Fractional diffusion-wave equation: hidden regularity for weak solutions

Local regularity for fractional heat equations

We prove the maximal local regularity of weak solutions to the parabolic problem associated with the fractional Laplacian with homogeneous Dirichlet boundary conditions on an arbitrary bounded open set $Ω\subset\mathbb{R}^N$. Proofs combine classical abstract regularity results for parabolic equations with some new local regularity results for the associated elliptic problems.

分数阶p-Laplace方程的Hölder正则性，再探

Hölder regularity for the fractional p-Laplacian, revisited

本文利用正性的聚类与重新中心化技术，为分数阶p-Laplace方程解的局部Hölder正则性提供了一个新的证明。

Existence and Regularity of solutions to 1-D Fractional Order Diffusion Equations

In this article we investigate the existence and regularity of 1-d steady state fractional order diffusion equations. Two models are investigated:the Riemann-Liouville fractional diffusion equation, and the Riemann-Liouville-Caputo fractional diffusion equation. For these models we explicitly show how the regularity of the solution depends upon the right hand side (rhs) function. We also establish for which Dirichlet and Neumann boundary conditions the models are well posed.

空间齐次分数阶Kramers-Fokker-Planck方程的正则性

Regularity of the spatially homogenous fractional Kramers-Fokker-Planck equation

研究空间齐次分数阶Kramers-Fokker-Planck方程的Cauchy问题，证明解在L2初值下具有Gevrey正则性和衰减估计。

Higher Regularity of the Free Boundary in the Obstacle Problem for the Fractional Laplacian

We prove a higher regularity result for the free boundary in the obstacle problem for the fractional Laplacian via a higher order boundary Harnack inequality.

Regularity of solutions to the parabolic fractional obstacle problem

In this paper we study a parabolic version of the fractional obstacle problem, proving almost optimal regularity for the solution. This problem is motivated by an American option model proposed by Menton which introduces, into the theory of option evaluation, discontinuous paths in the dynamics of the stock's prices.

Existence and regularity results for viscous Hamilton-Jacobi equations with Caputo time-fractional derivative

We study existence, uniqueness and regularity properties of classical solutions to viscous Hamilton-Jacobi equations with Caputo time-fractional derivative. Our study relies on a combination of a gradient bound for the time-fractional Hamilton-Jacobi equation obtained via nonlinear adjoint method and sharp estimates in Sobolev and Hölder spaces for the corresponding linear problem.

Regularity in a one-phase free boundary problem for the fractional Laplacian

For a one-phase free boundary problem involving a fractional Laplacian, we prove that "flat free boundaries" are $C^{1,α}$. We recover the regularity results of Caffarelli for viscosity solutions of the classical Bernoulli-type free boundary problem with the standard Laplacian.

Quantitative estimates on localized finite differences for the fractional Poisson problem, and applications to regularity and spectral stability

We establish new quantitative estimates for localized finite differences of solutions to the Poisson problem for the fractional Laplace operator with homogeneous Dirichlet conditions of solid type settled in bounded domains satisfying the Lipschitz cone regularity condition. We then apply these estimates to obtain (i)~regularity results for solutions of fractional Poisson problems in Besov spaces; (ii)~quantitative stability estimates for solutions of fractional Poisson problems with respect to domain perturbations; (iii)~quantitative stability estimates for eigenvalues and eigenfunctions of f…

Partial regularity for fractional harmonic maps into spheres

This article addresses the regularity issue for stationary or minimizing fractional harmonic maps into spheres of order $s\in(0,1)$ in arbitrary dimensions. It is shown that such fractional harmonic maps are $C^\infty$ away from a small closed singular set. The Hausdorff dimension of the singular set is also estimated in terms of $s\in(0,1)$ and the stationarity/minimality assumption.

On maximal regularity for the Cauchy-Dirichlet mixed parabolic problem with fractional time derivative

We prove two maximal regularity results in spaces of continuous and Hölder continuous functions, for a mixed linear Cauchy-Dirichlet problem with a fractional time derivative $\mathbb{D}_t^α$. This derivative is intended in the sense of Caputo and $α$ is taken in $(0, 2)$. In case $α= 1$, we obtain maximal regularity results for mixed parabolic problems already known in mathematica literature.

