We give the regularization for fractional integral by delta sequence and apply it to obtain existence-uniqueness theorems in Colombeau algebras for nonlinear equations with singularities: nonlinear system of integral equations with polar kernel and nonlinear parabolic equations (of ordinary type, with nonlinear conservative term and with Schrödinger kernel) with strongly singular initial data and non-Lipschitz nonlinearities. In a case of nonlinear parabolic equations we do in fact regularization of heat semigroup with delta sequence with respect to the time variable $t.$ We do the same for linear Schrödinger equation.
This is the first of a series of two papers which studies the fractional porous medium equation on a Riemannian manifold with isolated conical singularities. In this article, we show $R$-sectoriality for the fractional powers of possibly non-invertible $R$-sectorial operators. Applications concern existence, uniqueness and maximal $L^{q}$-regularity results for solutions of the fractional porous medium equation on manifolds with conical singularities. Space asymptotic behavior of the solutions close to the singularities is provided and its relation to the local geometry is established. Our met…
We introduce the study of isolated singularities for a semilinear equation involving the fractional Laplacian. In conformal geometry, it is equivalent to the study of singular metrics with constant fractional curvature. Our main ideas are: first, to set the problem into a natural geometric framework, and second, to perform some kind of phase portrait study for this non-local ODE.
In this paper we study removable singularities for solutions of the fractional heat equation in time varying domains. We introduce associated capacities and we study some of its metric and geometric properties.
We introduce a new regularization of the rotational Navier-Stokes equations that we call the Rotational Approximate Deconvolution Model (RADM). We generalize the deconvolution type model, studied by Berselli and Lewandowski [5], to the RADM model with fractional regularization where the convergence of the solution is studied with weaker conditions on the parameter regularization.
We construct solutions for the fractional Yamabe problem that are singular at a prescribed number of isolated points. This seems to be the first time that a gluing method is successfully applied to a non-local problem. The main step is an infinite-dimensional Lyapunov-Schmidt reduction method, that reduces the problem to an (infinite dimensional) Toda type system.
We study the isolated singularities of functions satisfying (E) (--$Δ$) s v$\pm$|v| p--1 v = 0 in $Ω$\{0}, v = 0 in R N \$Ω$, where 0 < s < 1, p > 1 and $Ω$ is a bounded domain containing the origin. We use the Caffarelli-Silvestre extension to R + x R N. We emphasize the obtention of a priori estimates, analyse the set of self-similar solutions via energy methods to characterize the singularities.
In this paper we study the existence of a least energy sign-changing solution to a nonlocal elliptic PDE involving singularity by using the Nehari manifold method, the constraint variational method and Brouwer degree theory.
In this paper, we study the local behaviors of nonnegative local solutions of fractional order semi-linear equations $(-Δ)^σu=u^{\frac{n+2σ}{n-2σ}}$ with an isolated singularity, where $σ\in (0,1)$. We prove that all the solutions are asymptotically radially symmetric. When $σ=1$, these have been proved in \cite{CGS} by Caffarelli, Gidas and Spruck.
In this paper, we study the local behavior of nonnegative solutions of fractional semi-linear equations $(-Δ)^σu = u^p$ with an isolated singularity, where $\sg \in (0, 1)$ and $\frac{n}{n-2\sg} < p < \frac{n+2\sg}{n-2\sg}$. We first use blow up method and a Liouville type theorem to derive an upper bound. Then we establish a monotonicity formula and a sufficient condition for removable singularity to give a classification of the isolated singularities. When $\sg=1$, this classification result has been proved by Gidas and Spruck (Comm. Pure Appl. Math. 34: 525-598, 1981).
We show that any open set that is a finite distance away from a Lipschitz subgraph will become a Lipschitz subgraph after flowing under fractional mean curvature flow for a finite, universal time. Our proof is quantitative and inherently nonlocal, as the corresponding statement is false for classical mean curvature flow. This is the first regularizing effect proven for weak solutions to nonlocal curvature flow.
