Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn the $8pi$-critical mass threshold of a Patlak-Keller-Segel-Navier-Stokes system
90
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this paper, we proposed a coupled Patlak-Keller-Segel-Navier-Stokes system, which has dissipative free energy. On the plane $rr^2$, if the total mass of the cells is strictly less than $8pi$, classical solutions exist for any finite time, and their $H^s$-Sobolev norms are almost uniformly bounded in time. For the radially symmetric solutions, this $8pi$-mass threshold is critical. On the torus $mathbb{T}^2$, the solutions are uniformly bounded in time under the same mass constraint.
rate research
Read More
We study the regularity and large-time behavior of a crowd of species driven by chemo-tactic interactions. What distinguishes the different species is the way they interact with the rest of the crowd: the collective motion is driven by different chemical reactions which end up in a coupled system of parabolic Patlak-Keller-Segel equations. We show that the densities of the different species diffuse to zero provided the chemical interactions between the different species satisfy certain sub-critical condition; the latter is intimately related to a log-Hardy-Littlewood-Sobolev inequality for systems due to Shafrir & Wolansky. Thus for example, when two species interact, one of which has mass less than $4pi$, then the 2-system stays smooth for all time independent of the total mass of the system, in sharp contrast with the well-known breakdown of one specie with initial mass$> 8pi$.
Perhaps the most classical diffusion model for chemotaxis is the Patlak-Keller-Segel system begin{equation} label{ks0} left{ begin{aligned} u_t =& Delta u - abla cdot(u abla v) quad inn R^2times(0,infty), v =& (-Delta_{R^2})^{-1} u := frac 1{2pi} int_{R^2} , log frac 1{|x-z|} u(z,t), dz, & qquad u(cdot ,0) = u_0 ge 0quadhbox{in } R^2. end{aligned} right. end{equation} We consider the {em critical mass case} $int_{R^2} u_0(x), dx = 8pi$ which corresponds to the exact threshold between finite-time blow-up and self-similar diffusion towards zero. We find a radial function $u_0^*$ with mass $8pi$ such that for any initial condition $u_0$ sufficiently close to $u_0^*$ the solution $u(x,t)$ of equ{ks0} is globally defined and blows-up in infinite time. As $tto+infty $ it has the approximate profile $$ u(x,t) approx frac 1{la^2} U_0left (frac {x-xi(t)}{la(t)} right ), quad U_0(y)= frac{8}{(1+|y|^2)^2}, $$ where $la(t) approx frac c{sqrt{log t}}, xi(t)to q $ for some $c>0$ and $qin R^2$
We are concerned with the Keller--Segel--Navier--Stokes system begin{equation*} left{ begin{array}{ll} rho_t+ucdot ablarho=Deltarho- ablacdot(rho mathcal{S}(x,rho,c) abla c)-rho m, &!! (x,t)in Omegatimes (0,T), m_t+ucdot abla m=Delta m-rho m, &!! (x,t)in Omegatimes (0,T), c_t+ucdot abla c=Delta c-c+m, & !! (x,t)in Omegatimes (0,T), u_t+ (ucdot abla) u=Delta u- abla P+(rho+m) ablaphi,quad ablacdot u=0, &!! (x,t)in Omegatimes (0,T) end{array}right. end{equation*} subject to the boundary condition $( ablarho-rho mathcal{S}(x,rho,c) abla c)cdot u!!=! abla mcdot u= abla ccdot u=0, u=0$ in a bounded smooth domain $Omegasubsetmathbb R^3$. It is shown that the corresponding problem admits a globally classical solution with exponential decay properties under the hypothesis that $mathcal{S}in C^2(overlineOmegatimes [0,infty)^2)^{3times 3}$ satisfies $|mathcal{S}(x,rho,c)|leq C_S $ for some $C_S>0$, and the initial data satisfy certain smallness conditions.
Based on some elementary estimates for the space-time derivatives of the heat kernel, we use a bootstrapping approach to establish the optimal decay rates for the $L^q(mathbb{R}^d)$ ($1leq qleqinfty$, $dinmathbb{N}$) norm of the space-time derivatives of solutions to the (modified) Patlak-Keller-Segel equations with initial data in $L^1(mathbb{R}^d)$, which implies the joint space-time analyticity of solutions. When the $L^1(mathbb{R}^d)$ norm of the initial datum is small, the upper bound for the decay estimates is global in time, which yields a lower bound on the growth rate of the radius of space-time analyticity in time. As a byproduct, the space analyticity is obtained for any initial data in $L^1(mathbb{R}^d)$. The decay estimates and space-time analyticity are also established for solutions bounded in both space and time variables. The results can be extended to a more general class of equations, including the Navier-Stokes equations.
We are concerned with the Cauchy problem of the full compressible Navier-Stokes equations satisfied by viscous and heat conducting fluids in $mathbb{R}^n.$ We focus on the so-called critical Besov regularity framework. In this setting, it is natural to consider initial densities $rho_0,$ velocity fields $u_0$ and temperatures $theta_0$ with $a_0:=rho_0-1indot B^{frac np}_{p,1},$ $u_0indot B^{frac np-1}_{p,1}$ and $theta_0indot B^{frac np-2}_{p,1}.$ After recasting the whole system in Lagrangian coordinates, and working with the emph{total energy along the flow} rather than with the temperature, we discover that the system may be solved by means of Banach fixed point theorem in a critical functional framework whenever the space dimension is $ngeq2,$ and $1<p<2n.$ Back to Eulerian coordinates, this allows to improve the range of $p$s for which the system is locally well-posed, compared to Danchin, Comm. Partial Differential Equations 26 (2001).
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