具有分数阶扩散的SQG在瞬时失去正则性时的全局唯一解

Global unique solutions with instantaneous loss of regularity for SQG with fractional diffusion

本文构造了在超临界分数阶扩散下耗散表面准地转方程（$α$-SQG）的全局唯一解，该解会瞬时失去正则性。

Regularity estimates for the solution and the free boundary to the obstacle problem for the fractional Laplacian

We use a characterization of the fractional Laplacian as a Dirichlet to Neumann operator for an appropriate differential equation to study its obstacle problem. We write an equivalent characterization as a thin obstacle problem. In this way we are able to apply local type arguments to obtain sharp regularity estimates for the solution and study the regularity of the free boundary.

Partial regularity for the steady hyperdissipative fractional Navier-Stokes equations

We extend the Caffarelli-Kohn-Nirenberg type partial regularity theory for the steady $5$-dimensional fractional Navier-Stokes equations with external force to the hyperdissipative setting. In our argument we use the methods of Colombo-De Lellis-Massaccesi to apply a blowup procedure adapted from work of Ladyzhenskaya-Seregin.

Hölder regularity for fractional $p$-Laplace equations

We give an alternative proof for Hölder regularity for weak solutions of nonlocal elliptic quasilinear equations modelled on the fractional p-Laplacian where we replace the discrete De Giorgi iteration on a sequence of concentric balls by a continuous iteration. This work can be viewed as the nonlocal counterpart to the ideas developed by Tiziano Granucci.

分数阶p-拉普拉斯算子的边界正则性及其应用综述

A survey on boundary regularity for the fractional p-Laplacian and its applications

该综述聚焦分数阶p-拉普拉斯椭圆方程的边界正则性，重点阐述狄利克雷问题解的边界 Hölder 连续性结果，并展示其在分数阶一般非线性椭圆方程（变分或拓扑方法处理）中的应用。

（分数阶）Kolmogorov方程的动力学极大$L^2$正则性

Kinetic maximal $L^2$-regularity for the (fractional) Kolmogorov equation

该研究引入带时间权重的动力学极大$L^2$正则性，用分数阶各向异性Sobolev空间刻画（分数阶）Kolmogorov方程解的正则性，并证明齐次方程解构成半流且具有瞬时正则化性质。

Partial regularity of stable solutions to the fractional Gel'fand-Liouville equation

We analyze stable weak solutions to the fractional Gel'fand problem \begin{equation*} (-Δ)^su=e^u\quad\mathrm{in}\quad Ω\subset\mathbb{R}^n. \end{equation*} We prove that the dimension of the singular set is at most $n-10s.$

Uniqueness and Regularity of the Fractional Harmonic Gradient Flow in $S^{n-1}$

In this paper, we study the fractional harmonic gradient flow on $S^1$ taking values in $S^{n-1} \subset \mathbb{R}^n$ for every $n \geq 2$, in particular addressing uniqueness and regularity of solutions in the so-called energy class with sufficiently small energy, adding to the existing body of knowledge which includes existence of solutions and certain bubbling phenomena.

A Hopf lemma and regularity for fractional $p-$Laplacians

In this paper, we study qualitative properties of the fractional $p$-Laplacian. Specifically, we establish a Hopf type lemma for positive weak super-solutions of the fractional $p-$Laplacian equation with Dirichlet condition. Moreover, an optimal condition is obtained to ensure $(-\triangle)_p^s u\in C^1(\mathbb{R}^n)$ for smooth functions $u$.

Fractional De Giorgi classes and applications to nonlocal regularity theory

We present some recent results obtained by the author on the regularity of solutions to nonlocal variational problems. In particular, we review the notion of fractional De Giorgi class, explain its role in nonlocal regularity theory, and propose some open questions in the subject.

Existence and regularity results for terminal value problem for nonlinear fractional wave equations

We consider the terminal value problem (or called final value problem, initial inverse problem, backward in time problem) of determining the initial value, in a general class of time-fractional wave equations with Caputo derivative, from a given final value. We are concerned with the existence, regularity of solutions upon the terminal value. Under several assumptions on the nonlinearity, we address and show the well-posedness (namely, the existence, uniqueness, and continuous dependence) for the terminal value problem. Some regularity results for the mild solution and its derivatives of first…

Global Hölder regularity for the fractional $p$-Laplacian

By virtue of barrier arguments we prove $C^α$-regularity up to the boundary for the weak solutions of a non-local nonlinear problem driven by the fractional $p$-Laplacian operator. The equation is boundedly inhomogeneous and the boundary conditions are of Dirichlet type. We employ different methods according to the singular ($p<2$) of degenerate ($p>2$) case.

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.

