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We develop a domain-decomposition model reduction method for linear steady-state convection-diffusion equations with random coefficients. Of particular interest to this effort are the diffusion equations with random diffusivities, and the…
This paper studies a longitudinal shape transformation model in which shapes are deformed in response to an internal growth potential that evolves according to an advection reaction diffusion process. This model extends prior works that…
We study solution techniques for an evolution equation involving second order derivative in time and the spectral fractional powers, of order $s \in (0,1)$, of symmetric, coercive, linear, elliptic, second-order operators in bounded domains…
The solution for non-linear, complex partial differential Equations (PDEs) is achieved through numerical approximations, which yield a linear system of equations. This approach is prevalent in Computational Fluid Dynamics (CFD), but it…
This paper presents a novel approach to rigorously solving initial value problems for semilinear parabolic partial differential equations (PDEs) using fully spectral Fourier-Chebyshev expansions. By reformulating the PDE as a system of…
Long simulation times in climate sciences typically require coarse grids due to computational constraints. Nonetheless, unresolved subscale information significantly influences the prognostic variables and can not be neglected for reliable…
In this paper, the initial value problem of the convection-diffusion equation of Burgers type is treated. In the asymptotic profile of solutions, the nonlinearity of the equation is reflected. Regarding the solutions to this model, the…
We analyze the propagation of Lipschitz continuity of solutions to various linear and nonlinear drift-diffusion systems, with and without incompressibility constraints. Diffusion is assumed to be either fractional or classical. Such…
Different relaxation approximations to partial differential equations, including conservation laws, Hamilton-Jacobi equations, convection-diffusion problems, gas dynamics problems, have been recently proposed. The present paper focuses onto…
Motivated by the modeling of three-dimensional fluid turbulence, we define and study a class of stochastic partial differential equations (SPDEs) that are randomly stirred by a spatially smooth and uncorrelated in time forcing term. To…
Fractional nonlinear differential equations present an interplay between two common and important effective descriptions used to simplify high dimensional or more complicated theories: nonlinearity and fractional derivatives. These…
We consider a class of macroscopic models for the spatio-temporal evolution of urban crime, as originally going back to Short et al. (Math. Mod. Meth. Appl. Sci. 18, 2008). The focus here is on the question how far a certain nonlinear…
It is well-known under the name of `periodic homogenization' that, under a centering condition of the drift, a periodic diffusion process on R^d converges, under diffusive rescaling, to a d-dimensional Brownian motion. Existing proofs of…
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 give sharp conditions for the large time asymptotic simplification of aggregation-diffusion equations with linear diffusion. As soon as the interaction potential is bounded and its first and second derivatives decay fast enough at…
We study nonlinear elliptic equations for operators corresponding to non-stable L\'evy diffusions. We include a sum of fractional Laplacians of different orders. Such operators are infinitesimal generators of non-stable (i.e., non…
We present regularity results for nonlinear drift-diffusion equations of porous medium type (together with their incompressible limit). We relax the assumptions imposed on the drift term with respect to previous results and additionally…
Describing particle transport at the macroscopic or mesoscopic level in non-ideal environments poses fundamental theoretical challenges in domains ranging from inter and intra-cellular transport in biology to diffusion in porous media. Yet,…
We prove that weakly continuous solutions to martingale problems admit a canonical regular conditional probability distribution. This allows for the construction of time consistent convex dynamic procedures in a non dominated setting.…
Following Assiotis (2020), we study general $\beta$-Hua-Pickrell diffusions of $N$ particles on $\mathbb R$ as solutions of the stochastic differential equations (SDEs) $$dX_{j,t}=\sqrt{2(1+X_{j,t}^2)}\,dB_{j,t}+\beta\left[b-a…