In this paper we consider a subcritical model that involves nonlocal diffusion and a classical convective term. In spite of the nonlocal diffusion, we obtain an Oleinik type estimate similar to the case when the diffusion is local. First we prove that the entropy solution can be obtained by adding a small viscous term $μu_{xx}$ and letting $μ\to 0$. Then, by using uniform Oleinik estimates for the viscous approximation we are able to prove the well-posedness of the entropy solutions with $L^1$-initial data. Using a scaling argument and hyperbolic estimates given by Oleinik's inequality, we obtain the first term in the asymptotic behavior of the nonnegative solutions. Finally, the large time behavior of changing sign solutions is proved using the classical flux-entropy method and estimates for the nonlocal operator.
This paper deals with a parabolic-elliptic chemotaxis system with nonlocal type of source in the whole space. It's proved that the initial value problem possesses a unique global solution which is uniformly bounded. Here we identify the exponents regimes of nonlinear reaction and aggregation in such a way that their scaling and the diffusion term coincide (see Introduction). Comparing to the classical KS model (without the source term), it's shown that how energy estimates give natural conditions on the nonlinearities implying the absence of blow-up for the solution without any restriction on …
In this paper we consider a model that involves nonlocal diffusion and a classical convective term. Using a scaling argument and a new compactness argument we obtain the first term in the asymptotic behavior of the solutions.
We study a 1D transport equation with nonlocal velocity with subcritical or supercritical dissipation. For all data in the weighted Sobolev space $H^{k}(w_{λ,κ}) \cap L^{\infty},$ with $k=\max(0,3/2-α)$ and $w_{λ, κ}$ is a given family of Muckenhoupt weights. We prove a global existence result in the subcritical case $α\in (1,2)$. We also prove a local existence theorem for large data in $H^{2}(w_{λ, κ})\cap L^{\infty}$ in the supercritical case $α\in (0,1)$. The proofs are based on the use of the weighted Littlewood-Paley theory, interpolation along with some new commutator estimates.
In this paper we consider a nonlocal viscous Burgers equation and study the well-posedness and asymptotic behaviour of its solutions. We prove that under the smallness assumption on the initial data the solutions behave as the self similar profiles of the Burgers equation with Dirac mass as the initial datum. The first term in the asymptotic expansion of the solutions is obtained by rescaling the solutions and proving the compactness of their trajectories.
We consider the "convection-diffussion" equation $u_t=J*u-u-uu_x,$ where $J$ is a probability density. We supplement this equation with step-like initial conditions and prove a convergence of corresponding solution towards a rarefaction wave, i.e. a unique entropy solution of the Riemann problem for the nonviscous Burgers equation. Methods and tools used in this paper are inspired by those used in [Karch, Miao and Xu, SIAM J. Math. Anal. {\bf 39} (2008), no. 5, 1536--1549.], where the fractal Burgers equation was studied.
In this paper we continue our study of the large time behavior of the bounded solution to the nonlocal diffusion equation with absorption \begin{align} \begin{cases} u_t = \mathcal{L} u-u^p\quad& \mbox{in}\quad \mathbb R^N\times(0,\infty),\\ u(x,0) = u_0(x)\quad& \mbox{in}\quad \mathbb R^N, \end{cases} \end{align} where $p>1$, $u_0\ge0$ and bounded and $$ \mathcal{L} u(x,t)=\int J(x-y)\left(u(y,t)-u(x,t)\right)\,dy $$ with $J\in C_0^{\infty}(\mathbb R^N)$, radially symmetric, $J\geq 0$ with $\int J=1$. Our assumption on the initial datum is that $0\le u_0\in L^\infty(\mathbb R^N)$ and $$ |x|^α…
We develop a general framework for finding error estimates for convection-diffusion equations with nonlocal, nonlinear, and possibly degenerate diffusion terms. The equations are nonlocal because they involve fractional diffusion operators that are generators of pure jump Levy processes (e.g. the fractional Laplacian). As an application, we derive continuous dependence estimates on the nonlinearities and on the Levy measure of the diffusion term. Estimates of the rates of convergence for general nonlinear nonlocal vanishing viscosity approximations of scalar conservation laws then follow as a …
In this paper, we consider the existence of nodal solutions with two bubbles to the slightly subcritical problem with the fractional Laplacian \begin{equation*} \left\{\aligned &(-Δ)^su=|u|^{p-1-\varepsilon}u\ \ \mbox{in}\ Ω&u=0\ \mbox{on}\ \partialΩ, \endaligned \right. \end{equation*} where $Ω$ is a smooth bounded domain in $\mathbb R^N$, $N>2s$, $0<s<1$, $ p=\frac{N+2s}{N-2s}$ and $\varepsilon>0$ is a small parameter, which can be seen as a nonlocal analog of the results of Bartsch, Micheletti and Pistoia (2006) \cite{Bartsch1}.
