For the original Navier-Stokes problem we construct next global solution of the Leray-Hopf type satisfying also Duhamel's integral formula. Focusing finally on the 3-D model with zero external force we estimate a time after which the latter solution regularizes to strong solution.

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We also estimate a time such that if a local strong solution exists until that time, then it exists for ever. Google Scholar. Carvalho , Abstract parabolic problems with critical nonlinearities and applications to Navier-Stokes and heat equations, Trans. Ball , Strongly continuous semigroups, weak solutions and the variation of constants formula, Proc. Brezis, Analyse Fonctionelle. Vasseur , Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann.

Cholewa , Continuation and asymptotics of solutions to semilinear parabolic equations with critical nonlinearities, J. Cholewa and T.

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Cholewa, C. Quesada and A. Dlotko, Navier-Stokes equation and its fractional approximations Appl. Sun , Quasi-geostrophic equation in R 2 , J. Differential Equations , , Foias, O.

## Navier–Stokes equations - Wikipedia

Manley, R. Rosa and R. Rational Mech. Morimoto , An L r -theorem of the Helmholtz decomposition of vector fields, J. Tokyo , 24 , Giga , Analyticity of the semigroup generated by the Stokes operator in L r spaces, Math. Differential Equations , 61 , Giga , Domains of fractional powers of the Stokes operator in L r spaces, Arch. Nonlinear Stud. Wakimoto , On existence of solutions of the Navier-Stokes equation in a time dependent domain, J. Tokyo, Sec. I , 24 , Fujita , On the nonstationary Navier-Stokes system, Rend.

Padova , 32 , Komatsu , Fractional powers of operators, Pacific J. Ladyzhenskaya , On some gaps in two of my papers on the Navier-Stokes equations and the way of closing them, J. Ladyzhenskaya , On the uniqueness and smoothness of generalized solutions of the Navier-Stokes equations, Zap. LOMI , , Translated from the Russian by Richard A. Silverman and John Chu. Mathematics and its Applications, Vol. Lions and E. Kalita, Navier-Stokes Equations.

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Sobolevskii, On equations of parabolic type in a Banach space, Trudy Moskov. Sohr, The Navier-Stokes Equations. Strauss , On continuity of functions with values in various Banach spaces, Pacific J.

Temam , On the Euler equations of incompressible perfect fluids, J. Functional Analysis , 20 , Favini, E.

- Lectures on Navier-Stokes Equations.
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Fan , Weak-strong uniqueness for the generalized Navier-Stokes equations, Appl. Mixed initial-boundary value problem for the three-dimensional Navier-Stokes equations in polyhedral domains. Conference Publications , , Special : Hantaek Bae.

Solvability of the free boundary value problem of the Navier-Stokes equations. Mehdi Badra. Abstract settings for stabilization of nonlinear parabolic system with a Riccati-based strategy. Yuning Liu , Wei Wang. Yoshikazu Giga. A remark on a Liouville problem with boundary for the Stokes and the Navier-Stokes equations.

Jing Wang , Lining Tong. Stability of boundary layers for the inflow compressible Navier-Stokes equations. Sylvie Monniaux. Various boundary conditions for Navier-Stokes equations in bounded Lipschitz domains. Pavel I.

## The Navier-Stokes Equations – A Millenium Problem

Plotnikov , Jan Sokolowski. Compressible Navier-Stokes equations. Stability estimates for Navier-Stokes equations and application to inverse problems. Joel Avrin. Guangrong Wu , Ping Zhang. Dougalis , D. Mitsotakis , J. In a first series of tests, we disregard inertia and the effects of rotation to consider a steady Stokes problem. Second, we also wish to retrieve the classical spectral convergence properties of spectral methods.

Note spectral convergence as N increases. The analytic reference solution is not shown here, at it is indistinguishable from the SEM solution. The black fit line corresponds to an exponent of 1.

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Motion is generated inside the tangent cylinder only, as the fluid located outside is insensitive to the super rotation by virtue of the Taylor—Proudman theorem. A shear layer is created along to accommodate the shear between the two regions. On the left contour plots of the meridional stream function are shown, while the right shows contour plots of the angular velocity.

Note the transition from a smooth, essentially viscous solution b to a solution dominated by rotation d. This classic kinematic flow problem in its asymptotic form was originally proposed by Proudman It was developed later by Stewarston , , who derived the exact structure of the shear layers alongside of. Since then it has been treated numerically by several groups Hollerbach ; Dormy et al. Its linear character, moreover, makes it an ideal test case to verify the numerical accuracy of our method, leaving aside complications that inevitably appear when one introduces non-linear effects.

First, we describe the reference numerical solution together with the changes in the physics of the solution, as we go from a slowly rotating system to a system rotating more rapidly. Our numerical reference solution is the spectral solution published by Hollerbach It is displayed in Fig.