16 problems

Consider the discrete coagulation-fragmentation equation with multiplicative coagulation and constant fragmentation kernels, with initial data $\rho_0$ satisfying the paper's initi…

Let $\Omega$ be the infinite-dimensional simplex of decreasing nonnegative sequences with sum $1$, and let $M$ be the transition kernel of the continuous coagulation-fragmentation…

Let solutions of the Lifshitz–Slyozov–Wagner (LSW) model evolve from admissible initial data. LSW self-similar attractor conjecture. The large-time behavior of solutions to the LSW…

Ratio-limit conjecture. The existence of the limit

Let $\sigma\in {\mathcal M}_1((0,1/2])$ and let $\beta_m,\beta_s\in(0,1]$. Let $K_{\sigma,\beta_m,\beta_s}$ denote the Markov kernel governing the coagulation-fragmentation process…

Let ${\mathcal A}$ be the class of probability measures appearing in Theorem, let $\mu$ be an invariant probability measure, and let the theorem's part b) assert uniqueness under t…

Consider the discrete coagulation-fragmentation equation with multiplicative coagulation and constant fragmentation kernels, and let $\rho_0$ be initial data satisfying the paper's…

Consider the coagulation-fragmentation system with density $\rho(s,t)$, coagulation operator $Q_C$, fragmentation operator $Q_F$, and kernels … for $s,\hat s>0$. Let … be the $k$-t…

Nonuniqueness conjecture. In this case, the uniqueness results for solutions in the infinite population regime will no longer hold.

Coagulation–transport conjecture. Coagulation overwhelms transport, so that the mass distribution densifies around large masses as $t\to\infty$.

Let $\operatorname{R}(W_{\phi})$ and $\operatorname{N}(W_{\phi})$ denote the range and null space of the operator $W_{\phi}$ on $L^1([0,L])$, and let…

Reversible nucleation conjecture. The reversible nucleation case can still be managed by a similar strategy, with more involved compactness estimates, and will yield a different bo…

Let $K$ be the coagulation kernel and $a$ the overall breakup rate, given by … … where $\alpha\leq\beta\leq 1$, $\lambda:=\alpha+\beta>1$, $\gamma>0$, and $\kappa,k>0$. Gelation th…

Let the fluctuation field be the empirical particle-density discrepancy from the profile predicted by the Smoluchowski coagulation-fragmentation-diffusion PDE, normalized by the sq…

Consider the self-similar equation … with normalization … where $J[\Phi]$ is the encounter convolution operator defined in the paper and $\lambda(\tau)$ is the corresponding ratio…

The processes considered are coagulation-fragmentation processes on the space of set partitions $\Omega_{[N]}$ of $[N]=\{1,2,\ldots,N\}$, with clusters represented by subsets of th…