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Journal of Mathematical Fluid Mechanics

, Volume 14, Issue 2, pp 311–324 | Cite as

Random Data Cauchy Theory for the Generalized Incompressible Navier–Stokes Equations

  • Ting Zhang
  • Daoyuan FangEmail author
Article

Abstract

In this paper, we consider the generalized Navier–Stokes equations where the space domain is \({\mathbb{T}^N}\) or \({\mathbb{R}^N, N\geq3}\) . The generalized Navier–Stokes equations here refer to the equations obtained by replacing the Laplacian in the classical Navier–Stokes equations by the more general operator (−Δ) α with \({\alpha\in (\frac{1}{2},\frac{N+2}{4})}\) . After a suitable randomization, we obtain the existence and uniqueness of the local mild solution for a large set of the initial data in \({H^s, s\in[-\alpha,0]}\) , if \({1 < \alpha < \frac{N+2}{4}, s\in(1-2\alpha,0]}\) , if \({\frac{1}{2} < \alpha\leq 1}\) . Furthermore, we obtain the probability for the global existence and uniqueness of the solution. Specially, our result shows that, in some sense, the Cauchy problem of the classical Navier–Stokes equation is local well-posed for a large set of the initial data in H −1+, exhibiting a gain of \({\frac{N}{2}-}\) derivatives with respect to the critical Hilbert space \({H^{\frac{N}{2}-1}}\) .

Mathematics Subject Classification (2010)

Primary 35Q30 Secondary 76D05 35A07 

Keywords

Generalized Navier–Stokes equations Existence and uniqueness 

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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Department of MathematicsZhejiang UniversityHangzhouChina

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