Navier-Stokes equations in a rotating frame in R3 with initial data nondecreasing at infinity
Navier-Stokes equations in a rotating frame in R3 with initial data nondecreasing at infinity
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DOI：
10.14492/hokmj/1285766360
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发表时间：
2006
影响因子：
0.5
通讯作者：
Y. Giga;K. Inui;K. Inui;A. Mahalov;Shin’ya Matsui
中科院分区：
文献类型：
--
作者：
Y. Giga;K. Inui;K. Inui;A. Mahalov;Shin’ya Matsui
考虑了具有常数Coriolis参数0和初始数据在无穷远处不减的三维旋转Navier-Stokes方程。与非旋转情况（Ω = 0）相反。证明了对于转动问题（Ω <$0），与线性问题（Stokes + Coriolis组合算子）对应的绿色函数不属于L1（R3）.而且。相应的积分算子在R3中的螺线管向量场空间L ∞ σ（R3）中是无界的，线性（Stokes+Coriolis）组合算子在L ∞ σ（R3）中不生成半群.证明了旋转Navier-Stokes方程在L ∞螺线管矢量场组成的空间L <$σ，α（R3）中的初速度场在时间上的局部唯一解，其中L ∞螺线管矢量场满足垂直平均性质，其斜压分量属于齐次Besov空间B 0 ∞，1，B 0 ∞，1小于L ∞，但仍包含各种周期和概周期函数.将初始数据限制在L ∞ σ（R3）的子空间L ∞ σ，α（R3）上，是组合线性算子（Stokes + Coriolis）生成半群的必要条件.利用旋转变换，我们还得到了具有初速度和涡度的三维Navier-Stokes方程V（0）= V0（y）+（Ω/2）e3 xy，curl V（0）= curl V0（y）+ Ω e3的局部时间可解性，其中V0（y）∈ L ∞ σ，α（R3）.
Three-dimensional rotating Navier-Stokes equations are considered with a constant Coriolis parameter 0 and initial data nondecreasing at infinity. In contrast to the non-rotating case (Ω = 0). it is shown for the problem with rotation (Ω ≠ 0) that Green's function corresponding to the linear problem (Stokes + Coriolis combined operator) does not belong to L 1 (R 3 ). Moreover. the corresponding integral operator is unbounded in the space L ∝ σ (R 3 ) of solenoidal vector fields in R 3 and the linear (Stokes+Coriolis) combined operator does not generate a semigroup in L ∞ σ (R 3 ). Local in time unique solvability of the rotating Navier-Stokes equations is proven for initial velocity fields in the space L ∝ σ,α (R 3 ) which consists of L ∞ solenoidal vector fields satisfying vertical averaging property such that their baroclinic component belongs to a homogeneous Besov space B 0 ∞,1 which is smaller than L ∞ but still contains various periodic and almost periodic functions. This restriction of initial data to L ∞ σ,α (R 3 ) which is a subspace of L ∞ σ (R 3 ) is essential for the combined linear operator (Stokes + Coriolis) to generate a semigroup. Using the rotation transformation, we also obtain local in time solvability of the classical 3D Navier-Stokes equations in R 3 with initial velocity and vorticity of the form V(0) = V 0 (y) + (Ω/2)e 3 x y, curl V(0) = curl V 0 (y) + Ωe 3 where V 0(y) ∈ L ∞ σ,α (R 3 ).