Stability behaviors of Leray weak solutions to the three-dimensional Navier–Stokes equations |
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| Institution: | 1. Departamento de Matematica Aplicada, Universidad Complutense, 28040 Madrid, Spain;2. Departamento de Análisis Económico: Economía Cuantitativa, Universidad Autónoma de Madrid, 28049 Madrid, Spain;1. Departamento de Computação e Matemática, Universidade de São Paulo (USP), FFCLRP, Av. dos Bandeirantes, 3900, CEP 14040-901, Ribeirão Preto(SP), Brazil;2. Faculty for Mathematics and Computer Science Technical University Bergakademie Freiberg Prüferstr. 9, 09596 Freiberg, Germany;1. Cadi Ayyad University, Ecole Nationale des Sciences Appliquées, Marrakech, Maroc;2. University of Granada, Departamento de Matemática Aplicada, 18071 Granada, Spain |
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| Abstract: | This paper is devoted to the investigation of stability behaviors of Leray weak solutions to the three-dimensional Navier–Stokes equations. For a Leray weak solution of the Navier–Stokes equations in a critical Besov space, it is shown that the Leray weak solution is uniformly stable with respect to a small perturbation of initial velocity and external forcing. If the perturbation is not small, the perturbed weak solution converges asymptotically to the original weak solution as the time tends to the infinity. Additionally, an energy equality and weak–strong uniqueness for the three-dimensional Navier–Stokes equations are derived. The findings are mainly based on the estimations of the nonlinear term of the Navier–Stokes equations in a Besov space framework, the use of special test functions and the energy estimate method. |
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| Keywords: | Navier–Stokes equations Stability behaviors Energy equality Weak–strong uniqueness |
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