TY - JOUR

T1 - Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations

AU - Cáceres, Mar'ia J.

AU - Cañizo, José A.

AU - Mischler, Stéphane

PY - 2011/10/1

Y1 - 2011/10/1

N2 - We study the asymptotic behavior of linear evolution equations of the type tâˆ‚g=Dg+Lgâˆ’Î»g, where L is the fragmentation operator, D is a differential operator, and Î» is the largest eigenvalue of the operator Dg+Lg. In the case Dg=âˆ’xâˆ‚g, this equation is a rescaling of the growth-fragmentation equation, a model for cellular growth; in the case Dg=âˆ’xâˆ‚(xg), it is known that Î»=1 and the equation is the self-similar fragmentation equation, closely related to the self-similar behavior of solutions of the fragmentation equation tâˆ‚f=Lf. By means of entropyâ€“entropy dissipation inequalities, we give general conditions for g to converge exponentially fast to the steady state G of the linear evolution equation, suitably normalized. In other words, the linear operator has a spectral gap in the natural L2 space associated to the steady state. We extend this spectral gap to larger spaces using a recent technique based on a decomposition of the operator in a dissipative part and a regularizing part.

AB - We study the asymptotic behavior of linear evolution equations of the type tâˆ‚g=Dg+Lgâˆ’Î»g, where L is the fragmentation operator, D is a differential operator, and Î» is the largest eigenvalue of the operator Dg+Lg. In the case Dg=âˆ’xâˆ‚g, this equation is a rescaling of the growth-fragmentation equation, a model for cellular growth; in the case Dg=âˆ’xâˆ‚(xg), it is known that Î»=1 and the equation is the self-similar fragmentation equation, closely related to the self-similar behavior of solutions of the fragmentation equation tâˆ‚f=Lf. By means of entropyâ€“entropy dissipation inequalities, we give general conditions for g to converge exponentially fast to the steady state G of the linear evolution equation, suitably normalized. In other words, the linear operator has a spectral gap in the natural L2 space associated to the steady state. We extend this spectral gap to larger spaces using a recent technique based on a decomposition of the operator in a dissipative part and a regularizing part.

U2 - 10.1016/j.matpur.2011.01.003

DO - 10.1016/j.matpur.2011.01.003

M3 - Article

VL - 96

SP - 334

EP - 362

JO - Journal de Mathematiques Pures et Appliquees

JF - Journal de Mathematiques Pures et Appliquees

IS - 4

ER -