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Register a new userOptimal decay rates and space-time analyticity of solutions to the Patlak-Keller-Segel equations
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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.
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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 $qin R^2$
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.
We address the analyticity and large time decay rates for strong solutions of the Hall-MHD equations. By Gevrey estimates, we show that the strong solution with small initial date in $H^r(mathbb{R}^3)$ with $r>f 52$ becomes analytic immediately after $t>0$, and the radius of analyticity will grow like $sqrt{t}$ in time. Upper and lower bounds on the decay of higher order derivatives are also obtained, which extends the previous work by Chae and Schonbek (J. Differential Equations 255 (2013), 3971--3982).
We show that the Keller-Segel model in one dimension with Neumann boundary conditions and quadratic cellular diffusion has an intricate phase transition diagram depending on the chemosensitivity strength. Explicit computations allow us to find a myriad of symmetric and asymmetric stationary states whose stability properties are mostly studied via free energy decreasing numerical schemes. The metastability behavior and staircased free energy decay are also illustrated via these numerical simulations.
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