Steady solutions with finite kinetic energy for a perturbed Navier-Stokes system in \(\mathbb R^3\). (English) Zbl 1181.35179
Summary: Consider a Navier-Stokes liquid filling the three-dimensional space exterior to a moving rigid body and subject to an external force. Using a coordinates system attached to the body, the equations of the fluid can be written in a time-independent domain, which results in a perturbed Navier-Stokes system where the extra terms depend on the velocity of the rigid body.
In this paper, we consider the related whole space problem and construct a strong solution with finite kinetic energy for the corresponding steady-state equations. For this, appropriate conditions on the external force have to be imposed (for instance, that it is a function with compact support and null average) together with a smallness condition involving the viscosity of the fluid. First, a linearized version of the problem is analysed by means of the Fourier transform, and then a strong solution to the full nonlinear problem is obtained by a fixed point procedure. We also show that such a solution satisfies the energy equation and is unique within a certain class.
In this paper, we consider the related whole space problem and construct a strong solution with finite kinetic energy for the corresponding steady-state equations. For this, appropriate conditions on the external force have to be imposed (for instance, that it is a function with compact support and null average) together with a smallness condition involving the viscosity of the fluid. First, a linearized version of the problem is analysed by means of the Fourier transform, and then a strong solution to the full nonlinear problem is obtained by a fixed point procedure. We also show that such a solution satisfies the energy equation and is unique within a certain class.
MSC:
| 35Q30 | Navier-Stokes equations |
| 74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 76D03 | Existence, uniqueness, and regularity theory for incompressible viscous fluids |
| 35D35 | Strong solutions to PDEs |
Keywords:
3-D Navier-Stokes equations; whole space; steady solutions; finite kinetic energy; rotation effectReferences:
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