This paper establishes several existence and uniqueness results for two families of active
scalar equations with velocity fields determined by the scalars through very singular …

We survey rigorous, formal and numerical results on the formation of point-like singularities
(or blow-up) for a wide range of evolution equations. We use a similarity transformation of …

Collective phenomena occur in nature, technology, and social behavior in a vast variety of
contexts. For example, birds and fish use their sensory abilities to communicate and align …

We study a pressureless Euler system with a non-linear density-dependent alignment term,
originating in the Cucker–Smale swarming models. The alignment term is dissipative in the …

Any classical solution of the two-dimensional incompressible Euler equation is global in
time. However, it remains an outstanding open problem whether classical solutions of the …

Formation of Singularities for a Transport Equation with Nonlocal Velocity Page 1 Annals of
Mathematics, 162 (2005), 1377-1389 Formation of singularities for a transport equation with …

arxiv:0812.4979v1 [math.AP] 29 Dec 2008 Page 1 arxiv:0812.4979v1 [math.AP] 29 Dec 2008
NONLINEAR DIFFUSION OF DISLOCATION DENSITY AND SELF-SIMILAR SOLUTIONS …

We present evidence on the global existence of solutions of De Gregorio's equation, based
on numerical computation and a mathematical criterion analogous to the Beale–Kato–Majda …

The question of spontaneous apparition of singularity in the 3D incompressible Euler
equations is one of the most important and challenging open problems in mathematical fluid …

We consider a one dimensional transport model with nonlocal velocity given by the Hilbert
transform and develop a global well-posedness theory of probability measure solutions …