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Linear inviscid damping and enhanced viscous dissipation of shear flows by using the conjugate operator method. (English) Zbl 1447.35247

The long time behavior of solutions to the two 2D linearized systems is studied. The first system is the Euler one linearized near a stationary shear flow. \[ \frac{\partial v}{\partial t}+(U_s\cdot \nabla)v+(v\cdot \nabla)U_s+\nabla p=0,\quad \mbox{div}\,v=0,\quad x\in \mathbb{T},\,y\in \mathbb{R},\tag{1} \] where \(U_s\) is a smooth stationary shear flow under the form \(U_s=(U(y),0)\). The second system is the linearized Navier-Stokes one. \[ \frac{\partial v}{\partial t}+(U_s\cdot \nabla)v+(v\cdot \nabla)U_s+\nabla p-\nu\Delta v=0,\quad \mbox{div}\,v=0,\quad x\in \mathbb{T},\,y\in \mathbb{R},\tag{2} \] where \(\nu>0\) is a small viscosity. The function \(U\) is assumed to be smooth and ensured a spectral stability of a self-adjoint operator \[ -\frac{\partial^2}{\partial y^2}+\frac{U^{\prime\prime}}{U}. \] It is proved that:
1)
let \(v=(v^1,v^2)\) is the solution to the system (1) and \((v^1_\alpha,v^2_\alpha)_{\alpha\in \mathbb{Z}}\) are the Fourier coefficients with respect to the variable \(x\in\mathbb{T}\), then for the large time \[ \Vert v^1_\alpha(t)\Vert\leq\frac{C}{|\alpha| t},\quad \Vert v^2_\alpha(t)\Vert\leq\frac{C|\alpha|}{(\alpha t)^2}, \]
2)
the solution to the problem (2) is damped at the time scale \(\nu^{-1/3}\).
The prove is based on the conjugate operator method.

MSC:

35Q30 Navier-Stokes equations
35Q31 Euler equations
76D05 Navier-Stokes equations for incompressible viscous fluids
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
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