In this paper, we give a classification of the isolated singularities of positive solutions to the semilinear fractional elliptic equations $$(E) \quad\quad (-Δ)^s u = |x|^θ u^{p}\quad {\rm in}\ \ B_1\setminus\{0\},\quad u= h\quad{\rm in}\ \ \mathbb{R}^N\setminus B_1,\quad $$ where $s\in(0,1)$, $θ>-2s$, $p>\frac{N+θ}{N-2s}$, $B_1$ is the unit ball centered at the origin of $\mathbb{R}^N$ with $N>2s$. $h$ is a nonnegative Hölder continuous function in $\mathbb{R}^N\setminus B_1$. Our analysis of isolated singularities of $(E)$ is based on an integral upper bounds and the study of the Poisson pr…
In this paper, we consider an inverse problem for a time-fractional diffusion equation with a nonlinear source. We prove that the considered problem is ill-posed, i.e. the solution does not depend continuously on the data. The problem is ill-posed in the sense of Hadamard. Under some weak {\color{black} a} priori assumptions on the sought solution, we propose a new regularization method for stabili{\color{black}z}ing the ill-posed problem. We also provide a numerical example to illustrate our results.
In this paper, we solve the fractional Lane-Emden equation in the Serrin's critical case for the fractional Laplacian by developing an innovative and self-contained approach that also applies to the classical setting ( Laplacian). We give a classification of the isolated singularities of positive solutions to the semilinear fractional elliptic equations $$(E) \qquad\qquad (-Δ)^s u = u^{\frac{N}{N-2s}}\quad {\rm in}\ \ Ω\setminus\{0\},\qquad u\geq 0\quad{\rm in}\ \ \mathbb{R}^N\setminusΩ,\qquad\qquad\quad $$ where $s\in(0,1)$, $Ω$ is a bounded domain containing the origin in $\mathbb{R}^N$ with…
We prove the existence of a solution of (--$Δ$) s u + f (u) = 0 in a smooth bounded domain $Ω$ with a prescribed boundary value $μ$ in the class of positive Radon measures for a large class of continuous functions f satisfying a weak singularity condition expressed under an integral form. We study the existence of a boundary trace for positive moderate solutions. In the particular case where f (u) = u p and $μ$ is a Dirac mass, we prove the existence of several critical exponents p.
We consider solutions of the linear heat equation in $\mathbb{R}^N$ with isolated singularities. It is assumed that the position of a singular point depends on time and is Hölder continuous with the exponent $α\in (0,1)$. We show that any isolated singularity is removable if it is weaker than a certain order depending on $α$. We also show the optimality of the removability condition by showing the existence of a solution with a nonremovable singularity. These results are applied to the case where the singular point behaves like a fractional Brownian motion with the Hurst exponent $H \in (0,1/2…
In this paper, we consider two Approximate Deconvolution Magnetohydrodynamics models which are related to Large Eddy Simulation. We first study existence and uniqueness of solutions in the double viscous case. Then, we study existence and uniqueness of solutions of the Approximate Deconvolution MHD model with magnetic diffusivity, but without kinematic viscosity. In each case, we give the optimal value of regularizations where we can prove global existence and uniqueness of the solutions. The second model includes the Approximate Deconvolution Euler Model as a particular case. Finally, an asym…
We study some qualitative properties (including removable singularities and superharmonicity) of non-negative solutions to $$ (-Δ)^γu=fu^p\quad\text{in }\mathbb R^n\setminusΣ$$ which are singular at $Σ$. Here $γ\in (0, \frac{n}{2})$. Among other things, we first prove that if $Σ$ is a compact set in $\mathbb R^n$ with Assouad dimension $\bf d$ (not necessarily an integer), ${\bf d}<n-2γ$, and $ u\in L_γ(\mathbb R^n)\cap L^p_{loc}({\mathbb R^n\setminusΣ})$ is a non-negative solution for some $$p>\frac{n-\bf d}{n-{\bf d}-2γ},$$ then $u\in L^p_{loc}(\mathbb R^n)$ and $u$ is a distributional solut…
We study variational models for dislocations in three dimensions in the line-tension scaling. We present a unified approach which allows to treat energies with subquadratic growth at infinity and other regularizations of the singularity near the dislocation lines. We show that the asymptotics via Gamma convergence is independent of the specific choice of the energy and of the regularization procedure.
