arXiv subjects

James P. Kelliher

Publications and source records attributed to James P. Kelliher.

22 records · Page 2Linked to original sources

arXiv2009

Infinite-energy 2D statistical solutions to the equations of incompressible fluids

We develop the concept of an infinite-energy statistical solution to the Navier-Stokes and Euler equations in the whole plane. We use a velocity formulation with enough generality to encompass initial velocities having bounded vorticity, which includes the important special case of vortex patch initial data. Our approach is to use well-studied properties of statistical solutions in a ball of radius R to construct, in the limit as R goes to infinity, an infinite-energy solution to the Navier-Stokes equations. We then construct an infinite-energy statistical solution to the Euler equations by making a vanishing viscosity argument.

math.AP↗

arXiv2008

Vanishing viscosity and the accumulation of vorticity on the boundary

We say that the vanishing viscosity limit holds in the classical sense if the velocity for a solution to the Navier-Stokes equations converges in the energy norm uniformly in time to the velocity for a solution to the Euler equations. We prove, for a bounded domain in dimension 2 or higher, that the vanishing viscosity limit holds in the classical sense if and only if a vortex sheet forms on the boundary.

math-ph↗

arXiv2004

The inviscid limit for two-dimensional incompressible fluids with unbounded vorticity

Chemin has shown that solutions of the Navier-Stokes equations in the plane for an incompressible fluid whose initial vorticity is bounded and lies in L^2 converge in the zero-viscosity limit in the L^2-norm to a solution of the Euler equations, convergence being uniform over any finite time interval. Yudovich, assuming an initial vorticity lying in L^p for all p >= q for some q, established the uniqueness of solutions to the Euler equations for an incompressible fluid in a bounded domain of n-space assuming a particular bound on the growth of the L^p-norm of the initial vorticity as p grows large. We combine these two approaches to establish, in the plane, the uniqueness of solutions to the Euler equations and the same zero-viscosity convergence as Chemin, but under Yudovich's assumptions on the vorticity with q = 2. The resulting bounded rate of convergence can be arbitrarily slow as a function of the viscosity.

math-ph↗

arXiv2004

Navier-Stokes equations with Navier boundary conditions for a bounded domain in the plane

We consider solutions to the Navier-Stokes equations with Navier boundary conditions in a bounded domain in the plane with a C^2-boundary. Navier boundary conditions can be expressed in the form w = (2 K - A) v . T and v . n = 0 on the boundary, where v is the velocity, w the vorticity, n a unit normal vector, T a unit tangent vector, and A is a bounded measurable function on the boundary. Such solutions have been considered for simply connected domains by Clopeau, Mikelić, and Robert under the assumption that the initial vorticity is bounded, and by Lopes Filho, Nussenzveig Lopes, and Planas under the assumption that the initial vorticity lies in L^p for some p > 2. We extend the results of these authors to non-simply connected domains. Assuming a particular bound on the growth of the L^p-norms of the initial vorticity with p, and also assuming that the boundary and the function A have fractionally greater smoothness, we obtain a bound on the rate of convergence in L^2 uniform over any finite time interval to the solution of the Euler equations in the vanishing viscosity limit. We also show that if the initial velocity is in H^3 and the boundary is C^3, then solutions to the Navier-Stokes equations with Navier boundary conditions converge in L^2 uniformly over any finite time interval to the solution to the Navier-Stokes equations with the usual no-slip boundary conditions as we let the function A grow large uniformly on the boundary.

math-ph↗