Global strict solutions to continuous coagulation–fragmentation equations with strong fragmentation

Banasiak, Jacek; Lamb, Wilson

doi:10.1017/s0308210509001255

Cited by 24 publications

(21 citation statements)

References 19 publications

(28 reference statements)

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“…(b) F is a bounded subset of L 1 (Ω) satisfying the following two properties: 5) and, for every ε > 0, there is Ω ε ∈ B such that µ(Ω ε ) < ∞ and…”

Section: 1mentioning

confidence: 99%

Weak Compactness Techniques and Coagulation Equations

Laurençot

2014

Lecture Notes in Mathematics

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Abstract. Smoluchowski's coagulation equation is a mean-field model describing the growth of clusters by successive mergers. Since its derivation in 1916 it has been studied by several authors, using deterministic and stochastic approaches, with a blossoming of results in the last twenty years. In particular, the use of weak L 1 -compactness techniques led to a mature theory of weak solutions and the purpose of these notes is to describe the results obtained so far in that direction, as well as the mathematical tools used.

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“…(b) F is a bounded subset of L 1 (Ω) satisfying the following two properties: 5) and, for every ε > 0, there is Ω ε ∈ B such that µ(Ω ε ) < ∞ and…”

Section: 1mentioning

confidence: 99%

Weak Compactness Techniques and Coagulation Equations

Laurençot

2014

Lecture Notes in Mathematics

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“…In general, due to the effect of gelation, the existence analysis of coagulationfragmentation models follows two main streams. In the first stream, several works have been devoted to the construction of mass-conserving solutions to models whose kernels satisfy 0 U Smo (ω 1 , ω 2 ) 2 + ω 1 + ω 2 with various assumptions on a and b (see [6,7,21,45,8,84] and the references therein). In the second stream, weak solutions which need not satisfy the mass conservation have been constructed (cf.…”

mentioning

confidence: 99%

On the Energy Cascade of 3-Wave Kinetic Equations: Beyond Kolmogorov–Zakharov Solutions

Soffer

Tran

2019

Commun. Math. Phys.

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In weak turbulence theory, the Kolmogorov-Zakharov spectra is a class of time-independent solutions to the kinetic wave equations. In this paper, we construct a new class of time-dependent isotropic solutions to the decaying turbulence problems (whose solutions are energy conserved), with general initial conditions. These solutions exhibit the interesting property that the energy is cascaded from small wavenumbers to large wavenumbers. We can prove that starting with a regular initial condition whose energy at the infinity wave number |p| = ∞ is 0, as time evolves, the energy is gradually accumulated at {|p| = ∞}. Finally, all the energy of the system is concentrated at {|p| = ∞} and the energy function becomes a Dirac function at infinity Eδ {|p|=∞} , where E is the total energy. The existence of this class of solutions is, in some sense, the first complete rigorous mathematical proof based on the kinetic description for the energy cascade phenomenon for waves with quadratic nonlinearities. We only represent in this paper the analysis of the statistical description of acoustic waves (and equivalently capillary waves). However, our analysis works for other cases as well.

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“…Coagulation-fragmentation equations are mean-field models describing the growth of clusters changing their sizes under the combined effects of (binary) merging and breakup. Denoting the size distribution function of the particles with mass x > 0 at time t > 0 by f = f (t, x) ≥ 0, the coagulation equation with multiple fragmentation reads for (t, x) ∈ (0, ∞) 2 . In (1.1), K denotes the coagulation kernel which is a non-negative and symmetric function K(x, y) = K(y, x) ≥ 0 of (x, y) accounting for the likelihood of two particles with respective masses x and y to merge.…”

Section: Introductionmentioning

confidence: 99%

Absence of Gelation and Self-Similar Behavior for a Coagulation-Fragmentation Equation

Laurençot¹,

Roessel²

2015

SIAM J. Math. Anal.

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Abstract. The dynamics of a coagulation-fragmentation equation with multiplicative coagulation kernel and critical singular fragmentation is studied. In contrast to the coagulation equation, it is proved that fragmentation prevents the occurrence of the gelation phenomenon and a mass-conserving solution is constructed. The large time behavior of this solution is shown to be described by a selfsimilar solution. In addition, the second moment is finite for positive times whatever its initial value. The proof relies on the Laplace transform which maps the original equation to a first-order nonlinear hyperbolic equation with a singular source term. A precise study of this equation is then performed with the method of characteristics.

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Global strict solutions to continuous coagulation–fragmentation equations with strong fragmentation

Cited by 24 publications

References 19 publications

Weak Compactness Techniques and Coagulation Equations

Weak Compactness Techniques and Coagulation Equations

On the Energy Cascade of 3-Wave Kinetic Equations: Beyond Kolmogorov–Zakharov Solutions

Absence of Gelation and Self-Similar Behavior for a Coagulation-Fragmentation Equation

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