Liouville type of theorems with weights for the Navier-Stokes equations and the Euler equations
نویسنده
چکیده
We study Liouville type of theorems for the Navier-Stokes and the Euler equations on R N , N ≥ 2. Specifically, we prove that if a weak solution (v, p) satisfies |v| 2 +|p| ∈ L 1 (0, T ; L 1 (R N , w 1 (x)dx)) and R N p(x, t)w 2 (x)dx ≥ 0 for some weight functions w 1 (x) and w 2 (x), then the solution is trivial, namely v = 0 almost everywhere on R N × (0, T). Similar results hold for the MHD Equations on R N , N ≥ 3.
منابع مشابه
Liouville type of theorems for the Euler and the Navier-Stokes equations
We prove Liouville type of theorems for weak solutions of the Navier-Stokes and the Euler equations. In particular, if the pressure satisfies p ∈ L1(0, T ;H1(RN )), then the corresponding velocity should be trivial, namely v = 0 on RN × (0, T ), while if p ∈ L1(0, T ;L1(RN )), then we have equipartition of energy over each component. Similar results hold also for the magnetohydrodynamic equations.
متن کامل2 8 Fe b 20 09 On the Liouville theorem for the Navier - Stokes and the Euler equations
In this paper we prove that any weak solution v to the incom-pressible Navier-Stokes/Euler equations in N 2 (s − 2). Similar result also holds for the viscous/inviscid MHD equations in R N with N ≥ 3.
متن کاملOptimization with the time-dependent Navier-Stokes equations as constraints
In this paper, optimal distributed control of the time-dependent Navier-Stokes equations is considered. The control problem involves the minimization of a measure of the distance between the velocity field and a given target velocity field. A mixed numerical method involving a quasi-Newton algorithm, a novel calculation of the gradients and an inhomogeneous Navier-Stokes solver, to find the opt...
متن کاملInvestigation of instable fluid velocity in pipes with internal nanofluid flow based on Navier-Stokes equations
In this article, the instable fluid velocity in the pipes with internal nanofluid is studied. The fluid is mixed by SiO2, AL2O3, CuO and TiO2 nanoparticles in which the equivalent characteristic of nanofluid is calculated by rule of mixture. The force induced by the nanofluid is assumed in radial direction and is obtained by Navier-Stokes equation considering viscosity of nanofluid. The displac...
متن کاملStability of two classes of improved backward Euler methods for stochastic delay differential equations of neutral type
This paper examines stability analysis of two classes of improved backward Euler methods, namely split-step $(theta, lambda)$-backward Euler (SSBE) and semi-implicit $(theta,lambda)$-Euler (SIE) methods, for nonlinear neutral stochastic delay differential equations (NSDDEs). It is proved that the SSBE method with $theta, lambdain(0,1]$ can recover the exponential mean-square stability with some...
متن کامل