We study the existence of global-in-time solutions for a nonlinear heat equation with nonlocal diffusion, power nonlinearity and suitably small data (either compared pointwisely to the singular solution or in the norm of a critical Morrey space). Then, asymptotics of subcritical solutions is determined. These results are compared with conditions on the initial data leading to a finite time blowup.
The Cauchy problem considered in this paper is the following \begin{align} \left\{ \begin{array}{ll} u_t=Δu+u^α\left(M_0- \int_{\mathbb{R}^n} u(x,t)dx\right),\quad & x \in \mathbb{R}^n, t>0, u(x,0)=U_0(x)\geq 0,\quad & x \in \mathbb{R}^n. \end{array} \right. \end{align} where $M_0>0, α>1, n \ge 3$. When the coefficient $M_0-\int_{\mathbb{R}^n} u(x,t) dx$ remains positive, \er{nkpp0} is analogous to \begin{align} \left\{ \begin{array}{ll} u_t=Δu+u^α,\quad & x \in \mathbb{R}^n, t>0, u(x,0)=U_0(x)\geq 0,\quad & x \in \mathbb{R}^n. \end{array} \right. \end{align} It is well known that when $1<α\le…
In this paper, we study a one dimensional nonlinear equation with diffusion $-ν(-\partial_{xx})^{\fracα{2}}$ for $0\leq α\leq 2$ and $ν>0$. We use a viscous-splitting algorithm to obtain global nonnegative weak solutions in space $L^1(\mathbb{R})\cap H^{1/2}(\mathbb{R})$ when $0\leqα\leq 2$. For subcritical $1<α\leq 2$ and critical case $α=1$, we obtain global existence and uniqueness of nonnegative spatial analytic solutions. We use a fractional bootstrap method to improve the regularity of mild solutions in Bessel potential spaces for subcritical case $1<α\leq 2$. Then, we show that the solu…
On the decay estimates of a nonlocal convection-diffusion Hamer system
Institut de Mathématiques de Toulouse, Université de Toulouse（图卢兹数学研究所，图卢兹大学）；Department of Mathematics, College of Sciences, University of Sharjah（沙迦大学理学院数学系）
Traveling Waves for Nonlocal Derivative Nonlinear Schrödinger Equations: A Variational Characterization
School of Computing and Artificial Intelligence (SCAI) Nazarbayev University（纳扎尔巴耶夫大学计算与人工智能学院）；Private Institution Nazarbayev University Research Administration（纳扎尔巴耶夫大学研究管理局私立机构）；School of Artificial Intelligence and Data Science Astana IT University（阿斯塔纳IT大学人工智能与数据科学学院）
In this paper we give a new compactness criterion in the Lebesgue spaces $L^p((0,T)\times Ω)$ and use it to obtain the first term in the asymptotic behaviour of the solutions of a nonlocal convection diffusion equation. We use previous results of Bourgain, Brezis and Mironescu to give a new criterion in the spirit of the Aubin-Lions-Simon Lemma.
We study a class of nonlinear non-autonomous nonlocal equations with subcritical and critical exponential nonlinearity. The involved potential can vanish at infinity.