The use of the fractional Laplacian in image denoising and regularization of inverse problems has enjoyed a recent surge in popularity, since for discontinuous functions it can behave less aggressively than methods based on $H^1$ norms, while being linear and computable with fast spectral numerical methods. In this work, we examine denoising and linear inverse problems regularized with fractional Laplacian in the vanishing noise and regularization parameter regime. The clean data is assumed piecewise constant in the first case, and continuous and satisfying a source condition in the second. In…
This paper considers optimal control of fractional parabolic PDEs with both state and control constraints. The key challenge is how to handle the state constraints. Similarly, to the elliptic case, in this paper, we establish several new mathematical tools in the parabolic setting that are of wider interest. For example, existence of solution to the fractional parabolic equation with measure data on the right-hand-side. We employ the Moreau-Yosida regularization to handle the state constraints. We establish convergence, with rate, of the regularized optimal control problem to the original one.…
We study the formation of singularities for the Euler-Alignment system with influence function $ψ=\frac{k_α}{|x|^α}$ in 1D. As in [20] the problem is reduced to the analysis of a nonlocal 1D equation. We show the existence of singularities in finite time for any $α$ in the range $0<α<2$ in both the real line and the periodic case.
Regularization operators for identifying the unknown source in the time-fractional convection-diffusion-reaction equation
Universidad Austral（澳大利亚大学）；Instituto de Tecnologías Emergentes y Ciencias Aplicadas (UNSAM-CONICET), Centro de Matemática Aplicada, Escuela de Ciencia y Tecnología, Universidad Nacional de General San Martín（通用圣马丁国立大学新兴技术与应用科学学院数学应用中心）
We consider the problem of constructing solutions to the fractional Yamabe problem that are singular at a given smooth sub-manifold, and we establish the classical gluing method of Mazzeo and Pacard for the scalar curvature in the fractional setting. This proof is based on the analysis of the model linearized operator, which amounts to the study of an ODE, and thus our main contribution here is the development of new methods coming from conformal geometry and scattering theory for the study of non-local ODEs. No traditional phase-plane analysis is available here. Instead, first, we provide a r…
Recently the authors have studied a state and control constrained optimal control problem with fractional elliptic PDE as constraints. The goal of this paper is to continue that program forward and introduce an algorithm to solve such optimal control problems. We shall employ the well-known Moreau-Yosida regularization to handle the state constraints. Similarly to the classical case, we establish the convergence (with rate) of the regularized control problem to the original one. We discretize the problem using a finite element method and establish convergence of our numerical scheme. We emphas…
This paper focuses on the regularization of backward time-fractional diffusion problem on unbounded domain. This problem is well-known to be ill-posed, whence the need of a regularization method in order to recover stable approximate solution. For the problem under consideration, we present a unified framework of regularization which covers some techniques such as Fourier regularization [19], mollification [12] and approximate-inverse [7]. We investigate a regularization technique with two major advantages: the simplicity of computation of the regularized solution and the avoid of truncation o…
We consider the problem of attainability of the best constant in the following critical fractional Hardy-Sobolev inequality: \begin{equation*} μ_{γ,s}(\R^n):= \inf\limits_{u \in H^{\fracα{2}} (\R^n)\setminus \{0\}} \frac{ \int_{\R^n} |({-}{ Δ})^{\fracα{4}}u|^2 dx - γ\int_{\R^n} \frac{|u|^2}{|x|^α}dx }{(\int_{\R^n} \frac{|u|^{2_α^*(s)}}{|x|^{s}}dx)^\frac{2}{2_α^*(s)}}, \end{equation*} where $0\leq s<α<2$, $n>α$, ${2_α^*(s)}:=\frac{2(n-s)}{n-α},$ and $γ\in \mathbb{R}$. This allows us to establish the existence of nontrivial weak solutions for the following doubly critical problem on $\R^n$, \beg…
In this paper, we classify the singularities of nonnegative solutions to fractional elliptic equation \begin{equation}\label{eq 0.1} \arraycolsep=1pt \begin{array}{lll} \displaystyle (-Δ)^αu=u^p\quad &{\rm in}\quad Ω\setminus\{0\},\\[2mm] \phantom{ (-Δ)^α} \displaystyle u=0\quad &{\rm in}\quad \mathbb{R}^N\setminusΩ, \end{array} \end{equation} where $p>1$, $Ω$ is a bounded, $C^2$ domain in $\mathbb{R}^N$ containing the origin, $N\ge2$ and the fractional Laplacian $(-Δ)^α$ is defined in the principle value sense. We obtain that any classical solution $u$ of (\ref{eq 0.1}) is a weak solution of …
A nonlinear parabolic equation of the fourth order is analyzed. The equation is characterized by a mobility coefficient that degenerates at 0. Existence of at least one weak solution is proved by using a regularization procedure and deducing suitable a-priori estimates. If a viscosity term is added and additional conditions on the nonlinear terms are assumed, then it is proved that any weak solution becomes instantaneously strictly positive. This in particular implies uniqueness for strictly positive times and further time-regularization properties. The long-time behavior of the problem is als…
We investigate the regularizing effect of adding small fractional Laplacian, with critical fractional exponent 1/2, to a general first order HJB equation. Our results include some regularity estimates for the viscosity solutions of such perturbations, making the solutions classically well-defined. Most importantly, we use these regularity estimates to study the vanishing viscosity approximation to first order HJB equations by 1/2-Laplacian and derive an explicit rate convergence for the vanishing viscosity limit.
