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Register a new userFine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates
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We are concerned with the long-time behavior of the growth-fragmentation equation. We prove fine estimates on the principal eigenfunctions of the growth-fragmentation operator, giving their first-order behavior close to 0 and $+infty$. Using these estimates we prove a spectral gap result by following the technique in [Caceres, Canizo, Mischler 2011, JMPA], which implies that solutions decay to the equilibrium exponentially fast. The growth and fragmentation coefficients we consider are quite general, essentially only assumed to behave asymptotically like power laws.
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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 consider a class of Fokker--Planck equations with linear diffusion and superlinear drift enjoying a formal Wasserstein-like gradient flow structure with convex mobility function. In the drift-dominant regime, the equations have a finite critical mass above which the measure minimising the associated entropy functional displays a singular component. Our approach, which addresses the one-dimensional case, is based on a reformulation of the problem in terms of the pseudo-inverse distribution function. Motivated by the structure of the equation in the new variables, we establish a general framework for global-in-time existence, uniqueness and regularity of monotonic viscosity solutions to a class of nonlinear degenerate (resp. singular) parabolic equations, using as a key tool comparison principles and maximum arguments. We then focus on a specific equation and study in more detail the regularity and dynamics of solutions. In particular, blow-up behaviour, formation of condensates (i.e. Dirac measures at zero) and long-time asymptotics are investigated. As a consequence, in the mass-supercritical case, solutions will blow up in $L^infty$ in finite time and---understood in a generalised, measure sense---they will eventually have condensate. We further show that the singular part of the measure solution does in general interact with the density and that condensates can be transient. The equations considered are motivated by a model for bosons introduced by Kaniadakis and Quarati (1994), which has a similar entropy structure and a critical mass if $dge3$.
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.
Consider a bounded solution of the focusing, energy-critical wave equation that does not scatter to a linear solution. We prove that this solution converges in some weak sense, along a sequence of times and up to scaling and space translation, to a sum of solitary waves. This result is a consequence of a new general compactness/rigidity argument based on profile decomposition. We also give an application of this method to the energy-critical Schrodinger equation.
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.
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