On the regularity of the Non-dynamic Parabolic Fractional Obstacle Problem

In the class of the so called non-dynamic Fractional Obstacle Problems of parabolic type, it is shown how to obtain higher regularity as well as optimal regularity of the space derivatives of the solution. Furthermore, at free boundary points of positive parabolic density, it is proven that the time derivative of the solution is Hölder continuous. Finally, at regular free boundary points, space-time regularity of the corresponding free boundary is obtained for any fraction $s\in(0,1)$.

A note on global regularity for the weak solutions of fractional p-Laplacian equations

We consider a boundary value problem driven by the fractional p-Laplacian operator with a bounded reaction term. By means of barrier arguments, we prove Hölder regularity up to the boundary for the weak solutions, both in the singular (1<p<2) and the degenerate (p>2) case.

Hölder regularity for the time fractional Schrödinger equation

In this paper, we investigate that the Hölder regularity of solutions to the time fractional Schrödinger equation of order $1<α<2$, which interpolates between the Schrödinger and wave equations. This is inspired by Hirata and Miao's work which studied the fractional diffusion-wave equation. First, we give the asymptotic behavior for the oscillatory distributional kernels and their Bessel potentials by using Fourier analytic techniques. Then, the space regularity is derived by employing some results on singular Fourier multipliers. Using the asymptotic behavior for the above kernels, we prove t…

Regularity of the free boundary for the obstacle problem for the fractional Laplacian with drift

We establish the $C^{1+γ}$-Hölder regularity of the regular free boundary in the stationary obstacle problem defined by the fractional Laplace operator with drift in the subcritical regime. Our method of the proof consists in proving a new monotonicity formula and an epiperimetric inequality. Both tools generalizes the original ideas of G. Weiss for the classical obstacle problem to the framework of fractional powers of the Laplace operator with drift. Our study continues the earlier research, where two of us established the optimal interior regularity of solutions.

Some existence and regularity results for a non-local transport-diffusion equation with fractional derivatives in time and space

We study the existence of global weak solutions of a nonlinear transport-diffusion equation with a fractional derivative in the time variable and under some extra hypotheses, we also study some regularity properties for this type of solutions. In the system considered here, the diffusion operator is given by a fractional Laplacian and the nonlinear drift is assumed to be divergence free and it is assumed to satisfy some general stability and boundedness properties in Lebesgue spaces.

A Hopf's Lemma and the Boundary Regularity for the Fractional P-Laplacian

We begin the paper with a Hopf's lemma for a fractional p-Laplacian problem on a half-space. Specifically speaking, we show that the derivative of the solution along the outward normal vector is strictly positive on the boundary of the half-space. Next we show that positive solutions to a fractional p-Laplacian equation possess certain Holder continuity up to the boundary.

On the Hölder regularity for solutions of integro-differential equations like the anisotropic fractional Laplacian

In this paper we study integro-differential equations like the anisotropic fractional Laplacian. As in [Silvestre, Indiana University Mathematics Journal 55, 2006], we adapt the De Giorgi technique to achieve the $C^γ$-regularity for solutions of class $C^{2}$ and use the geometry found in [Caffarelli, Leitão, and Urbano, Math. Ann. 360, 2014] to get an ABP estimate, a Harnack inequality and the interior $C^{1, γ}$ regularity for viscosity solutions.

A one-phase problem for the fractional Laplacian: regularity of flat free boundaries

We consider a one-phase free boundary problem involving a fractional Laplacian $(-Δ)^α$, $0<α<1,$ and we prove that ``flat free boundaries" are $C^{1,γ}$. We thus extend the known result for the case $α=1/2.$

Hölder regularity for parabolic fractional $p$-Laplacian

Local Hölder regularity is established for certain weak solutions to a class of parabolic fractional $p$-Laplace equations with merely measurable kernels. The proof uses DeGiorgi's iteration and refines DiBenedetto's intrinsic scaling method. The control of a nonlocal integral of solutions in the reduction of oscillation plays a crucial role and entails delicate analysis in this intrinsic scaling scenario. Dispensing with any logarithmic estimate and any comparison principle, the proof is new even for the linear case.

