اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدOn a multi-dimensional transport equation with nonlocal velocity
55
0
0.0
(
0
)
تأليف
Hongjie Dong
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We study a multi-dimensional nonlocal active scalar equation of the form $u_t+vcdot abla u=0$ in $mathbb R^+times mathbb R^d$, where $v=Lambda^{-2+alpha} abla u$ with $Lambda=(-Delta)^{1/2}$. We show that when $alphain (0,2]$ certain radial solutions develop gradient blowup in finite time. In the case when $alpha=0$, the equations are globally well-posed with arbitrary initial data in suitable Sobolev spaces.
قيم البحث
اقرأ أيضاً
We prove existence, uniqueness, regularity and separation properties for a nonlocal Cahn-Hilliard equation with a reaction term. We deal here with the case of logarithmic potential and degenerate mobility as well an uniformly lipschitz in $u$ reaction term $g(x,t,u).$
We consider the 1D transport equation with nonlocal velocity field: begin{equation*}label{intro eq} begin{split} &theta_t+utheta_x+ u Lambda^{gamma}theta=0, & u=mathcal{N}(theta), end{split} end{equation*} where $mathcal{N}$ is a nonlocal operator.
In this paper, we show the existence of solutions of this model locally and globally in time for various types of nonlocal operators.
In this paper we study a distributed optimal control problem for a nonlocal convective Cahn--Hilliard equation with degenerate mobility and singular potential in three dimensions of space. While the cost functional is of standard tracking type, the c
ontrol problem under investigation cannot easily be treated via standard techniques for two reasons: the state system is a highly nonlinear system of PDEs containing singular and degenerating terms, and the control variable, which is given by the velocity of the motion occurring in the convective term, is nonlinearly coupled to the state variable. The latter fact makes it necessary to state rather special regularity assumptions for the admissible controls, which, while looking a bit nonstandard, are however quite natural in the corresponding analytical framework. In fact, they are indispensable prerequisites to guarantee the well-posedness of the associated state system. In this contribution, we employ recently proved existence, uniqueness and regularity results for the solution to the associated state system in order to establish the existence of optimal controls and appropriate first-order necessary optimality conditions for the optimal control problem.
We study a class of nonlocal reaction-diffusion equations with a harvesting term where the nonlocal operator is given by a Bernstein function of the Laplacian. In particular, it includes the fractional Laplacian, fractional relativistic operators, su
m of fractional Laplacians of different order etc. We study existence, uniqueness and multiplicity results of the solutions to the steady state equation. We also consider the parabolic counterpart and establish the long time asymptotic of the solutions. Our proof techniques rely on both analytic and probabilistic arguments.
We study the radially symmetric high dimensional Fisher-KPP nonlocal diffusion equation with free boundary, and reveal some fundamental differences from its one dimensional version considered in cite{cdjfa} recently. Technically, this high dimensiona
l problem is much more difficult to treat since it involves two kernel functions which arise from the original kernel function $J(|x|)$ in rather implicit ways. By introducing new techniques, we are able to determine the long-time dynamics of the model, including firstly finding the threshold condition on the kernel function that governs the onset of accelerated spreading, and the determination of the spreading speed when it is finite. Moreover, for two important classes of kernel functions, sharp estimates of the spreading profile are obtained. More precisely, for kernel functions with compact support, we show that logarithmic shifting occurs from the finite wave speed propagation, which is strikingly different from the one dimension case; for kernel functions $J(|x|)$ behaving like $|x|^{-beta}$ for $xinR^N$ near infinity, we obtain the rate of accelerated spreading when $betain (N, N+1]$, which is the exact range of $beta$ where accelerated spreading is possible. These sharp estimates are obtained by constructing subtle upper and lower solutions, based on careful analysis of the involved kernel functions.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
