Related papers: Exact solutions of a nonlinear diffusion equation …
We analyze the static response to kinetic perturbations of nonequilibrium steady states that can be modeled as diffusions. We demonstrate that kinetic response is purely a nonequilibirum effect, measuring the degree to which the…
We analyze the problem for viscous incompressible heat-conduc\-ting fluid in a finite cylinder with large inflow and outflow, modelled with Navier-Stokes equations coupled with the heat equation. We prove energy estimate without…
This investigation deals with some exact solutions of the equations governing the steady plane motions of an incompressible third grade fluid by using complex variables and complex functions. Some of the solutions admit, as particular…
A statistical method for calculating equilibrium solutions of the shallow water equations, a model of essentially 2-d fluid flow with a free surface, is described. The model contains a competing acoustic turbulent {\it direct} energy…
We investigate a nonlinear parabolic reaction-diffusion equation describing the oxygen concentration in encapsulated pancreatic cells with a general core-shell geometry. This geometry introduces a discontinuous diffusion coefficient as the…
We prove stability results for nonlinear diffusion equations of the porous medium and fast diffusion types with respect to the nonlinearity power $m$: solutions with fixed data converge in a suitable sense to the solution of the limit…
We describe an exact and highly efficient numerical algorithm for solving a special but important class of convection-diffusion equations. These equations occur in many problems in physics, chemistry, or biology, and they are usually hard…
We derive one dimensional (1D) analytical solutions for the transport equations of incompressible magnetohydrodynamic (MHD) turbulence developed by Zank et al. [2012], Adhikari et al. [2023], including the Els\"asser energies and the…
Zank et al. developed models describing the transport of low frequency incompressible and nearly incompressible turbulence in inhomogeneous flows. The formalism was based on expressing the fluctuating variables in terms of the Els\"assar…
This paper is devoted to the study of some nonlinear parabolic equations with discontinuous diffusion intensities. Such problems appear naturally in physical and biological models. Our analysis is based on variational techniques and in…
The clearing up of a wave nature of the energy and mass transfer phenomena in classical expressions of the molecular-kinetic theory has allowed to find a quantitative measure of intensity of processes of a thermal conductivity, viscosity…
We introduce a notion of viscosity solutions for a nonlinear degenerate diffusion equation with a drift potential. We show that our notion of solutions coincide with the weak solutions defined via integration by parts. As an application of…
In this paper we analyze a nonlinear parabolic equation characterized by a singular diffusion term describing very fast diffusion effects. The equation is settled in a smooth bounded three-dimensional domain and complemented with a general…
We survey recent results in the mathematical literature on the equations of incompressible fluid dynamics, highlighting common themes and how they might contribute to the understanding of some phenomena in the theory of fully developed…
Exact solutions for nonlinear Arrhenius reaction-diffusion are constructed in $n$ dimensions. A single relationship between nonlinear diffusivity and the nonlinear reaction term leads to a nonclassical Lie symmetry whose invariant solutions…
We construct non-negative weak solutions of fast diffusion equations with a divergence type of drift term satisfying the $L^q$-energy inequality and speed estimate in Wasserstein spaces under some integrability conditions on the drift term.…
We show how the nonlinear interaction effects `volume filling' and `adhesion' can be incorporated into the fractional subdiffusive transport of cells and individual organisms. To this end, we use microscopic random walk models with…
Our goal is to develop a flux limiter of the Flux-Corrected Transport method for a nonconservative convection-diffusion equation. For this, we consider a hybrid difference scheme that is a linear combination of a monotone scheme and a…
We present a unique method for solving for the Reynolds stress in turbulent canonical flows, based on the momentum balance for a control volume moving at the local mean velocity. A differential transform converts this momentum balance to a…
A full symmetry classification is given for models of energy transport in radiant plasma when the mass density is spatially variable and the diffusivity is nonlinear. A systematic search for conservation laws also leads to some potential…