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On propagation of wave perturbations in shear flows with free boundary. (Russian) Zbl 1073.76542

The authors study a mathematical model of long-wave propagation of perturbations which describes a plane-parallel flow of a layer of a viscous incompressible liquid with free boundary in gravity field. From the mathematical viewpoint, this problem reduces to the following initial-boundary value problem: \[ \begin{gathered} u_t +uu_x +vu_y +gh_x =\nu u_{yy}, \quad u_x+v_y=0,\quad h_t+\Bigg(\int\limits_0^h u \,dy\Bigg)_{\!x}=0, \\ \nu u(x,0,t)=0, \quad v(x,0,t)=0,\quad \nu u_y(x,h,t)=0,\quad u(x,y,0)=u^0(x,y),\quad h(x,0)=h^0(x). \end{gathered} \] Here \(u,v\) are the components of the velocity vector, \(t\) is time, \(x,y\) are the cartesian coordinates in the plane, \(h(x,t)\) is the depth of the liquid layer, \(g\) is gravity acceleration, \(\nu\) is the viscosity coefficient. The authors prove uniqueness of a solution to the Cauchy problem for equations of long waves on the surface of an ideal vortex liquid, and uniqueness of a solution to the initial-boundary value problem for a linearized model of long-wave propagation in a viscous liquid.

MSC:

76D05 Navier-Stokes equations for incompressible viscous fluids
76D33 Waves for incompressible viscous fluids
35R35 Free boundary problems for PDEs
35Q72 Other PDE from mechanics (MSC2000)
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