Strong-viscosity Solutions: Semilinear Parabolic PDEs and Path-dependent PDEs

The aim of the present work is the introduction of a viscosity type solution, called strong-viscosity solution to distinguish it from the classical one, with the following peculiarities: it is a purely analytic object; it can be easily adapted to more general equations than classical partial differential equations. First, we introduce the notion of strong-viscosity solution for semilinear parabolic partial differential equations, defining it, in a few words, as the pointwise limit of classical solutions to perturbed semilinear parabolic partial differential equations; we compare it with the standard definition of viscosity solution. Afterwards, we extend the concept of strong-viscosity solution to the case of semilinear parabolic path-dependent partial differential equations, providing an existence and uniqueness result.Comment: arXiv admin note: text overlap with arXiv:1401.503

On the vanishing viscosity limit in a disk

We say that the solution u to the Navier-Stokes equations converges to a solution v to the Euler equations in the vanishing viscosity limit if u converges to v in the energy norm uniformly over a finite time interval. Working specifically in the unit disk, we show that a necessary and sufficient condition for the vanishing viscosity limit to hold is the vanishing with the viscosity of the time-space average of the energy of u in a boundary layer of width proportional to the viscosity due to modes (eigenfunctions of the Stokes operator) whose frequencies in the radial or the tangential direction lie between L and M. Here, L must be of order less than 1/(viscosity) and M must be of order greater than 1/(viscosity)

Non-Smooth Stochastic Lyapunov Functions With Weak Extension of Viscosity Solutions

This paper proposes a notion of viscosity weak supersolutions to build a bridge between stochastic Lyapunov stability theory and viscosity solution theory. Different from ordinary differential equations, stochastic differential equations can have the origins being stable despite having no smooth stochastic Lyapunov functions (SLFs). The feature naturally requires that the related Lyapunov equations are illustrated via viscosity solution theory, which deals with non-smooth solutions to partial differential equations. This paper claims that stochastic Lyapunov stability theory needs a weak extension of viscosity supersolutions, and the proposed viscosity weak supersolutions describe non-smooth SLFs ensuring a large class of the origins being noisily (asymptotically) stable and (asymptotically) stable in probability. The contribution of the non-smooth SLFs are confirmed by a few examples; especially, they ensure that all the linear-quadratic-Gaussian (LQG) controlled systems have the origins being noisily asymptotically stable for any additive noises

Gravitational Instability in Presence of Bulk Viscosity: the Jeans Mass and the Quasi-Isotropic Solution

This paper focuses on the analysis of the gravitational instability in presence of bulk viscosity both in Newtonian regime and in the fully-relativistic approach. The standard Jeans Mechanism and the Quasi-Isotropic Solution are treated expressing the bulk-viscosity coefficient $\zeta$ as a power-law of the fluid energy density $\rho$, i.e., \zeta=\zo\rho^{s}. In the Newtonian regime, the perturbation evolution is founded to be damped by viscosity and the top-down mechanism of structure fragmentation is suppressed. The value of the Jeans Mass remains unchanged also in presence of viscosity. In the relativistic approach, we get a power-law solution existing only in correspondence to a restricted domain of \zo.Comment: 3 pages, Proceedings of The XII Marcel Grossmann Meetin

Continuous dependence estimate for a degenerate parabolic-hyperbolic equation with Levy noise

In this article, we are concerned with a multidimensional degenerate parabolic-hyperbolic equation driven by Levy processes. Using bounded variation (BV) estimates for vanishing viscosity approximations, we derive an explicit continuous dependence estimate on the nonlinearities of the entropy solutions under the assumption that Levy noise depends only on the solution. This result is used to show the error estimate for the stochastic vanishing viscosity method. In addition, we establish fractional BV estimate for vanishing viscosity approximations in case the noise coefficients depend on both the solution and spatial variable.Comment: 31 Pages. arXiv admin note: text overlap with arXiv:1502.0249

Correlation of turbulent trailing vortex decay data

A correlation function, derived on the basis of self similar variable eddy viscosity decay, is introduced and utilized to correlate aircraft trailing vortex velocity data from ground and flight experiments. The correlation function collapses maximum tangential velocity data from scale model and flight tests to a single curve. The resulting curve clearly shows both the inviscid plateau and the downstream decay regions. A comparison between experimental data and numerical solution shows closer agreement with the variable eddy viscosity solution than the constant viscosity analytical solution

The vanishing viscosity limit for Hamilton-Jacobi equations on Networks

For a Hamilton-Jacobi equation defined on a network, we introduce its vanishing viscosity approximation. The elliptic equation is given on the edges and coupled with Kirchhoff-type conditions at the transition vertices. We prove that there exists exactly one solution of this elliptic approximation and mainly that, as the viscosity vanishes, it converges to the unique solution of the original problem