Navier-Stokes Equations In R3 Research Articles

Decay rates of second order derivatives of axisymmetric D-solutions to the stationary Navier-Stokes equations

In this paper, we are interested in the decay properties of the axisymmetric solutions to the stationary Navier-Stokes equations in R3 with finite Dirichlet integral. We establish some new a priori decay rates for the second order derivatives of the velocity. The result is obtained by combining the weighted energy estimates, the Brezis-Galleout inequality and the scaling argument.

A Liouville theorem of Navier-Stokes equations with two periodic variables

In this paper, we study the Liouville theorem of the stationary Navier-Stokes equations in R3. When the solution is periodic in two variables, we can prove that actually the solution is trivial (constant vector) under the assumption that one component of the velocity, vanishing at infinity, has finite Dirichlet integral and the other two components can have some growth with respect to the distance to the origin.

Note on the Liouville type problem for the stationary Navier-Stokes equations in [formula omitted

In this brief note we are concerned on the Liouville type problem for the stationary Navier-Stokes equations in R3. We deduce an asymptotic formula for an integral involving the head pressure, Q=12|v|2+p, and its derivative over domains enclosed by level surfaces of Q. This formula provides us with new sufficient condition for the triviality of solution to the Navier-Stokes equations.

On the supnorm form of Leray’s problem for the incompressible Navier-Stokes equations

We show that t3/4 u(⋅,t) L∞(R3)→0 as t → ∞ for all Leray-Hopf’s global weak solutions u(⋅, t) of the incompressible Navier-Stokes equations in ℝ3. It is also shown that t u(⋅,t)−eΔtu0 L∞(R3)→0 as t → ∞, where eΔt is the heat semigroup, as well as other fundamental new results. In spite of the complexity of the questions, our approach is elementary and is based on standard tools like conventional Fourier and energy methods.

Some Serrin-type regularity criteria for weak solutions to the Navier-Stokes equations

We consider the regularity criteria for a weak solution \documentclass[12pt]{minimal}\begin{document}$\bm {u}=(u_1,u_2,u_3)$\end{document}u=(u1,u2,u3) to the Navier-Stokes equations in \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^3$\end{document}R3. Denoting by \documentclass[12pt]{minimal}\begin{document}$\bm {\omega }=(\omega _1,\omega _2,\omega _3)=\mbox{curl }\bm {u}$\end{document}ω=(ω1,ω2,ω3)=curlu the vorticity, we then prove that \documentclass[12pt]{minimal}\begin{document}$\bm {u}$\end{document}u is smooth, provided u3, ω3; or ∂3u3, ω3 are in some Serrin-type integrability classes.

On regularity criteria in terms of pressure for the Navier-Stokes equations in ℝ³

In this paper we establish a Serrin-type regularity criterion on the gradient of pressure for the weak solutions to the Navier-Stokes equations in R 3 \mathbb {R}^3 . It is proved that if the gradient of pressure belongs to L α , γ L^{\alpha ,\gamma } with 2 / α + 3 / γ ≤ 3 2/\alpha +3/\gamma \leq 3 , 1 ≤ γ ≤ ∞ 1\leq \gamma \leq \infty , then the weak solution is actually regular. Moreover, we give a much simpler proof of the regularity criterion on the pressure, which was showed recently by Berselli and Galdi (Proc. Amer. Math. Soc. 130 (2002), no. 12, 3585–3595).

A probabilistic representation of solutions of the incompressible Navier-Stokes equations in R3

A new probabilistic representation is presented for solutions of the incompressible Navier-Stokes equations in R3 with given forcing and initial velocity. This representation expresses solutions as scaled conditional expectations of functionals of a Markov process indexed by the nodes of a binary tree. It gives existence and uniqueness of weak solutions for all time under relatively simple conditions on the forcing and initial data. These conditions involve comparison of the forcing and initial data with majorizing kernels.

L∞ Stability for Weak Solutions of the Navier-Stokes Equations in R3 with Singular Initial Data in Morrey Spaces

In this paper we study the L∞ stability of weak solutions of the Navier-Stokes equations when the initial data is a measure belonging to a suitable Morrey space. It is also assumed that these measures are sufficiently small in the corresponding norms. The physically relevant cases of vortex filament, vortex ring and vortex sheet structure of vorticity are included in this analysis.

Parametrix and the asymptotics of localized solutions of the Navier-Stokes equations in R3, linearized on a smooth flow

Parametrix and the asymptotics of localized solutions of the Navier-Stokes equations in R3, linearized on a smooth flow

Group-theoretic analysis of the Navier-Stokes equations in the rotationally symmetric case and some new exact solutions

In the paper one investigates the group-theoretic properties of the Navier-Stokes equations in R3 in the case of rotational symmetry. One computes the fundamental algebra of the Navier-Stokes equations in the case of rotational symmetry. This algebra is infinite-dimensional. For it one constructs the so-called optimal solutions of order 1 and 2. For the subalgebras of these systems one finds the corresponding reduced systems of equations for the functions which define the invariant solutions of the initial equations. In a series of cases these reduced systems can be explicitly solved and one finds new exact solutions of the Navier-Stokes equations.

The convergence with vanishing viscosity of nonstationary Navier-Stokes flow to ideal flow in $R\sb{3}$

It is shown here that a unique solution to the Navier-Stokes equations exists in R3 for a small time interval independent of the viscosity and that the solutions for varying viscosities converge uniformly to a function that is a solution to the equations for ideal flow in R3. The existence of the solutions is shown by transforming the Navier-Stokes equations to an equivalent system solvable by applying fixed point methods with estimates derived from using semigroup theory. Introduction. We wish to find a solution, local in time, to the Cauchy problem for the Navier-Stokes equations for viscous incompressible flow in R3 and show that the solutions of the Navier-Stokes equations for various viscosities converge, as the viscosity goes to zero, to a function that is a solution to the Euler equations for an ideal (inviscid) fluid. The Navier-Stokes equations are (E') av/lt + (v * grad) v-vAv = -grad P + B, VV = v O, with constraints lim v(x, t) = 0 and v(x, 0) = Cx), Ixl +o where x = (x1, X2, X3) is a point in R3; t is in some time interval [0, T]; the velociy v(x, t)=(v1(x, t), v2(x, t), v3(x, t)); the pressure is P(x, t); the force is B(x, t) = (B1(x, t), B2(x, t), B3(x, t)); and the constant v>0 is the viscosity (the coefficient of kinematic viscosity). The Euler equations for ideal flow differ from the Navier-Stokes equations (E') only in that the viscosity term vAv does not occur in the Euler equations. Uniqueness and existence of a solution to the Navier-Stokes equations in R3 has been shown for both bounded and unbounded domains: in both cases existence has been shown only for a sufficiently small time interval. The first results are those of C. W. Oseen [11] and Jean Leray [8]. The time interval where the solution is shown to exist must be small enough to satisfy a condition of form T? Kv, where K is an appropriate constant and v is the viscosity. Thus the length of the time interval Received by the editors January 28, 1970 and, in revised form, June 25, 1970. AMS 1968 subject classifications. Primary 7635, 7646; Secondary 3536, 4750.