Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userGlobal existence of a non-local semilinear parabolic equation with advection and applications to shear flow
105
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this paper, we consider the following non-local semi-linear parabolic equation with advection: for $1 le p<1+frac{2}{N}$, begin{equation*} begin{cases} u_t+v cdot abla u-Delta u=|u|^p-int_{mathbb T^N} |u|^p quad & textrm{on} quad mathbb T^N, u textrm{periodic} quad & textrm{on} quad partial mathbb T^N end{cases} end{equation*} with initial data $u_0$ defined on $mathbb T^N$. Here $v$ is an incompressible flow, and $mathbb T^N=[0, 1]^N$ is the $N$-torus with $N$ being the dimension. We first prove the local existence of mild solutions to the above equation for arbitrary data in $L^2$. We then study the global existence of the solutions under the following two scenarios: (1). when $v$ is a mixing flow; (2). when $v$ is a shear flow. More precisely, we show that under these assumptions, there exists a global solution to the above equation in the sense of $L^2$.
rate research
Read More
In this paper, we consider the advective Cahn-Hilliard equation in 2D with shear flow: $$ begin{cases} u_t+v_1(y) partial_x u+gamma Delta^2 u=gamma Delta(u^3-u) quad & quad textrm{on} quad mathbb T^2; u textrm{periodic} quad & quad textrm{on} quad partial mathbb T^2, end{cases} $$ where $mathbb T^2$ is the two-dimensional torus. Under the assumption that the shear has a finite number of critical points and there are linearly growing modes only in the direction of the shear, we show the global existence of solutions with arbitrary initial $H^2$ data. The main difficulty of this paper is to handle the high-regularity and non-linearity underlying the term $Delta(u^3)$ in a proper way. For such a purpose, we modify the methods by Iyer, Xu, and Zlatov{s} in 2021 under a shear flow setting.
We consider an evolution problem associated to the Kazdan-Warner equation on a closed Riemann surface $(Sigma,g)$ begin{align*} -Delta_{g}u=8pileft(frac{he^{u}}{int_{Sigma}he^{u}{rm d}mu_{g}}-frac{1}{int_{Sigma}{rm d}mu_{g}}right) end{align*} where the prescribed function $hgeq0$ and $max_{Sigma}h>0$. We prove the global existence and convergence under additional assumptions such as begin{align*} Delta_{g}ln h(p_0)+8pi-2K(p_0)>0 end{align*} for any maximum point $p_0$ of the sum of $2ln h$ and the regular part of the Green function, where $K$ is the Gaussian curvature of $Sigma$. In particular, this gives a new proof of the existence result by Yang and Zhu [Proc. Amer. Math. Soc. 145 (2017), no. 9, 3953-3959] which generalizes existence result of Ding, Jost, Li and Wang [Asian J. Math. 1 (1997), no. 2, 230-248] to the non-negative prescribed function case.
The local and global existence of the Cauchy problem for semilinear heat equations with small data is studied in the weighted $L^infty (mathbb R^n)$ framework by a simple contraction argument. The contraction argument is based on a weighted uniform control of solutions related with the free solutions and the first iterations for the initial data of negative power.
We consider a class of semilinear nonlocal problems with vanishing exterior condition and establish a Ambrosetti-Prodi type phenomenon when the nonlinear term satisfies certain conditions. Our technique makes use of the probabilistic tools and heat kernel estimates.
Considered herein is a multi-component Novikov equation, which admits bi-Hamiltonian structure, infinitely many conserved quantities and peaked solutions. In this paper, we deduce two blow-up criteria for this system and global existence for some two-component case in $H^s$. Finally we verify that the system possesses peakons and periodic peakons.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
