Jan Burczak

Boundedness of large-time solutions to a chemotaxis model with nonlocal and semilinear flux

A semilinear version of parabolic-elliptic Keller--Segel system with the \emph{critical} nonlocal diffusion is considered in one space dimension. We show boundedness of weak solutions under very general conditions on our semilinearity. It can degenerate, but has to provide a stronger dissipation for large values of a solution than in the critical linear case or we need to assume certain (explicit) data smallness. Moreover, when one considers a~logistic term with a parameter

On a generalized doubly parabolic Keller–Segel system in one spatial dimension

r

Global solutions for a supercritical drift–diffusion equation

, we obtain our results even for diffusions slightly weaker than the critical linear one and for arbitrarily large initial datum, provided

Critical Keller–Segel meets Burgers on : large-time smooth solutions

r\ge 1

On the generalized Buckley-Leverett equation

. For a mild logistic dampening, we can improve the smallness condition on the initial datum up to

Evolutionary, symmetric

\sim {1}/({1-r})

p

.

-Laplacian. Interior regularity of time derivatives and its consequences

We study a doubly parabolic Keller-Segel system in one spatial dimension, with diffusions given by fractional laplacians. We obtain several local and global well-posedness results for the subcritical and critical cases (for the latter we need certain smallness assumptions). We also study dynamical properties of the system with added logistic term. Then, this model exhibits a spatio-temporal chaotic behavior, where a number of peaks emerge. In particular, we prove the existence of an attractor and provide an upper bound on the number of peaks that the solution may develop. Finally, we perform a numerical analysis suggesting that there is a finite time blow up if the diffusion is weak enough, even in presence of a damping logistic term. Our results generalize on one hand the results for local diffusions, on the other the results for the parabolic-elliptic fractional case.

A Unified Theory for Some Non-Newtonian Fluids Under Singular Forcing

Abstract We study the global existence of solutions to a one-dimensional drift–diffusion equation with logistic term, generalizing the classical parabolic–elliptic Keller–Segel aggregation equation arising in mathematical biology. In particular, we prove that there exists a global weak solution, if the order of the fractional diffusion α ∈ ( 1 − c 1 , 2 ] , where c 1 > 0 is an explicit constant depending on the physical parameters present in the problem (chemosensitivity and strength of logistic damping). Furthermore, in the range 1 − c 2 α ≤ 2 with 0 c 2 c 1 , the solution is globally smooth. Let us emphasize that when α 1 , the diffusion is in the supercritical regime.

Quantitative robustness of regularity for 3D Navier–Stokes system in Ḣα-spaces

We show that solutions to the parabolic–elliptic Keller–Segel system on with critical fractional diffusion remain smooth for any initial data and any positive time. This disproves, at least in the periodic setting, the large-data-blowup conjecture by Bournaveas and Calvez [15]. As a tool, we show smoothness of solutions to a modified critical Burgers equation via a generalization of the ingenious method of moduli of continuity by Kiselev, Nazarov and Shterenberg [35] over a setting where the considered equation has no scaling. This auxiliary result may be interesting by itself. Finally, we study the asymptotic behavior of global solutions corresponding to small initial data, improving the existing results.We show that solutions to the parabolic-elliptic Keller-Segel system on