Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAsymptotic behavior of the growth-fragmentation equation with bounded fragmentation rate
308
0
0.0
(
0
)
Authors
Etienne Bernard
Ask ChatGPT about the research
No Arabic abstract
We are interested in the large time behavior of the solutions to the growth-fragmentation equation. We work in the space of integrable functions weighted with the principal dual eigenfunction of the growth-fragmentation operator. This space is the largest one in which we can expect convergence to the steady size distribution. Although this convergence is known to occur under fairly general conditions on the coefficients of the equation, we prove that it does not happen uniformly with respect to the initial data when the fragmentation rate in bounded. First we get the result for fragmentation kernels which do not form arbitrarily small fragments by taking advantage of the Dyson-Phillips series. Then we extend it to general kernels by using the notion of quasi-compactness and the fact that it is a topological invariant.
rate research
Read More
The objective is to prove the asynchronous exponential growth of the growth-fragmentation equation in large weighted $L^1$ spaces and under general assumptions on the coefficients. The key argument is the creation of moments for the solutions to the Cauchy problem, resulting from the unboundedness of the total fragmentation rate. It allows us to prove the quasi-compactness of the associated (rescaled) semigroup, which in turn provides the exponential convergence toward the projector on the Perron eigenfunction.
We consider the fragmentation equation $dfrac{partial}{partial t}f (t, x) = --B(x)f (t, x) + int_{ y=x}^{ y=infty} k(y, x)B(y)f (t, y)dy,$ and address the question of estimating the fragmentation parameters-i.e. the division rate $B(x)$ and the fragmentation kernel $k(y, x)$-from measurements of the size distribution $f (t, $times$)$ at various times. This is a natural question for any application where the sizes of the particles are measured experimentally whereas the fragmentation rates are unknown, see for instance (Xue, Radford, Biophys. Journal, 2013) for amyloid fibril breakage. Under the assumption of a polynomial division rate $B(x) = alpha x^{gamma}$ and a self-similar fragmentation kernel $k(y, x) = frac{1}{y} k_0 (x/ y)$, we use the asymptotic behaviour proved in (Escobedo, Mischler, Rodriguez-Ricard, Ann. IHP, 2004) to obtain uniqueness of the triplet $(alpha, gamma, k _0)$ and a representation formula for $k_0$. To invert this formula, one of the delicate points is to prove that the Mellin transform of the asymptotic profile never vanishes, what we do through the use of the Cauchy integral.
The small and large size behavior of stationary solutions to the fragmentation equation with size diffusion is investigated. It is shown that these solutions behave like stretched exponentials for large sizes, the exponent in the exponential being solely given by the behavior of the overall fragmentation rate at infinity. In contrast, the small size behavior is partially governed by the daughter fragmentation distribution and is at most linear, with possibly non-algebraic behavior. Explicit solutions are also provided for particular fragmentation coefficients.
We give here an explicit formula for the following critical case of the growth-fragmentation equation $$frac{partial}{partial t} u(t, x) + frac{partial}{partial x} (gxu(t, x)) + bu(t, x) = balpha^2 u(t, alpha x), qquad u(0, x) = u_0 (x),$$ for some constants $g > 0$, $b > 0$ and $alpha > 1$ - the case $alpha = 2$ being the emblematic binary fission case. We discuss the links between this formula and the asymptotic ones previously obtained in (Doumic, Escobedo, Kin. Rel. Mod., 2016), and use them to clarify how periodicity may appear asymptotically.
We study the kinetics of nonlinear irreversible fragmentation. Here fragmentation is induced by interactions/collisions between pairs of particles, and modelled by general classes of interaction kernels, and for several types of breakage models. We construct initial value and scaling solutions of the fragmentation equations, and apply the non-vanishing mass flux criterion for the occurrence of shattering transitions. These properties enable us to determine the phase diagram for the occurrence of shattering states and of scaling states in the phase space of model parameters.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
