arXiv subjects

Ram Gopal Jaiswal

Publications and source records attributed to Ram Gopal Jaiswal.

5 recordsLinked to original sources

arXiv2026

The Diffusive Exchange Driven Growth Model with unbounded kernels

We study the discrete diffusive exchange-driven growth (EDG) equations on a bounded smooth domain of arbitrary dimension subject to homogeneous Neumann boundary conditions. The system belongs to the class of infinite systems of semilinear partial differential equations with nonlinear source terms of quadratic type. Global-in-time existence of solutions is established for separable exchange kernels of the form $K_{i,j}=b_i a_j$, where the donor rates exhibit at most linear growth while the receiver rates are sublinear. The analysis is based on a uniform Fisher information estimate obtained from an entropy-entropy dissipation identity. This estimate yields renormalized solutions to a truncated system with finitely many species. A compactness argument then enables passage to the limit in the exchange operator, leading to the existence of global-in-time solutions for the full system.

math.AP↗

arXiv2026

Existence for the Discrete Nonlinear Fragmentation Equation with Degenerate Diffusion

A mathematical model for the discrete nonlinear fragmentation (collision-induced breakage) equation with diffusion is studied. The existence of global weak solutions is established in arbitrary spatial dimensions without assuming a strictly positive lower bound on the diffusion coefficients, extending previous results that were restricted to one-dimensional domains and relied on uniformly positive diffusion. The analysis is carried out under boundedness assumptions on the collision and breakage kernels. The proof is based on the construction of a suitable regularized system, combined with weak $L^2$ a priori estimates and compactness arguments in $L^1$, which allow the passage to the limit in the nonlinear fragmentation operator.

math.AP↗

arXiv2025

Mass-conserving weak solutions to the continuous nonlinear fragmentation equation in the presence of mass transfer

A mathematical model for the continuous nonlinear fragmentation equation is considered in the presence of mass transfer. In this paper, we demonstrate the existence of mass-conserving weak solutions to the nonlinear fragmentation equation with mass transfer for collision kernels of the form $Φ(x,y) = κ(x^{σ_1} y^{σ_2} + y^{σ_1} x^{σ_2})$, $κ>0$, $0 \leq {σ_1} \leq {σ_2} \leq 1$, and ${σ_1} \neq 1$ for $(x, y) \in \mathbb{R}_+^2$, with integrable daughter distribution functions, thereby extending previous results obtained by Giri \& Lauren\c cot (2021). In particular, the existence of at least one global weak solution is shown when the collision kernel exhibits at least linear growth, and one local weak solution when the collision kernel exhibits sublinear growth. In both cases, finite superlinear moment bounds are obtained for positive times without requiring the finiteness of initial superlinear moments. Additionally, the uniqueness of solutions is confirmed in both cases.

math.AP↗

arXiv2024

Mass-Conserving Self-Similar Solutions to Collision-Induced Breakage Equations

Existence of mass-conserving self-similar solutions to collision-induced breakage equation is shown for a specific class of homogeneous collision kernels and breakage functions. The proof mainly relies on a dynamical approach and compactness method to constructing mass-conserving stationary solutions for an evolution problem, which induces mass-conserving self-similar solutions to collision-induced breakage equation. Furthermore, we also determine lower and upper bound of the scaling profile.

math.AP↗

arXiv2024

The continuous collision-induced nonlinear fragmentation equation with non-integrable fragment daughter distributions

Existence, non-existence, and uniqueness of mass-conserving weak solutions to the continuous collision-induced nonlinear fragmentation equations are established for the collision kernels $Φ$ satisfying $Φ(x_1,x_2)={x_1}^{λ_1} {x_2}^{λ_2}+{x_2}^{λ_1} {x_1}^{λ_2}$, $(x_1,x_2)\in(0,\infty)^2$, with ${λ_1} \leq {λ_2}\leq 1$, and non-integrable fragment daughter distributions. In particular, global existence of mass-conserving weak solutions is shown when $1\leλ:={λ_1}+{λ_2}\le2$ with $λ_1\ge k_0$, the parameter $k_0\in(0,1)$ being related to the non-integrability of the fragment daughter distribution. The existence of at least one mass-conserving weak solution is also demonstrated when $2k_0 \le λ< 1$ with $λ_1\ge k_0$ but its maximal existence time is shown to be finite. Uniqueness is also established in both cases. The last result deals with the non-existence of mass-conserving weak solutions, even on a small time interval, for power law fragment daughter distribution when $λ_1<k_0$. It is worth mentioning that the previous literature on the nonlinear fragmentation equation does not treat non-integrable fragment daughter distribution functions.

math.AP↗