Self-similar asymptotic behavior for the solutions of a linear coagulation equation

Niethammer, Barbara; Nota, Alessia; Throm, Sebastian; Velázquez, Juan J. L.

doi:10.1016/j.jde.2018.07.059

Cited by 17 publications

(10 citation statements)

References 20 publications

(34 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…These solutions might have a finite monomer density (that is, ∞ 0 x f (x, t) dx < ∞) as in [18,21], or infinite monomer density (that is, ∞ 0 x f (x, t) dx = ∞) as in [3,4,34,35]. Similar strategies can be applied to other kinetic equations [25,26,33].…”

Section: Introductionmentioning

confidence: 99%

Stationary Non-equilibrium Solutions for Coagulation Systems

Ferreira

Lukkarinen

Nota

et al. 2021

Arch Rational Mech Anal

Self Cite

View full text Add to dashboard Cite

We study coagulation equations under non-equilibrium conditions which are induced by the addition of a source term for small cluster sizes. We consider both discrete and continuous coagulation equations, and allow for a large class of coagulation rate kernels, with the main restriction being boundedness from above and below by certain weight functions. The weight functions depend on two power law parameters, and the assumptions cover, in particular, the commonly used free molecular and diffusion limited aggregation coagulation kernels. Our main result shows that the two weight function parameters already determine whether there exists a stationary solution under the presence of a source term. In particular, we find that the diffusive kernel allows for the existence of stationary solutions while there cannot be any such solutions for the free molecular kernel. The argument to prove the non-existence of solutions relies on a novel power law lower bound, valid in the appropriate parameter regime, for the decay of stationary solutions with a constant flux. We obtain optimal lower and upper estimates of the solutions for large cluster sizes, and prove that the solutions of the discrete model behave asymptotically as solutions of the continuous model.

show abstract

Section: Introductionmentioning

confidence: 99%

Stationary Non-equilibrium Solutions for Coagulation Systems

Ferreira

Lukkarinen

Nota

et al. 2021

Arch Rational Mech Anal

Self Cite

View full text Add to dashboard Cite

show abstract

“…We observe that, to prove the existence of stationary solutions, finding fixed points for the corresponding evolution semigroup has often been used in the study of coagulation equations. We refer for instance to [5,11,18,19,20]. Similar ideas have been used also to construct stationary solutions of more general classes of kinetic equations.…”

Section: Continuous Coagulation Equationmentioning

confidence: 99%

Multicomponent coagulation systems: existence and non-existence of stationary non-equilibrium solutions

Ferreira,

Lukkarinen,

Nota

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

We study multicomponent coagulation via the Smoluchowski coagulation equation under non-equilibrium stationary conditions induced by a source of small clusters. The coagulation kernel can be very general, merely satisfying certain power law asymptotic bounds in terms of the total number of monomers in a cluster. The bounds are characterized by two parameters and we extend previous results for one-component systems to classify the parameter values for which the above stationary solutions do or do not exist. Moreover, we also obtain criteria for the existence or non-existence of solutions which yield a constant flux of mass towards large clusters.

show abstract

“…where the constant B2 = 3kBT/2μ, and μ is the gas viscosity. For the nonlinear partial integrodifferential structure of SCE, only a limited number of known analytical solution exist for simple coagulation kernel [2][3][4][5], If the collision frequency function is a homogeneous function of its arguments, the SCE can be converted into an ordinary integrodifferential equation by a similarity transformation [6]. The previous studies indicate that the particle size distribution (PSD) of an aged coagulating system approaches a universal asymptotic form called the self-preserving PSD [7][8][9][10][11].…”

Section: Introductionmentioning

confidence: 99%

Particle Size Distribution of Smoluchowski Coagulation Equation for Brownian Motion at Equilibrium

Xie¹

2021

Preprint

View full text Add to dashboard Cite

The information entropy for Smoluchowski coagulation equation is proposed based on statistical mechanics. And the normalized particle size distribution is a lognormal function at equilibrium from the principle of maximum entropy and moment constraint. The geometric mean volume and standard deviation in the distribution function are determined as simple constant. The results reveal that the assumption that algebraic mean volume be unit in self-preserving hypothesis is reasonable in some sense. Based on the present definition of information entropy, the Cercignani’s conjecture holds naturally for Smoluchowski coagulation equation. Together with the proof that the conjecture is also true for Boltzmann equation, Cercignani’s conjecture will holds for any two-body collision systems, which will benefit the understanding of Brownian motion and molecule kinematic theory, such as the stability of the dissipative system, and the mathematical theory of convergence to thermodynamic equilibrium.

show abstract

Self-similar asymptotic behavior for the solutions of a linear coagulation equation

Cited by 17 publications

References 20 publications

Stationary Non-equilibrium Solutions for Coagulation Systems

Stationary Non-equilibrium Solutions for Coagulation Systems

Multicomponent coagulation systems: existence and non-existence of stationary non-equilibrium solutions

Particle Size Distribution of Smoluchowski Coagulation Equation for Brownian Motion at Equilibrium

Contact Info

Product

Resources

About