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The concept of continuous topological evolution, based upon Cartans methods of exterior differential systems, is used to develop a topological theory of non-equilibrium thermodynamics, within which there exist processes that exhibit continuous topological change and thermodynamic irreversibility. The technique furnishes a universal, topological foundation for the partial differential equations of hydrodynamics and electrodynamics; the technique does not depend upon a metric, connection or a variational principle. Certain topological classes of solutions to the Navier-Stokes equations are shown to be equivalent to thermodynamically irreversible processes.
A coupled forward-backward stochastic differential system (FBSDS) is formulated in spaces of fields for the incompressible Navier-Stokes equation in the whole space. It is shown to have a unique local solution, and further if either the Reynolds numb
In a particle physics dynamics, we assume a uniform distribution as the physical measure and a measure-theoretic definition of entropy on the velocity configuration space. This distribution is labeled as the physical solution in the remainder of the
We prove that the steady--state Navier--Stokes problem in a plane Lipschitz domain $Omega$ exterior to a bounded and simply connected set has a $D$-solution provided the boundary datum $a in L^2(partialOmega)$ satisfies ${1over 2pi}|int_{partialOmega
In this paper, we investigate numerically a diffuse interface model for the Navier-Stokes equation with fluid-fluid interface when the fluids have different densities cite{Lowengrub1998}. Under minor reformulation of the system, we show that there is
We accomplish two major tasks. First, we show that the turbulent motion at large scales obeys Gaussian statistics in the interval 0 < Rlambda < 8.8, where Rlambda is the microscale Reynolds number, and that the Gaussian flow breaks down to yield plac