The singular coagulation equation with multiple fragmentation

Saha, Jitraj; Kumar, Jitendra

doi:10.1007/s00033-014-0452-3

Cited by 19 publications

(15 citation statements)

References 17 publications

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“…The proof is standard and is similar with the existence proof of Kierkels and Velázquez [41]. It is the classical regularized, kernel cutting off strategy, commonly used for the coagulation-fragmentation equation (see, for instance, [21,78]) and the classical homogeneous Boltzmann equation (see, for instance, [3]).…”

Section: Existence Of Weak Solutions and Energy Cascadementioning

confidence: 84%

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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Section: Existence Of Weak Solutions and Energy Cascadementioning

confidence: 84%

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

Soffer

Tran

2019

Commun. Math. Phys.

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“…Our methods do not yield the uniqueness of solutions, which is not clear under our mild assumptions on the initial datum as it need not have finite negative moments. For uniqueness results for coagulation-fragmentation equations with singular kernels see [4], [16].…”

Section: Given An Initial Valuementioning

confidence: 99%

Well-posedness for boundary value problems for coagulation-fragmentation equations

Erneta¹

2020

Kinetic &Amp; Related Models

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We investigate a coagulation-fragmentation equation with boundary data, establishing the well-posedness of the initial value problem when the coagulation kernels are bounded at zero and showing existence of solutions for the singular kernels relevant in the applications. We determine the large time asymptotic behavior of solutions, proving that solutions converge exponentially fast to zero in the absence of fragmentation and stabilize toward an equilibrium if the boundary value satisfies a detailed balance condition. Incidentally, we obtain an improvement in the regularity of solutions by showing the finiteness of negative moments for positive time.

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“…The formulation (7)(8) is based on the Leibnitz integration rule and the mass conservation property (4). It is pertinent to mention that literature reveals most theoretical research works have comprised only with linear breakage [25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42] and only few investigations have been presented to nonlinear breakage due to interactions between fragments, such as collision-induced fragmentation. 18,[20][21][22][23][24]36 These circumstances are motivated the present study.…”

Section: Introductionmentioning

confidence: 99%

“…It is pertinent to mention that literature reveals most theoretical research works have comprised only with linear breakage and only few investigations have been presented to nonlinear breakage due to interactions between fragments, such as collision‐induced fragmentation . These circumstances are motivated the present study.…”

Section: Introductionmentioning

confidence: 99%

An existence‐uniqueness result for the pure binary collisional breakage equation

Paul

Kumar

2018

Math Methods in App Sciences

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The study of collision‐induced breakage phenomenon in the particulate process has much current interest. This is an important process arising in many engineering disciplines. In this work, the existence of continuous solution of the pure collisional breakage model is developed beneath some restrictions on the breakage kernels. Furthermore, the mass conservation and uniqueness of solution are investigated in the absence of “shattering transition.” The underlying theory is based on the compactness result of Arzelà‐Ascoli and Banach contraction mapping principle.

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The singular coagulation equation with multiple fragmentation

Cited by 19 publications

References 17 publications

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

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

Well-posedness for boundary value problems for coagulation-fragmentation equations

An existence‐uniqueness result for the pure binary collisional breakage equation

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