Vanishing viscosity limits for axisymmetric flows with boundary
Vanishing viscosity limits for axisymmetric flows with boundary
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DOI：
10.1016/j.matpur.2020.01.005
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发表时间：
2018-06
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影响因子：
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通讯作者：
K. Abe
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文献类型：
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作者：
K. Abe
我们构造了无限长圆柱体中欧拉方程Π={x∈R3|xh=(x1，x2)，r=|xh|<；1}的整体弱解，当初始涡度ω0=ω0θeθ满足ω0θ/r∈∈q[3/2，3)时，所构造的解对于Π‾中的空间变量是Hölder连续的，如果此外还有ω0θ/r∈S对于S∈(3，∞)，且当S=∞时是唯一的.证明是用零粘性方法得到的。我们证明了满足Neumann边界条件的N-S方程对L p中的无涡的轴对称数据是整体适定的，对于所有p∈[3，∞]。如果ω0θ/r∈L q对于q∈[3/2，2]，则能量耗散趋于零；对于t∈[0，∞]，若另外ω0θ/r∈L∞，N-S流局部一致收敛到L 2中的欧拉流。特别地，L 2-收敛蕴含着弱解的能量相等。
We construct global weak solutions of the Euler equations in an infinite cylinder Π={x∈ R 3| x h=(x 1, x 2), r=| x h|< 1} for axisymmetric initial data without swirl when initial vorticity ω 0= ω 0 θ e θ satisfies ω 0 θ/r∈ L q for q∈[3/2, 3). The solutions constructed are Hölder continuous for spatial variables in Π‾ if in addition that ω 0 θ/r∈ L s for s∈(3,∞) and unique if s=∞. The proof is by a vanishing viscosity method. We show that the Navier-Stokes equations subject to the Neumann boundary condition is globally well-posed for axisymmetric data without swirl in L p for all p∈[3,∞). It is also shown that the energy dissipation tends to zero if ω 0 θ/r∈ L q for q∈[3/2, 2], and Navier-Stokes flows converge to Euler flow in L 2 locally uniformly for t∈[0,∞) if additionally ω 0 θ/r∈ L∞. The L 2-convergence in particular implies the energy equality for weak solutions.