16 problems

Let $K:(0,\infty)^2\to[0,\infty)$ be a coagulation kernel. It is $\gamma$-homogeneous when … for all $\lambda,x,y>0$, and gelation is the loss of mass from finite-mass particles th…

Assume the critical parameter conditions considered in the paper, in particular $\gamma+2\lambda=1$ and $\gamma\leq -1$, and consider the associated standard self-similar solutions…

Consider the critical parameter regime … and let $\Phi$ be the profile of a standard self-similar solution of the coagulation equation with growth. Singular self-similar solutions…

Let $f$ solve the coagulation equation under the complementary parameter conditions … Let equation (B6) denote the corresponding small-cluster-size equation with constant…

Let $\tau\geq 0$ denote time, let $xi>0$ denote cluster size, and let $\phi(\tau,\xi)\geq 0$ be the cluster-size density satisfying Smoluchowski's coagulation equation … Assume tha…

Phase-transition timing conjecture. The phase transition occurs strictly before the mean free time.

Let $K:(0,\b7infty)\times(0,\infty)\to[0,\infty)$ be a coagulation kernel and let $\kappa>0$. A kernel is called non-gelling if all solutions conserve mass, and gelling if mass con…

Fix a homogeneity exponent $\gamma>1$. For parameters $a,b$ satisfying … consider the self-similar profile equation … Here a solution is physically relevant when it corresponds to…

Class-II self-similar-solution conjecture. A similar one-parameter family of self-similar solutions with finite mass should exist for nonsolvable class-II kernels, including one so…

Consider solutions of the coagulation equation with integrable initial data, and let the transition region denote the spatial region connecting the parts of the asymptotic $N$-wave…

Let $K$ be a class-II kernel, meaning that $K(\xi,1)\to k_0>0$ as $\xi\to0$. Let $G_\rho$ denote a solution of the self-similar profile equation, with parameter $\rho>0$. Existence…

Let $H=L^{1}([x_{min},x_{max}],\mathbb{R}_{+})$, let $H^{N}$ be the piecewise-constant approximation space generated by the basis functions $\beta_i^N$, and let $f^{N}(t,x)$ solve…

Fix $v\in(0,1)$ and consider nonnegative self-similar profiles $F$ with … where … and the mass condition is … so that $m_1=m_0/v$. The profile described in Theorem P.1 is nonnegati…

Let $f(t,x)\ge 0$ be the number density of droplets of volume $x>0$ governed by the nonlocal transport equation … with conserved mass $\int_0^\infty x f(t,x)\,dx=m_1$. A self-simil…

Let $a$ be a coagulation kernel of the form … with $-1<\alpha\leq\beta<1$ and $\lambda=\alpha+\beta\in(-1,1)$. Let $g_1$ and $g_2$ be finite-mass self-similar profiles for Smolucho…

Dynamical scaling hypothesis. In general conditions, solutions eventually become close to a self-similar solution of Smoluchowski's coagulation equation in the long-time limit. Thi…