Regularity Issue of the Navier-Stokes Equations Involving the Combination of Pressure and Velocity Field
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Regularity Issue of the Navier-Stokes Equations Involving the Combination of Pressure and Velocity Field Zhengguang Guo · Peter Wittwer · Weiming Wang
Received: 24 November 2011 / Accepted: 10 April 2012 / Published online: 25 April 2012 © Springer Science+Business Media B.V. 2012
Abstract We establish some regularity criteria for the incompressible Navier-Stokes equations in a bounded three-dimensional domain concerning the quotients of the pressure, the velocity field and the pressure gradient. Keywords Navier-Stokes equations · Regularity criterion · A priori estimates Mathematics Subject Classification 35B45 · 35B65 · 76D05 1 Introduction In this article, we consider the following initial boundary value problem for the incompressible Navier-Stokes equations in Ω × (0, T ) ⎧ ∂u + u · ∇u + ∇p = u, in Ω × (0, T ) ⎪ ∂t ⎪ ⎪ ⎪ ⎨ div u = 0, in Ω × (0, T ) (1) ⎪ on ∂Ω × (0, T ) ⎪ u = 0, ⎪ ⎪ ⎩ in Ω u(x, 0) = u0 (x), where u = u(x, t) ∈ R3 is the velocity field, p(x, t) is a scalar pressure, u0 (x) with div u0 = 0 in the sense of distributions is the initial velocity field, and Ω is a bounded domain with smooth boundary ∂Ω. Z. Guo () · W. Wang College of Mathematics and Information Science, Wenzhou University, Wenzhou 325035, Zhejiang, P.R. China e-mail: [email protected] W. Wang e-mail: [email protected] P. Wittwer Département de Physique Théorique, Université de Genève, Genève, Switzerland e-mail: [email protected]
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The study of the incompressible Navier-Stokes equations in three space dimensions has a long history. In the pioneering work [12] and [20], Leray and Hopf proved the existence of weak solutions u(x, t) ∈ L∞ (0, T ; L2 (R3 )) ∩ L2 (0, T ; H 1 (R3 )) for given u0 (x) ∈ L2 (R3 ). However, it is not known yet whether or not the solution develops singularities in finite time even if the initial datum is C ∞ -smooth. Therefore the study of the regularity of solutions becomes interesting and attracts many researchers’ interest. On one hand, in [23], Scheffer began to study the partial regularity theory of the Navier-Stokes equations. Additional results were obtained by Caffarelli, Kohn and Nirenberg in [4]. Further result can be found in [26] and references therein. On the other hand, the regularity of a given weak solution u can be shown under additional conditions. In 1962, Serrin [24] proved that if u is a Leray-Hopf weak solution belonging to Lα,γ ≡ Lα (0, T ; Lγ (R3 )) with 2/α+ 3/γ ≤ 1, 2 < α < ∞, 3 < γ < ∞, then the solution u(x, t) ∈ C ∞ (R3 × (0, T ]). From then on, there are many results with additional criterion added on u, see for instance [9, 17, 18, 25, 27]. It is well-known that if (u, p) solves the Navier-Stokes equations, then so does (uλ , pλ ) for all λ > 0 in R3 , with uλ (x, t) = λu λx, λ2 t , pλ (x, t) = λ2 p λx, λ2 t . The class of Serrin’s type is important form a view point of scaling invariance which implies that uλ Lα,γ = uLα,γ holds true for all λ > 0 if and only if 2/α + 3/γ = 1 and we say that the norm uLα,γ has the scaling dimension ze
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