Related papers: Mass-conserving solutions to the Smoluchowski coag…
In this paper, we would like to consider the Cauchy problem for a multi-component weakly coupled system of semi-linear $\sigma$-evolution equations with double dissipation for any $\sigma\ge 1$. The first main purpose is to obtain the…
We study hydrodynamic limits of the cluster coagulation model; a coagulation model introduced by Norris [$\textit{Comm. Math. Phys.}$, 209(2):407-435 (2000)]. In this process, pairs of particles $x,y$ in a measure space $E$, merge to form a…
Black holes are inevitable mathematical outcome of spacetime-energy coupling in general relativity. Currently these objects are of vital importance for understanding numerous phenomena in astrophysics and cosmology. However, neither theory…
We report surprising steady oscillations in aggregation-fragmentation processes. Oscillating solutions are observed for the class of aggregation kernels $K_{i,j} = i^{\nu}j^{\mu} + j^{\nu}i^{\mu}$ homogeneous in masses $i$ and $j$ of…
We study the global in time existence and long time asymptotics of solutions to the parabolic-elliptic Patlak-Keller-Segel system for the multi-species populations in the whole Euclidean space $\mathbb{R}^2.$ We prove that at the borderline…
A mathematical model for collision-induced breakage is considered. Existence of weak solutions to the continuous nonlinear collision-induced breakage equation is shown for a large class of unbounded collision kernels and daughter…
We introduce and study a discrete random model for Smoluchowski's equation with limited aggregations. The latter is a model of coagulation introduced by Bertoin which may exhibit gelation. In our model, a large number of particles are…
The initial purpose of this work is to provide a probabilistic explanation of a recent result on a version of Smoluchowski's coagulation equations in which the number of aggregations is limited. The latter models the deterministic evolution…
We consider the following fully parabolic Keller--Segel system with logistic source $$ \left\{\begin{array}{ll} u_t=\Delta u-\chi\nabla\cdot(u\nabla v)+ au-\mu u^2,\quad x\in \Omega, t>0, \disp{v_t=\Delta v- v +u},\quad x\in \Omega, t>0,…
For the discrete Cucker-Smale's flocking model with a singular communication weight $\psi(s) = s^{-\alpha}$, with $0<\alpha<1/2$ , we prove that the velocity component of certain type of weak solutions is absolutly continuous. This result…
We prove the convergence of solutions of nonlocal conservation laws to their local entropic counterpart for a fundamentally extended class of nonlocal kernels when these kernels approach a Dirac distribution. The nonlocal kernels are…
We establish a weak-strong uniqueness principle for the two-phase Mullins-Sekerka equation in the plane: As long as a classical solution to the evolution problem exists, any weak De Giorgi type varifold solution (see for this notion the…
We present the convergence analysis of the cell average technique, introduced in [12], to solve the nonlinear continuous Smoluchowski coagulation equation. It is shown that the technique is second order accurate on uniform grids and first…
In this paper we study a class of coagulation equations including a source term that injects in the system clusters of size of order one. The coagulation kernel is homogeneous, of homogeneity $\gamma < 1$, such that $K(x,y)$ is…
Stochastic wave equations appear in several models for evolutionary processes subject to random forces, such as the motion of a strand of DNA in a liquid or heat flow around a ring. Semilinear stochastic wave equations can typically not be…
Strong convergence rates for (temporal, spatial, and noise) numerical approximations of semilinear stochastic evolution equations (SEEs) with smooth and regular nonlinearities are well understood in the scientific literature. Weak…
The possible occurrence of instantaneous gelation to Oort-Hulst-Safronov (OHS) coagulation equation is investigated for a certain class of unbounded coagulation kernels. The existence of instantaneous gelation is confirmed by showing the…
The compressible Euler-Riesz equations are fundamental with wide applications in astrophysics, plasma physics, and mathematical biology. In this paper, we are concerned with the global existence and nonlinear stability of finite-energy…
We extend known existence and uniqueness results of weak measure solutions for systems of non-local continuity equations beyond the usual Lipschitz regularity. Existence of weak measure solutions holds for uniformly continuous vector fields…
A stochastic version of the porous medium equation with coloured noise is studied. The corresponding Kolmogorov equation is solved in the space $L^2(H,\nu)$ where $\nu$ is an infinitesimally excessive measure. Then a weak solution is…