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…
The aim of this paper is to study a concentration-compactness principle for inhomogeneous fractional Sobolev space $H^s (\mathbb{R}^N)$ for $0<s\leq N/2.$ As an application we establish Palais-Smale compactness for the Lagrangian associated to the fractional Schrödinger equation $(-Δ)^{s} u + a(x)u= f(x,u)$ for $0<s<1.$ Moreover, we prove the existence of nontrivial nonnegative solutions to this class of elliptic equations for a wide class of possible singular potentials $a(x)$; not necessarily bounded away from zero. We consider possible oscillatory nonlinearities and that may not satisfy the…
This paper is concerned with the uniqueness of solutions to the following nonlocal semi-linear elliptic equation \begin{equation}\label{ellip}\tag{$\ast$} Δu-βu+λ\frac{e^u}{\int_Ωe^u}=0~\mathrm{in}~Ω, \end{equation} where $Ω$ is a bounded domain in $\mathbb{R}^2$ and $β, λ$ are positive parameters. The above equation arises as the stationary problem of the well-known classical Keller-Segel model describing chemotaxis. For equation \eqref{ellip} with Neumann boundary condition, we establish an integral inequality and prove that the solution of (\ref{ellip}) is unique if $0<λ\leq 8π$ and $u$ sat…
Existence and uniqueness of solutions for nonlocal Cahn-Hilliard equations with degenerate potential is shown. The nonlocality is described by means of a symmetric singular kernel not falling within the framework of any previous existence theory. A convection term is also taken into account. Building upon this novel existence result, we prove convergence of solutions for this class of nonlocal Cahn-Hilliard equations to their local counterparts, as the nonlocal convolution kernels approximate a Dirac delta. Eventually, we show that, under suitable assumptions on the data, the solutions to the …
We study the existence of large solutions for nonlocal Dirichlet problems posed on a bounded, smooth domain, associated to fully nonlinear elliptic equations of order $2s$, with $s\in (1/2,1)$, and a coercive gradient term with subcritical power $0<p<2s$. Due to the nonlocal nature of the diffusion, new blow-up phenomena arise within the range $0<p<2s$, involving a continuum family of solutions and/or solutions blowing-up to $-\infty$ on the boundary. This is in striking difference with the local case studied by Lasry-Lions for the case subquadratic case $1<p<2$.
A unified clasification of Liouville properties and nontrivial solution for fractional elliptic equations with general Hénon-type superquadratic and gradient growth
In this paper, we are concerned with static Schrödinger-Hartree and Schrödinger-Maxwell equations with combined nonlinearities. We derive the explicit forms for positive solution $u$ in the critical case and non-existence of nontrivial nonnegative solutions in the subcritical cases (see Theorem \ref{Thm0} and \ref{Thm1}). The arguments used in our proof is a variant (for nonlocal nonlinearity) of the direct moving spheres method for fractional Laplacians in \cite{CLZ}. The main ingredients are the variants (for nonlocal nonlinearity) of the maximum principles, i.e., \emph{Narrow region princip…
In this paper, we study the existence of three solutions for a Kirchhoff equation involving the nonlocal fractional p-Laplacian considering Sobolev and Hardy nonlinearities at subcritical and critical growths. The proof is based on Mountain Pass Theorem, Ekeland's variational principle and constrained minimization in Nehari sets.
We study the nonlocal scalar field equation with a vanishing parameter \[ \left\{\begin{array}{lll} (-Δ)^s u+εu &=|u|^{p-2}u -|u|^{q-2}u \quad\text{in}\quad\mathbb{R}^N \\ u >0, & u \in H^s(\mathbb{R}^N), \end{array} \right. \] where $s\in(0,1)$, $N>2s$, $q>p>2$ are fixed parameters and $ε>0$ is a vanishing parameter. For $ε>0$ small, we prove the existence of a ground state solution and show that any positive solution of above problem is a classical solution and radially symmetric and symmetric decreasing. We also obtain the decay rate of solution at infinity. Next, we study the asymptotic be…
We study the propagation of wave packets for nonlinear nonlocal Schrodinger equations in the semi-classical limit. When the kernel is smooth, we construct approximate solutions for the wave functions in subcritical, critical and supercritical cases (in terms of the size of the initial data). The validity of the approximation is proved up to Ehrenfest time. For homogeneous kernels, we establish similar results in subcritical and critical cases. Nonlinear superposition principle for two nonlinear wave packets is also considered.