We study pathwise regularization by noise for equations on the plane in the spirit of the framework outlined by Catellier and Gubinelli (Stochastic Process. Appl., 2016). To this end, we extend the notion of non-linear Young equations to a two dimensional domain and prove existence and uniqueness of such equations. This concept is then used in order to prove regularization by noise for stochastic equations on the plane. The statement of regularization by noise is formulated in terms of the regularity of the local time associated to the perturbing stochastic field. For this, we provide two quan…
We study necessary conditions and sufficient conditions for the existence of local-in-time solutions of the Cauchy problem for superlinear fractional parabolic equations. Our conditions are sharp and clarify the relationship between the solvability of the Cauchy problem and the strength of the singularities of the initial measure.
We prove regularization properties in short time for inhomogeneous kinetic equations whose collision kernel behaves like a fractional power of the Laplacian in velocity. We treat a fractional Kolmogorov equation and the linearized Boltzmann equation without cutoff (for hard potentials).
In this paper, we study the asymptotic behavior of positive solutions of the fractional Hardy-Hénon equation $$ (-Δ)^σu = |x|^αu^p ~~~~~~~~~~~ in ~~ B_1 \backslash \{0\} $$ with an isolated singularity at the origin, where $σ\in (0, 1)$ and the punctured unit ball $B_1 \backslash \{0\} \subset \mathbb{R}^n$ with $n \geq 2$. When $-2σ< α< 2σ$ and $\frac{n+α}{n-2σ} < p < \frac{n+2σ}{n-2σ}$, we give a classification of isolated singularities of positive solutions, and in particular, this implies sharp blow up estimates of singular solutions. Further, we describe the precise asymptotic behavior of…
This paper is devoted to study the nonlinear sequential fractional boundary value problems involving generalized $ψ$-Caputo fractional derivatives with nonlocal boundary conditions. We investigate the Green function and some of its properties, from which we obtain a new Lyapunov-type inequality for our problem. A lower bound for the possible eigenvalues of our problem is derived. Furthermore, we apply some properties of the Green function to obtain some existence results for our problem. It is worth mentioning that our results still work with some source functions including singularities.
The formation of singularities in finite time in non-local Burgers' equations, with time-fractional derivative, is studied in detail. The occurrence of finite time singularity is proved, revealing the underlying mechanism, and precise estimates on the blow-up time are provided. The employment of the present equation to model a problem arising in job market is also analyzed.
In this article we study inverse source problems for time-fractional diffusion equations from \textit{a posteriori} boundary measurement. Using the memory effect of these class of equations, we solve these inverse problems for several class of space or time dependent source terms. We prove also the unique determination of a general class of space-time dependent separated variables source terms from such measurement. Our approach is based on the study of singularities of the Laplace transform in time of boundary traces of solutions of time-fractional diffusion equations.
From fractional Lane-Emden-Serrin equation -- existence, multiplicity and local behaviors via classical ODE -- to fractional Yamabe metrics with singularity of "maximal" dimension
In this paper, the space-fractional Schrödinger equations with singular potentials are studied. Delta-like or even higher-order singularities are allowed. By using the regularising techniques, we introduce a family of 'weakened' solutions, calling them very weak solutions. The existence, uniqueness and consistency results are proved in an appropriate sense. Numerical simulations are done, and a particle accumulating effect is observed in the singular cases. From the mathematical point of view, a "splitting of the strong singularity" phenomena is also observed.
