Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2411.13838

Mathematical Analysis of Regularity, Bifurcations, and Turbulence in Fluid Dynamics via Sobolev, Besov, and Triebel-Lizorkin Spaces

Rômulo Damasclin Chaves dos Santos
View original source ↗Read source document ↗Export metadata ↓

Abstract

This article presents a comprehensive mathematical framework for the study of regularity, bifurcations, and turbulence in fluid dynamics, leveraging the power of Sobolev and Besov function spaces. We delve into the detailed definitions, properties, and notations of these spaces, illustrating their relevance in the context of partial differential equations governing fluid flow. The work emphasizes the intricate connections between Sobolev, Besov, and Triebel-Lizorkin spaces, highlighting their interplay in the analysis of fluid systems. We propose new regularity criteria for solutions to the Navier-Stokes equations, based on the interaction of low and high-frequency modes in turbulent regimes. These criteria offer a novel perspective on the conditions under which singularities may form, providing critical insights into the structure of turbulent flows. The article further explores the applications of these function spaces to the analysis of bifurcations in fluid systems, offering a deeper understanding of the mechanisms that lead to complex flow phenomena such as turbulence. Through the development of rigorous theorems and proofs, the paper aims to bridge the gap between abstract mathematical theory and practical fluid dynamics. In particular, the results contribute to ongoing efforts in solving the Navier-Stokes existence and smoothness problem, a key challenge in the field, and have potential implications for the Millennium Prize Problem. The conclusion underscores the significance of these findings, offering a pathway for future research in the analysis of fluid behavior at both small and large scales.

Explore related subjects

math.APphysics.flu-dyn

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Rômulo Damasclin Chaves dos Santos. 2024-11-21. Mathematical Analysis of Regularity, Bifurcations, and Turbulence in Fluid Dynamics via Sobolev, Besov, and Triebel-Lizorkin Spaces. https://arxiv.org/abs/2411.13838

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

KEEP EXPLORING

Related papers

arXiv2026

Around the Quantum Lenard-Balescu equation

In the mean-field regime, a gas of quantum particles with Boltzmann statistics can be described by the Hartree-Fock equation. This dynamics becomes trivial if the initial distribution of particle is invariant by translation. However, the first correction is given on time of order $O(N)$ by the quantum Lenard--Balescu equation. In the first part of the present article, we justify this equation until time of order $O((\log N)^{1-δ})$ (for any $δ\in(0,1)$). A similar phenomenon exists in the classical setting (with a similar validity time obtained by Duerinckx \cite{Duerinckx}). In a second time, we prove the convergence for dimension $d\geq 2$ of the solutions of the quantum Lenard--Balescu equation to the solutions of its classical counterpart in the semi-classical limit. This problem can be interpreted as a grazing collision limit: the quantum Lenard--Balescu equation looks like a cut-off Boltzmann equation, when the classical one looks like the Landau equation.

math.AP↗
arXiv2026

Almost Periodic Solutions of The Cubic Defocusing Nonlinear Schrödinger Equation

This paper addresses the Cauchy problem for the cubic defocusing nonlinear Schrödinger equation (NLS) with almost periodic initial data. We prove that for small analytic quasiperiodic initial data satisfying Diophantine frequency conditions, the Cauchy problem admits a solution that is almost periodic in both space and time, and that this solution is unique among solutions locally bounded in a suitable sense. The analysis combines direct and inverse spectral theory. In the inverse spectral theory part, we prove existence, almost periodicity, and uniqueness for solutions with initial data whose associated Dirac operator has purely a.c.\ spectrum that is not too thin. This resolves novel challenges presented by the NLS hierarchy, such as an additional degree of freedom and an additional commuting flow. In the direct spectral theory part, for Dirac operators with small analytic quasiperiodic potentials with Diophantine frequency conditions, we prove pure a.c.\ spectrum, exponentially decaying spectral gaps, and spectral thickness conditions (homogeneity and Craig-type conditions).

math.AP↗
arXiv2026

Large-Amplitude Steady Solitary Water Waves with General Vorticity

We study two-dimensional steady solitary gravity water waves with general vorticity, allowing for overhanging free-surface profiles. The main challenges arise from the free boundary, the unbounded fluid domain, and the inherent complexity of the general vorticity setting. To address these, we introduce a conformal reformulation that reduces the problem to an equivalent system on a fixed strip, consisting of an overdetermined elliptic problem coupled with an elliptic boundary-value problem. This framework enables a local analysis without imposing restrictive assumptions on the vorticity. Using the center-manifold construction of Chen, Walsh and Wheeler \cite{chennonlinearity}, we first establish the existence of small-amplitude solitary waves for such non-trivial vorticity distributions. Subsequently, via a global analytic bifurcation argument, we prove the existence of continuous branches of large-amplitude solitary waves for analytic vorticity functions. Along these global solution curves, the free surfaces may develop overhanging profiles and need not remain graphs over the horizontal coordinate. Our results provide a new constructive framework for large-amplitude solitary waves with general vorticity, extending the existing theory beyond the classical settings of constant vorticity \cite{susannaarma} and non-overhanging profiles \cite{milesjma}.

math.AP↗