}|u(x)|^3dx$ or $r^{-2}\int_{B_r^+}|\nabla u(x)|^2dx$ is sufficiently small, which implies that the 2D Hausdorff measure of the set of singular points near the boundary is zero. This generalizes recent interior regularity results by Dong-Strain \cite{DS}.

Quantitative partial regularity of the Navier-Stokes equations and applications

We prove a logarithmic improvement of the Caffarelli-Kohn-Nirenberg partial regularity theorem for the Navier-Stokes equations. The key idea is to find a quantitative counterpart for the absolute continuity of the dissipation energy using the pigeonhole principle. Based on the same method, for any suitable weak solution, we show the existence of intervals of regularity in one spatial direction with length depending exponentially on the natural local energies of the solution. Then, we give two applications of the latter result in the axially symmetric case. The first one is a local quantitative…

Hölder regularity of the nonlinear stochastic time-fractional slow and fast diffusion equations on $\mathbb{R}^d$

In this paper, we use local fraction derivative to show the Hölder continuity of the solution to the following nonlinear time-fractional slow and fast diffusion equation: $$\left(\partial^β+\fracν{2}(-Δ)^{α/2}\right)u(t,x) = I_t^γ\left[σ\left(u(t,x)\right)\dot{W}(t,x)\right],\quad t>0,\: x\in\mathbb{R}^d,$$ where $\dot{W}$ is the space-time white noise, $α\in(0,2]$, $β\in(0,2)$, $γ\ge 0$ and $ν>0$, under the condition that $2(β+γ)-1-dβ/α>0$. The case when $β+γ\le 1$ has been obtained in \cite{ChHuNu19}. In this paper, we have removed this extra condition, which in particular includes all cases…

关于三维 Navier–Stokes 方程的一个随时间变化分量的正则性准则

On the one time-varying component regularity criteria for 3-D Navier-Stokes equations

针对三维 Navier–Stokes 方程，本文证明局部强解在一个随时间变化方向上的分量满足齐次 Sobolev 空间积分条件时可延拓过 T。

不可压缩欧拉方程在端点临界Triebel-Lizorkin空间 $F^{d+1}_{1, \infty}(\mathbb{R}^d)$ 中解的时间正则性

Temporal regularity of the solution to the incompressible Euler equations in the end-point critical Triebel-Lizorkin space $F^{d+1}_{1, \infty}(\mathbb{R}^d)$

本文证明不可压缩欧拉方程的解在端点临界Triebel-Lizorkin空间中存在时间不连续性，从而说明该柯西问题不适定，并进一步讨论了相关空间中解的连续性与弱型连续性。

On the regularity of the free boundary in the optimal partial transport problem

This paper concerns the regularity and geometry of the free boundary in the optimal partial transport problem for general cost functions. More specifically, we prove that a $C^1$ cost implies a locally Lipschitz free boundary. As an application, we address a problem discussed by Caffarelli and McCann \cite{CM} regarding cost functions satisfying the Ma-Trudinger-Wang condition (A3): if the non-negative source density is in some $L^p(\mathbb{R}^n)$ space for $p \in (\frac{n+1}{2},\infty]$ and the positive target density is bounded away from zero, then the free boundary is a semiconvex $C_{loc}^…

Upper and lower $\dot{H}^{m}$ estimates for solutions to parabolic equations

In this article we prove results concerning upper and lower decay estimates for homogeneous Sobolev norms of solutions to a rather general family of parabolic equations. Following the ideas of Kreiss, Hagstrom, Lorenz and Zingano, we use eventual regularity of solutions to directly work with smooth solutions in physical space, bootstrapping decay estimates from the $L^2$ norm to higher order derivatives. Besides obtaining upper and lower bounds through this method, we also obtain reverse results: from higher order derivatives decay estimates, we deduce bounds for the $L^2$ norm. We use these g…

Blowup criterion for Navier-Stokes equation in critical Besov space with spatial dimensions $d \geq 4$

This paper is concerned with the blowup criterion for mild solution to the incompressible Navier-Stokes equation in higher spatial dimensions $d \geq 4$. By establishing an $ε$ regularity criterion, we show that if the mild solution $u$ with initial data in $\dot B^{-1+d/p}_{p,q}(\mathbb{R}^d) $, $d<p,\,q<\infty$ becomes singular at a finite time $T_*$, then $$ \limsup_{t\to T_*} \|u(t)\|_{\dot B^{-1+d/p}_{p,q}(\mathbb{R}^d)} = \infty. $$ The corresponding result in 3D case has been obtained by I.Gallagher, G.S.KochandF.Planchon. As a by-product, we also prove a regularity criterion for the Le…