极小化内蕴分数阶调和映射的定量正则性

Quantitative regularity for minimizing intrinsic fractional harmonic maps

本文研究极小化内蕴分数阶调和映射，结合修正Luckhaus引理与定量分层理论证明紧性、奇异集体积估计及改进的维数界。

Structure and regularity of the singular set in the obstacle problem for the fractional Laplacian

We study the singular part of the free boundary in the obstacle problem for the fractional Laplacian, \ $\min\bigl\{(-Δ)^su,\,u-φ\bigr\}=0$ in $\mathbb R^n$, for general obstacles $φ$. Our main result establishes the complete structure and regularity of the singular set. To prove it, we construct new monotonicity formulas of Monneau-type that extend those in \cite{GP} to all $s\in(0,1)$.

Optimal regularity of solutions to the obstacle problem for the fractional Laplacian with drift

We prove existence, uniqueness and optimal regularity of solutions to the stationary obstacle problem defined by the fractional Laplacian operator with drift, in the subcritical regime. We localize our problem by considering a suitable extension operator introduced by L. Caffarelli and L. Silvestre. The structure of the extension equation is different from the one considered by L. Caffarelli, S. Salsa and L. Silvestre in their study of the obstacle problem for the fractional Laplacian without drift, in that the obstacle function has less regularity, and exhibits some singularities. To take int…

Interior regularity results for fractional elliptic equations that degenerate with the gradient

In this paper we obtain interior regularity estimates for viscosity solutions of nonlocal Dirichlet problems that degenerate when the gradient of the solution vanishes. Interior Hölder estimates are obtained when the order of the fractional diffusion is less or equal than one, and Lipschitz estimates when it is bigger than one. In the latter case, the estimates are robust enough to conclude interior $C^{1, α}$ regularity by an improvement of the flatness procedure, which is possible when the nonlocal term is close enough to a second-order diffusion.

Comparison and regularity results for the fractional Laplacian via symmetrization methods

In this paper we establish a comparison result through symmetrization for solutions to some boundary value problems involving the fractional Laplacian. This allows to get sharp estimates for the solutions, obtained by comparing them with solutions of suitable radial problems. Furthermore, we use such result to prove a priori estimates for solutions in terms of the data, providing several regularity results which extend the well known ones for the classical Laplacian.

Global regularity of the three-dimensional fractional micropolar equations

The global well-posedness of the smooth solution to the three-dimensional (3D) incompressible micropolar equations is a difficult open problem. This paper focuses on the 3D incompressible micropolar equations with fractional dissipations $( Δ)^αu$ and $(-Δ)^βw$.Our objective is to establish the global regularity of the fractional micropolar equations with the minimal amount of dissipations. We prove that, if $α\geq \frac{5}{4}$, $β\geq 0$ and $α+β\geq\frac{7}{4}$, the fractional 3D micropolar equations always possess a unique global classical solution for any sufficiently smooth data. In addit…

Interior regularity for fractional systems

We study the regularity of solutions of elliptic fractional systems of order 2s, $s \in (0, 1)$, where the right hand side f depends on a nonlocal gradient and has the same scaling properties as the nonlocal operator. Under some structural conditions on the system we prove interior Hölder estimates in the spirit of [1]. Our results are stable in s allowing us to recover the classic results for elliptic systems due to S. Hildebrandt and K. Widman [6] and M. Wiegner [9].

Optimal regularity and nondegeneracy of a free boundary problem related to the fractional Laplacian

We discuss the optimal regularity and nondegeneracy of a free boundary problem related to the fractional Laplacian. This work is related to, but addresses a different problem from, recent work of Caffarelli, Roquejoffre, and Sire. A variant of the boundary Harnack inequality is also proved, where it is no longer required that the function be 0 along the boundary.

Well-posedness and regularity of the Cauchy problem for nonlinear fractional in time and space equations

The purpose is to study the Cauchy problem for non-linear in time and space pseudo-differential equations. These include the fractional in time versions of HJB equations governing the controlled scaled CTRW. As a preliminary step which is of independent interest we analyse the corresponding linear equation proving its well-posedness and smoothing properties.

A Parabolic Problem with a Fractional-Time Derivative

We study regularity for a parabolic problem with fractional diffusion in space and a fractional time derivative. Our main result is a De Giorgi-Nash-Moser Holder regularity theorem for solutions in a divergence form equation. We also prove results regarding existence, uniqueness, and higher regularity in time.

