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Free boundary problem for a viscous compressible flow with a surface tension. (English) Zbl 0752.35096

Constantin Carathéodory: an international tribute. Vol. II, 1270-1303 (1991).
[For the entire collection see Zbl 0728.00004.]
The paper considers the motion of an isolated mass of a viscous compressible fluid subject to external forces \(\underline f(\underline x,t)\) and self gravitation forces of potential \(U=\int_{\Omega_ t}\rho(y,t)/| x-y| dy\) where \(\Omega_ z\) is the domain occupied by the fluid at time \(t\). Inside \(\Omega_ t\) the velocity vector field \(\underline v(\underline x,t)\) and the density \(\rho(x,t)>0\) satisfy the Navier Stokes equations and on \(\partial\Omega_ Z\) the following condition holds: \(\tilde T\underline\eta=-p_ 0(x,t)\underline\eta+Z\sigma H\underline\eta\) where \(p_ 0\) is the given external pressure, \(\sigma\) is the surface term, \(H\) is the mean curvature, and \(\tilde T\) is the stress tension, related to the pressure, velocity and strain term \(\tilde S\left(S_{ij}={\partial v_ i\over\partial x_ j}+{\partial v_ j\over\partial x_ i}\right)\) by the relationship \(\tilde T=(-p(\rho)+\mu'\text{div} \underline v)\tilde I+\mu\tilde S\).
Passing to Lagrangian coordinates, the problem is transformed into an initial boundary value problem on a fixed domain \(\Omega\) and the motion of the free boundary \(\partial\Omega_ t\) is incorporated in the differential operators appearing in the problem. A local theorem of existence and uniqueness is proved under suitable assumption on the data.

MSC:

35R35 Free boundary problems for PDEs
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
35Q30 Navier-Stokes equations
35K55 Nonlinear parabolic equations

Biographic References:

Carathéodory, Constantin

Citations:

Zbl 0728.00004
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