Finite time singularities in a class of hydrodynamic models

Finite time singularities in a class of hydrodynamic models.

Models of inviscid incompressible fluid are considered, with the kinetic energy (i.e., the Lagrangian functional) taking the form L approximately integral k(alpha)/vk/2dk in 3D Fourier representation, where alpha is a constant, 0<alpha<1. Unlike the case alpha=0 (the usual Eulerian hydrodynamics), a finite value of alpha results in a finite energy for a singular, frozen-in vortex filament. This...

Hydrodynamic Singularities

The equations of hydrodynamics are nonlinear partial differential equations, so they include the possibility of forming singularities in finite time. This means that hydrodynamic fields become infinite or at least non-smooth at points, lines, or even fractal objects. This mathematical possibility is the price one has to pay for the enormous conceptual simplification a continuum theory brings fo...

Finite-time Euler singularities: A Lagrangian perspective

Finite Time Singularities for a Class of Generalized Surface Quasi-geostrophic Equations

We propose and study a class of generalized surface quasi-geostrophic equations. We show that in the inviscid case certain radial solutions develop gradient blow-up in finite time. In the critical dissipative case, the equations are globally well-posed with arbitrary H1 initial data.

Finite Time Singularities for Hyperbolic Systems

In this paper, we study the formation of finite time singularities in the form of super norm blowup for a spatially inhomogeneous hyperbolic system. The system is related to the variational wave equations as those in [18]. The system posses a unique C solution before the emergence of vacuum in finite time, for given initial data that are smooth enough, bounded and uniformly away from vacuum. At...