Lebesgue空间中分数阶扩散-波动方程的适定性与正则性理论

Well-posedness and regularity theory for the fractional diffusion-wave equation in Lebesgue spaces

本文研究半线性分数阶扩散-波动方程，在拉普拉斯算子相关分数幂尺度下给出线性算子族估计，分析局部mild解的存在唯一性与最大区间延拓性，还探讨空间正则性及解对初值的连续依赖性。

The existence and regularity theory for abstract semilinear time-fractional evolution equations

In this paper, we investigate abstract time-fractional evolution equations with nonlinear perturbations. We construct solutions of Lipschitz perturbation problems in arbitrary large time interval independent of the Lipschitz constants. We will extend well-known results for standard evolution equations such as the blow-up alternative, to the time-fractional evolution equations. We also prove the differentiability with respect to time of the solution when the perturbation is sufficiently smooth. The differentiability enables us to use the maximum principle. The theory on general Banach spaces en…

Higher Regularity for the fractional thin obstacle problem

In this article we investigate the higher regularity properties of the regular free boundary in the fractional thin obstacle problem. Relying on a Hodograph-Legendre transform, we show that for smooth or analytic obstacles the regular free boundary is smooth or analytic, respectively. This leads to the analysis of a fully nonlinear, degenerate (sub)elliptic operator which we identify as a (fully nonlinear) perturbation of the fractional Baouendi-Grushin Laplacian. Using its intrinsic geometry and adapted function spaces, we invoke the analytic implicit function theorem to deduce analyticity of…

Regularity theory for time-fractional advection-diffusion-reaction equations

We investigate the behavior of the time derivatives of the solution to a linear time-fractional, advection-diffusion-reaction equation, allowing space- and time-dependent coefficients as well as initial data that may have low regularity. Our focus is on proving estimates that are needed for the error analysis of numerical methods. The nonlocal nature of the fractional derivative creates substantial difficulties compared with the case of a classical parabolic PDE. In our analysis, we rely on novel energy methods in combination with a fractional Gronwall inequality and certain properties of frac…

On fractional elliptic equations in Lipschitz sets and epigraphs: regularity, monotonicity and rigidity results

We consider a nonlocal equation set in an unbounded domain with the epigraph property. We prove symmetry, monotonicity and rigidity results. In particular, we deal with halfspaces, coercive epigraphs and epigraphs that are flat at infinity.

A semilinear integro-differential equation: global existence and hidden regularity

Here we show a hidden regularity result for nonlinear wave equations with an integral term of convolution type and Dirichlet boundary conditions. Under general assumptions on the nonlinear term and on the integral kernel we are able to state results about global existence of strong and mild solutions without any further smallness on the initial data. Then we define the trace of the normal derivative of the solution showing a regularity result. In such a way we extend to integrodifferential equations with nonlinear term well-known results available in the literature for linear wave equations wi…

Higher Hölder regularity for the fractional $p-$Laplacian in the superquadratic case

We prove higher Hölder regularity for solutions of equations involving the fractional $p-$Laplacian of order $s$, when $p\ge 2$ and $0<s<1$. In particular, we provide an explicit Hölder exponent for solutions of the non-homogeneous equation with data in $L^q$ and $q>N/(s\,p)$, which is almost sharp whenever $s\,p\leq (p-1)+N/q$. The result is new already for the homogeneous equation.

Interior and boundary regularity results for strongly nonhomogeneous $p,q$-fractional problems

In this article, we deal with the global regularity of weak solutions to a class of problems involving the fractional $(p,q)$-Laplacian, denoted by $(-Δ)^{s_1}_{p}+(-Δ)^{s_2}_{q}$, for $s_2, s_1\in (0,1)$ and $1<p,q<\infty$. We establish completely new Hölder continuity results, up to the boundary, for the weak solutions to fractional $(p,q)$-problems involving singular as well as regular nonlinearities. Moreover, as applications to boundary estimates, we establish new Hopf type maximum principle and strong comparison principle in both situations.

混合阶分数阶各向异性$p$-拉普拉斯算子的正则性估计

Regularity estimates for fractional orthotropic $p$-Laplacians of mixed order

该研究针对带各向异性奇异核的非线性积分-微分算子，推导得出Sobolev型不等式、弱Harnack不等式与局部Hölder估计三类鲁棒正则性结果。

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 existence and regularity of a terminal value problem for the time fractional diffusion equation

In this paper we consider a final value problem for a diffusion equation with time-space fractional differentiation on a bounded domain $D$ of $ \mathbb{R}^{k}$, $k\ge 1$, which includes the fractional power $\mathcal L^β$, $0<β\le 1$, of a symmetric uniformly elliptic operator $\mathcal L$ defined on $L^2(D)$. A representation of solutions is given by using the Laplace transform and the spectrum of $\mathcal L^β$. We establish some existence and regularity results for our problem in both the linear and nonlinear case.