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Motivated by the statistical description of turbulence, we study statistical conservation laws in the form of kinetic-type PDEs for joint probability density functions (PDFs) and cumulative distribution functions (CDFs) associated with…
In this paper we address the regularity issues of drift-diffusion equation with nonlocal diffusion, where the diffusion operator is in the realm of stable-type L\'evy operator and the velocity field is defined from the considered quantity…
We obtain new $L^1$ contraction results for bounded entropy solutions of Cauchy problems for degenerate parabolic equations. The equations we consider have possibly strongly degenerate local or non-local diffusion terms. As opposed to…
A framework of finite-velocity model based Boltzmann equation has been developed for convection-diffusion equations. These velocities are kept flexible and adjusted to control numerical diffusion. A flux difference splitting based kinetic…
The aim of the paper is to construct and justify asymptotic approximations for solutions to quasilinear convection-diffusion problems with a predominance of nonlinear convective flow in a thin cylinder, where an inhomogeneous nonlinear…
A recent large deflection cantilever model is considered. The principal nonlinear effects come through the beam's inextensibility---local arc length preservation---rather than traditional extensible effects attributed to fully restricted…
This study handles spatial three-dimensional solution of the nonlinear diffusion equation without particular initial conditions. The functional behavior of the equation and the concentration have been studied in new ways. An auxiliary…
We introduce non-trivial contributions to diffusion constant in generic many-body systems arising from quadratic fluctuations of ballistically propagating, i.e. convective, modes. Our result is obtained by expanding the current operator in…
We establish a general nonlocal approximation principle for the entropy solutions of scalar conservation laws on $\mathbb{R}$. More precisely, we show that the entropy solution to a nonnegative initial datum can be obtained as a weak-star…
Difference schemes for the time-fractional diffusion equation with variable coefficients and nonlocal boundary conditions containing real parameters $\alpha$ and $\beta$ are considered. By the method of energy inequalities, for the solution…
We consider boundary value problems with Riemann-Liouville fractional derivatives of order $s\in (1, 2)$ with non-constant diffusion and reaction coefficients. A variational formulation is derived and analyzed leading to the well-posedness…
In this paper, we investigate the solutions for a generalized fractional diffusion equation that extends some known diffusion equations by taking a spatial time-dependent diffusion coefficient and an external force into account, which…
We investigate the role of the form of the spatial diffusion coefficient in shock acceleration of fast particles. Referring to non-classical diffusion and using the results of numerical (hybrid) simulations tailored for the downstream shock…
We study the existence of global weak solutions of a nonlinear transport-diffusion equation with a fractional derivative in the time variable and under some extra hypotheses, we also study some regularity properties for this type of…
Central limit theorems play an important role in the study of statistical inference for stochastic processes. However, when the nonparametric local polynomial threshold estimator, especially local linear case, is employed to estimate the…
We obtain the comparison principle for discontinuous viscosity sub- and supersolutions of nonlocal Hamilton-Jacobi equations, with superlinear and coercive gradient terms. The nonlocal terms are integro-differential operators in L\'evy…
This is a study of a class of nonlocal nonlinear diffusion equations. We present a strong maximum principle for nonlocal time-dependent Dirichlet problems. Results are for bounded functions of space, rather than (semi)-continuous functions.…
The frequency-dependent attenuation typically obeys an empirical power law with an exponent ranging from 0 to 2. The standard time-domain partial differential equation models can describe merely two extreme cases of frequency independent…
This article presents new gradient estimates for positive solutions to the nonlinear fast diffusion equation on smooth metric measure spaces, involving the $f$-Laplacian. The gradient estimates of interest are mainly of…
By use of Lagrangian tracers propagated on 2D simulations of Scrape-Off Layer (SOL) turbulence, we are able to determine the non-local fractional-advection, fractional-diffusion equation (FADE) coefficients for a number of equilibrium…