scholarly journals Mass-conserving solutions to the Smoluchowski coagulation equation with singular kernel

2019 ◽  
Vol 150 (4) ◽  
pp. 1805-1825 ◽  
Author(s):  
Prasanta Kumar Barik ◽  
Ankik Kumar Giri ◽  
Philippe Laurençot
Keyword(s):  
Weak Solutions ◽  
Linear Growth ◽  
Weak Compactness ◽  
Singular Kernel ◽  
Initial Condition ◽  
Existence Proof ◽  
Global Weak Solutions ◽  
Smoluchowski Coagulation Equation ◽  
Coagulation Equation ◽  
Coagulation Kernel

AbstractGlobal weak solutions to the continuous Smoluchowski coagulation equation (SCE) are constructed for coagulation kernels featuring an algebraic singularity for small volumes and growing linearly for large volumes, thereby extending previous results obtained in Norris (1999) and Cueto Camejo & Warnecke (2015). In particular, linear growth at infinity of the coagulation kernel is included and the initial condition may have an infinite second moment. Furthermore, all weak solutions (in a suitable sense) including the ones constructed herein are shown to be mass-conserving, a property which was proved in Norris (1999) under stronger assumptions. The existence proof relies on a weak compactness method in L1 and a by-product of the analysis is that both conservative and non-conservative approximations to the SCE lead to weak solutions which are then mass-conserving.

scholarly journals Localization in Stationary Non-equilibrium Solutions for Multicomponent Coagulation Systems

2021 ◽  
Author(s):  
Marina A. Ferreira ◽  
Jani Lukkarinen ◽  
Alessia Nota ◽  
Juan J. L. Velázquez
Keyword(s):  
Source Term ◽  
Equilibrium Solutions ◽  
Equilibrium Conditions ◽  
Measure Concentration ◽  
Localization Property ◽  
Smoluchowski Coagulation Equation ◽  
Coagulation Equation ◽  
Asymptotic Scaling ◽  
Coagulation Kernel ◽  
Non Equilibrium

AbstractWe consider the multicomponent Smoluchowski coagulation equation under non-equilibrium conditions induced either by a source term or via a constant flux constraint. We prove that the corresponding stationary non-equilibrium solutions have a universal localization property. More precisely, we show that these solutions asymptotically localize into a direction determined by the source or by a flux constraint: the ratio between monomers of a given type to the total number of monomers in the cluster becomes ever closer to a predetermined ratio as the cluster size is increased. The assumptions on the coagulation kernel are quite general, with isotropic power law bounds. The proof relies on a particular measure concentration estimate and on the control of asymptotic scaling of the solutions which is allowed by previously derived estimates on the mass current observable of the system.

scholarly journals Global classical and weak solutions of the prion proliferation model in the presence of chaperone in a banach space

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Kapil Kumar Choudhary ◽  
Rajiv Kumar ◽  
Rajesh Kumar
Keyword(s):  
Differential Equation ◽  
System Theory ◽  
Banach Space ◽  
Weak Solutions ◽  
Existence And Uniqueness ◽  
Weak Compactness ◽  
Evolution System ◽  
Global Classical Solution ◽  
Global Weak Solutions ◽  
Degradation Rates

<p style='text-indent:20px;'>The present work is based on the coupling of prion proliferation system together with chaperone which consists of two ODEs and a partial integro-differential equation. The existence and uniqueness of a positive global classical solution of the system is proved for the bounded degradation rates by the idea of evolution system theory in the state space <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R} \times \mathbb{R} \times L_{1}(Z,zdz). $\end{document}</tex-math></inline-formula> Moreover, the global weak solutions for unbounded degradation rates are discussed by weak compactness technique.</p>

L 1 and L ∞ stability of transition densities of perturbed diffusions

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Ilya Bitter ◽  
Valentin Konakov
Keyword(s):  
Linear Growth ◽  
Stability Result ◽  
Regularity Conditions ◽  
Initial Condition ◽  
Weak Regularity ◽  
Transition Densities ◽  
The Stability ◽  
Drift Terms

Abstract In this paper, we derive a stability result for L 1 {L_{1}} and L ∞ {L_{\infty}} perturbations of diffusions under weak regularity conditions on the coefficients. In particular, the drift terms we consider can be unbounded with at most linear growth, and the estimates reflect the transport of the initial condition by the unbounded drift through the corresponding flow. Our approach is based on the study of the distance in L 1 {L_{1}} - L 1 {L_{1}} metric between the transition densities of a given diffusion and the perturbed one using the McKean–Singer parametrix expansion. In the second part, we generalize the well-known result on the stability of diffusions with bounded coefficients to the case of at most linearly growing drift.

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