Related papers: On the supercritical fractional diffusion equation…
Convection-diffusion of heat transfer is one of the important phenomena in fluid flow and industrial problems. The involved parameters, boundary conditions, and material properties are greatly affecting the same. As such, the uncertainness…
We derive a fundamental solution $\mathscr{E}$ to a space-fractional diffusion problem on the half-line. The equation involves the Caputo derivative. We establish properties of $\mathscr{E}$ as well as formulas for solutions to the…
Superluminal propagation is an intrinsic problem in the diffusion equation and has not been effectively addressed for a long time. In this work, a rigorous solution to this issue is obtained under the assumption that particles undergo a…
We consider weak solutions of the fractional heat equation posed in the whole $n$-dimensional space, and establish their asymptotic convergence to the fundamental solution as $t\to\infty$ under the assumption that the initial datum is an…
In this note, we propose a discrete model to study one-dimensional transport equations with non-local drift and supercritical dissipation. The inspiration for our model is the equation $$ \theta_t + (H\theta) \theta_x +(-\Delta)^\alpha…
We consider diffusion operator $-\Delta + b \cdot \nabla$ in $\mathbb R^d$, $d \geq 3$, with drift $b$ in a large class of locally unbounded vector fields that can have critical-order singularities. Covering the entire range of admissible…
The diffusion system with time-fractional order derivative is of great importance mathematically due to the nonlocal property of the fractional order derivative, which can be applied to model the physical phenomena with memory effects. We…
This paper is devoted to the construction and analysis of the finite element approximations for the $H(D)$ convection-diffusion problems, where $D$ can be chosen as ${\rm grad}$, ${\rm curl}$ or ${\rm div}$ in 3D case. An essential feature…
By using Duhamel's formula, we prove sharp two-sided estimates for the heat kernel of spectral fractional Laplacian with time-dependent gradient perturbation in bounded $C^{1,1}$ domains. Moreover, we also obtain gradient estimate as well…
We obtain necessary conditions and sufficient conditions on the existence of solutions to the Cauchy problem for a fractional semilinear heat equation with an inhomogeneous term. We identify the strongest spatial singularity of the…
In this note, we connect two seemingly unrelated objects: On the one hand is a two-dimensional drift-diffusion process $X$ with divergence-free and time-independent drift $b$. The drift is given by a stationary Gaussian ensemble, and we…
Existence and uniqueness of a specific self-similar solution is established for the following reaction-diffusion equation with Hardy singular potential $$ \partial_tu=\Delta u^m+|x|^{-2}u^p, \qquad (x,t)\in \real^N\times(0,\infty), $$ in…
We establish a reduction principle to derive Li-Yau inequalities for non-local diffusion problems in a very general framework, which covers both the discrete and continuous setting. Our approach is not based on curvature-dimension…
We analyze the phase diagram of N=4 supersymmetric Yang-Mills theory with fundamental matter in the presence of a background magnetic field and nonzero baryon number. We identify an isolated quantum critical point separating two differently…
On a smooth bounded domain \Omega \subset R^N we consider the Schr\"odinger operators -\Delta -V, with V being either the critical borderline potential V(x)=(N-2)^2/4 |x|^{-2} or V(x)=(1/4) dist (x,\partial\Omega)^{-2}, under Dirichlet…
How far the Hadley circulation's ascending branch extends into the summer hemisphere is a fundamental but incompletely understood characteristic of Earth's climate. Here, we present a predictive, analytical theory for this ascending edge…
We study the nearly critical behaviour of holographic superfluids at finite temperature and chemical potential in their probe limit. This allows us to examine the coupled dynamics of the full complex order parameter with the charge density…
This paper is devoted to studying propagation phenomena in integro-differential equations with a weakly degenerate non-linearity. The reaction term can be seen as an intermediate between the classical logistic (or Fisher-KPP) non-linearity…
Diffusion may obliterate fluctuation signals of the QCD phase transition in nuclear collisions at SPS and RHIC energies. We propose a hyperbolic diffusion equation to study the dissipation of net charge fluctuations. This equation is needed…
We consider solutions of the linear heat equation in $\mathbb{R}^N$ with isolated singularities. It is assumed that the position of a singular point depends on time and is H\"older continuous with the exponent $\alpha \in (0,1)$. We show…