Energy Superposition and Regularity for 3D Navier-Stokes Equations in the Largest Critical Space

We show that a Leray-Hopf weak solution to the 3D Navier-Stokes Cauchy problem belonging to the space $L^\infty(0,T; B^{-1}_{\infty,\infty}(\mathbb R^3))$ is regular in $(0,T]$. As a consequence, it follows that any Leray-Hopf weak solution to the 3D Navier-Stokes equations is regular while it is temporally bounded in the largest critical space $\dot{B}^{-1}_{\infty,\infty}(\mathbb R^3)$ as well as in any critical spaces. For the proof we present a new elementary method which is to superpose the energy norm of high frequency parts in an appropriate way to generate higher order norms. Thus, starting from the energy estimates of high frequency parts of a weak solution, one can obtain its estimates of higher order norms. By a linear energy superposition we get very simple and short proofs for known regularity criteria for Leray-Hopf weak solutions in endpoint Besov spaces $B^{\sigma}_{\infty,\infty}$ for $\sigma\in [-1,0)$, the extension of Prodi-Serrin conditions. The main result of the paper is proved by applying technique of a nonlinear energy superposition and linear energy superpositions, repeatedly. The energy superposition method developed in the paper can also be applied to other supercritical nonlinear PDEs.Comment: 19 page

Stokes operator and stability of stationary Navier-Stokes flows in infinite cylindrical domains

infinite cylindrical domains have been attracting great attention due to its theoretical and practical significance. However, in most cases, stationary Navier-Stokes problems were dealt with whereas instationary Navier-Stokes problems have been less studied. The Lq-approach to instationary Navier-Stokes problems is very important and convenient to analyze existence, uniqueness as well as strong energy inequality and partial regularity for solutions; to this end, the study of the Stokes operator is fundamental. The aim of this dissertation is to get resolvent estimates, maximal regularity and boundedness of H∞-calculus of Stokes operators in infinite cylindrical domains and to apply them to the stability of stationary Navier-Stokes flows in infinite cylindrical domains. We start with a Stokes resolvent system in an infinite straight cylinder. The Stokes resolvent system on an infinite straight cylinder is reduced by the (one-dimensional) partial Fourier transform along the axis of the cylinder to a parametrized Stokes system with the Fourier variable as a parameter. Using the Fourier multiplier theory in weighted spaces we get estimates for the parametrized Stokes system with bound constants independent of parameters. Based on these estimates resolvent estimate and maximal regularity of the Stokes operator in weighted Lebesgue spaces on an infinite straight cylinder are shown using the techniques of operator-valued Fourier multiplier theory and Schauder decomposition in Banach spaces with UMD property. Next we consider the Stokes operator in general infinite cylinders with several exits to infinity. A resolvent estimate of the Stokes operator in Lq-space is obtained using cut-off techniques based on the result of generalized Stokes resolvent system in an infinite straight cylinder. In particular, the Stokes operator is shown to generate a bounded and exponentially decaying analytic semigroup in any Lq-space on a general infinite cylinder. Moreover, it is proved that the Stokes operator admits a bounded H∞-calculus in any Lq-space on an infinite cylinder with several exits to infinity. As an application of the obtained properties of the Stokes operator we study stability of stationary Navier-Stokes flows in infinite cylindrical domains. First, existence and uniqueness for stationary Navier-Stokes systems in infinite cylinders are shown. Then the exponential stability of the stationary Navier-Stokes flow is proved based on Lr-Lq estimates of the perturbed Stokes semigroup