Journal of Differential Equations 

TL;DR: In this paper, the Navier-Stokes equations for the flow of an incompressible, viscous fluid are derived and analyzed, and the main result is that depending on geometric and material parameters, the equations governing granular flow may lead to a violent instability analogous to that for u, = u XI up ;

TL;DR: In this article, the authors considered the classical parabolic-parabolic Keller-Segel system with homogeneous Neumann boundary conditions in a smooth bounded domain and proved that for each q > n 2 and p > n one can find e 0 > 0 such that if the initial data ( u 0, v 0 ) satisfy L q ( Ω ) e and ∇ v 0 ‖ L p (Ω) e then the solution is global in time and bounded and asymptotically behaves like the solution of a discoupled system of linear parabolic

TL;DR: In this paper, a local regular solution for the Navier-Stokes system was constructed for a class of semilinear parabolic equations with dimensionless or scaling invariant norm, where p and q are chosen so that the norm of Lq(0, T; Lp) is dimensionless.