Qualitative properties of solutions to vorticity equation for a viscous incompressible fluid on a rotating sphere
- PDF / 394,183 Bytes
- 14 Pages / 547.087 x 737.008 pts Page_size
- 44 Downloads / 195 Views
Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
Qualitative properties of solutions to vorticity equation for a viscous incompressible fluid on a rotating sphere Yuri N. Skiba Abstract. A nonlinear vorticity equation describing the behavior of a viscous incompressible fluid on a rotating sphere is considered. The viscosity term is modeled by a real degree of the Laplace operator. The smoothness of external forcing is established that guarantee the existence of a limited attractive set in the space of solutions. Theorems on the existence and uniqueness of non-stationary and stationary weak solutions are given. Sufficient conditions for the global asymptotic stability of solutions are obtained. An example is constructed that shows that, in contrast to the stationary forcing, the Hausdorff dimension of the global attractor generated by a quasiperiodic (in time) and polynomial (in space) forcing can be arbitrarily large, even if the generalized Grashof number is limited. Mathematics Subject Classification. 76D17, 76E05, 76E20. Keywords. Viscous incompressible fluid, Vorticity equation on a sphere, Asymptotic behavior of solutions, Conditions for global asymptotic stability, Dimension of global attractor.
1. Spaces of functions on a sphere Let x = (λ, μ) be a point of the unit sphere S = {x ∈ R3 : |x| = 1} where λ ∈ [0, 2π) is the longitude, μ = sin φ, the latitude φ ∈ [−π/2, π/2], and hence μ ∈ [−1, 1]. Let us denote by C∞ (S) the space of infinitely differentiable functions on S and by f, g = f (x) g(x) dS and f = f, f 1/2 (1) S
the inner product and norm of any functions f (x) and g(x) of C∞ (S), respectively. Here dS = dλdμ, and g(x) is the complex conjugate of g(x). It is well known that the spherical harmonics 1/2 (n − m)! m Yn (λ, μ) = Kn Pnm (μ) eimλ , n ≥ 0, |m| ≤ n (n + m)! where 1/2 2n + 1 (2) Kn = 4π form the orthonormal system in C∞ (S): Ynm , Ylk = δmk δnl where δmk is the Kronecker delta and m/2 n 1 − μ2 dn+m 2 m μ −1 Pn (μ) = n n+m 2 n! dμ is the associated Legendre function of degree n and zonal number m. Each spherical harmonic Ynm is the eigenfunction of the eigenvalue problem −ΔYnm = χn Ynm , χn = n (n + 1) , |m| ≤ n 0123456789().: V,-vol
159
Page 2 of 14
Y. N. Skiba
ZAMP
for symmetric and positive definite spherical Laplace operator ∂ ∂ ∂2 1 2 −Δ=− 1−μ − ∂μ ∂μ 1 − μ2 ∂λ2
(3)
Lemma 1. Let n be a natural and Kn be a constant (2). Then n n 2 2 |Ynm (x)| = Kn2 and |∇Ynm (x)| = χn Kn2 m= −n
m= −n
Proof. Let ω be an angle between two unit radius vectors x 1 , x 2 corresponding to points x1 , x2 ∈ S. Then, according to well-known formula, n Pn (x 1 · x 2 ) = Kn−2 Ynm (x1 ) Ynm (x2 ) m= −n
where x 1 · x 2 = cos ω is the scalar product of vectors x 1 and x 2 , and Pn is the Legendre polynomial [2,6]. In particular, if x1 = x2 = x then Pn (1) = 1 and we obtain the first equality of Lemma 1. Further, let us replace f and g in the identity Δ(f g) = f Δg + gΔf + 2∇f · ∇g by Ynm (λ, μ) and Ynm (λ, μ). If we summarize the result over m from −n to n and use the first e
Data Loading...
