The Navier-Stokes Equations in the Critical Lebesgue Space

By Hongjie Dong, Dapeng Du. 2009

53 citations recorded by the source corpus.

We study regularity criteria for the d-dimensional incompressible Navier-Stokes equations. We prove in this paper that if $${u \in L_\infty^tL_{d}^x((0,T)\times \mathbb{R}^d)}$$ is a Leray-Hopf weak solution, then u is smooth and unique in $${(0, T)\times \mathbb{R}^d}$$ . This generalizes a result by Escauriaza, Seregin and Šverák [5]. Additionally, we show that if T = ∞ then u goes to zero as t goes to infinity.

Subfields: Applied Mathematics, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/ecf8009ff6fc5d94ecb0ed54810303b377e83853

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In Applied Mathematics

In Control and Systems Engineering

Citations in the corpus

Necessary and Sucient Conditions for the Fractional Gagliardo-Nirenberg Inequalities and Applications to Navier-Stokes and Generalized Boson Equations (2011, background): “Dong and Du [19] generalized their results in higher spatial dimensions n > 3.”
Quantitative bounds for critically bounded solutions to the Navier-Stokes equations (2019, background): “It may be possible to obtain a higher-dimensional analogue of our results by finding quantitative versions of the argument in [5], but we do not pursue this question here.”
Necessary and Sufficient Conditions for the Fractional Gagliardo-Nirenberg Inequalities and Applications to Navier-Stokes and Generalized Boson Equations (Harmonic Analysis and Nonlinear Partial Differential Equations) (2011, background): “Dong and Du [19] generalized their results in higher spatial dimensions n\geq 3 .”