Lorentz spaces in action on pressureless systems arising from models of collective behavior

We are concerned with global-in-time existence and uniqueness results for models of pressureless gases that come up in the description of phenomena in astrophysics or collective behavior. The initial data are rough: in particular, the density is only bounded. Our results are based on interpolation and parabolic maximal regularity, where Lorentz spaces play a key role. We establish a novel maximal regularity estimate for parabolic systems in $$L_{q,r}(0,T;L_p(\Omega ))$$ L q , r ( 0 , T ; L p ( Ω ) ) spaces.

Local regularity and compactness for the p-harmonic map heat flows

We study a geometric analysis and local regularity for the evolution of {{p}}-harmonic maps, called {{p}}-harmonic map heat flows. Our main result is to establish a criterion for a uniform local regularity estimate for regular {{p}}-harmonic map heat flows, devising some new monotonicity-type formulas of a local scaled energy. The regularity criterion obtained is almost optimal, comparing with that of the corresponding stationary case. As application we show a compactness of regular {{p}}-harmonic map heat flows with energy bound.