Equations of fluid dynamics-free surface problems. (English) Zbl 0595.76068
We shall discuss several problems in fluid dynamics, especially free surface problems. Firstly we consider an initial value problem for the compressible Euler equation describing motions of a gaseous star. It has an existence theorem which is local in time - a restricted result - and it must be solved in general as a free boundary problem. Secondly we consider a one-dimensional viscous gas motion under gravity with an interface with vacuum. A generalized solution exists globally in time.
Thirdly we study water waves, i.e., free surface problems of an incompressible Euler equation under gravity. Local existence theorems give mathematical justifications of several approximation theories, i.e., Friedrichs expansion, shallow water wave equations, Boussinesq’s equation and the Korteweg-de Vries equation.
Lastly we consider free surface problems of the Navier-Stokes equation under gravity. A global in time solution exists and decays to a state of equilibrium with the rate \(t^{-1/2}\) as the time tends to infinity. A global existence theorem without consideration of surface tension is not known.
Thirdly we study water waves, i.e., free surface problems of an incompressible Euler equation under gravity. Local existence theorems give mathematical justifications of several approximation theories, i.e., Friedrichs expansion, shallow water wave equations, Boussinesq’s equation and the Korteweg-de Vries equation.
Lastly we consider free surface problems of the Navier-Stokes equation under gravity. A global in time solution exists and decays to a state of equilibrium with the rate \(t^{-1/2}\) as the time tends to infinity. A global existence theorem without consideration of surface tension is not known.
MSC:
| 76Nxx | Compressible fluids and gas dynamics |
| 35Qxx | Partial differential equations of mathematical physics and other areas of application |
| 76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 85A05 | Galactic and stellar dynamics |
Keywords:
local time existence; free surface problems; initial value problem; compressible Euler equation; gaseous star; free boundary problem; one- dimensional viscous gas motion under gravity; interface with vacuum; incompressible Euler equation; local existence theorems; approximation theories; Friedrichs expansion; shallow water wave equations; Boussinesq’s equation; Korteweg-de Vries equation; Navier-Stokes equation; global in time solution; state of equilibrium; global existence theoremReferences:
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