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Well-posedness and ill-posedness for the 3D generalized Navier-Stokes equations in \(\dot{F}^{-\alpha,r}_{\frac{3}{\alpha-1}}\). (English) Zbl 1368.76013

Summary: In this paper, we study the Cauchy problem of the 3-dimensional (3D) generalized Navier-Stokes equations (gNS) in the Triebel-Lizorkin spaces \(\dot{F}^{-\alpha,r}_{q_\alpha}\) with \((\alpha, r) \in (1, \frac{5}{4}) \times [1, \infty]\) and \(q_\alpha = \frac{3}{\alpha - 1}\). Our work establishes a dichotomy of well-posedness and ill-posedness depending on \(r\). Specifically, by combining the new endpoint bilinear estimates in \(L^{q_\alpha}_x L^2_T\) and \(L^\infty_T \dot{F}^{-\alpha,1}_{q_\alpha}\) and characterization of the Triebel-Lizorkin spaces via fractional semigroup, we prove well-posedness of the gNS in \(\dot{F}^{-\alpha,r}_{q_\alpha}\) for \(r \in [1, 2]\). Meanwhile, for any \(r \in (2, \infty]\), we show that the solution to the gNS can develop norm inflation in the sense that arbitrarily small initial data in \(\dot{F}^{-\alpha,r}_{q_\alpha}\) can produce arbitrarily large solution after arbitrarily short time.

MSC:

76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76D05 Navier-Stokes equations for incompressible viscous fluids
35Q35 PDEs in connection with fluid mechanics
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35R11 Fractional partial differential equations
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