Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0209208

Convergence results for a coarsening model using global linearization

Th. Gallay·A. Mielke
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study a coarsening model describing the dynamics of interfaces in the one-dimensional Allen-Cahn equation. Given a partition of the real line into intervals of length greater than one, the model consists in constantly eliminating the shortest interval of the partition by merging it with its two neighbors. We show that the mean-field equation for the time-dependent distribution of interval lengths can be explicitly solved using a global linearization transformation. This allows us to derive rigorous results on the long-time asymptotics of the solutions. If the average length of the intervals is finite, we prove that all distributions approach a uniquely determined self-similar solution. We also obtain global stability results for the family of self-similar profiles which correspond to distributions with infinite expectation.

Explore related subjects

math.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Th. Gallay, A. Mielke. 2002-09-17. Convergence results for a coarsening model using global linearization. https://arxiv.org/abs/math/0209208

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Existence and multiplicity of $L^2$-Normalized solutions for the periodic Schrödinger system of Hamiltonian type

In this paper, we study the following nonlinear Schrödinger system of Hamiltonian type \begin{equation*} \left\{\begin{array}{l} -Δu+V(x)u=\partial_v H(x,u,v)+ωv,\ x \in \mathbb{R}^N, \\ -Δv+V(x)v=\partial_u H(x,u,v)+ωu,\ x \in \mathbb{R}^N, \\ \displaystyle\int_{\R^N}|z|^2dx=a^2, \end{array}\right. \end{equation*} where the potential function $V(x)$ is periodic, $z:=(u,v):\R^N\rightarrow \R\times\R$, $ω\in \R$ appears as a Lagrange multiplier, $a>0$ is a prescribed constant. The existence and multiplicity of $L^2$-normalized solutions for the above Schrödinger system are obtained. The proof combines a constrained variational reduction, compactness estimates, and a Ljusternik--Schnirelmann type multiplicity theorem. In addition, a bifurcation result is also given.

math.AP↗
arXiv2026

Propagation of chaos for the Landau equation via microcanonical binary collisions

We develop a fully constructive, conservative, and collision-level realization of Kac's program for the spatially homogeneous Landau equation across the full interaction range, including the Coulomb case. Our model is the microcanonical binary-collision (MBC) process: a reversible pure-jump $N$-particle Markov process, realizing the grazing-collision mechanism via small conservative rotations of relative velocities. Under suitable assumptions on the initial data, we prove propagation of chaos in the joint mean-field and grazing-collision limit ($N\to\infty, h\to0$), identifying any limit point with the unique solution in the relevant weak-solution class. The empirical measures converge in quadratic Wasserstein distance, while fixed-order marginals converge strongly in $L^1$ at every fixed time. Furthermore, we rigorously derive the Landau master equation as the fixed-$N$ grazing-collision limit of the MBC process. The proof combines Fisher-information dissipation, compactness, and uniqueness; for $γ\in[-3,-2)$, second-order information along collision directions yields the time-integrable $L^\infty$ bound needed for uniqueness.

math.AP↗
arXiv2026

A Generalized Framework for Singular Fractional Burgers Equations with Stochastic Forcing

In this paper, we investigate the regularization effect of fractional stochastic forcing on Burgers-type equations with fractional dissipation, with an application to the Degasperis--Procesi (DP) equation. In particular, we consider the perturbation induced by the singular noise $|D|^{1/2}ξ$ and establish local well-posedness in the negative Sobolev space $H^{-1/4+δ}$ for some small $δ>0$. Due to the singular nature of the nonlinear interactions, classical solution theories cannot be directly applied. Inspired by the framework developed in \cite{hairer2013solving,gubinelli2017kpz}, we introduce a generalized solution theory based on an enhanced structure and derive the effective equations satisfied by these generalized solutions. Our main contribution is the establishment of a general framework for describing singular PDEs driven by rough data. Moreover, we prove the convergence of the associated non-Gaussian rough structures in the fractional dissipation setting. As an application, we apply this framework to the stochastic Degasperis--Procesi equation and obtain its local well-posedness in the low-regularity regime.

math.AP↗