Explicit solution of the Cauchy problem to the equation for groundwater motion with a~free surface

Kh. G. Umarov

Ufa mathematical journal, Tome 3 (2011) no. 2, pp. 79-84

Résumé

A linear partial differential equation modelling evolution of a free surface of the filtered fluid $$ \lambda u_t-\Delta_2u_t=\alpha\Delta_2u-\beta\Delta^2_2u+f $$ is considered. Here $u(x,y,t)$ is the searched function characterizing the fluid pressure, $f=f(x,y,t)$ is the given function calculating an external influence on the filtration flow, $\Delta_2=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}$ is the Laplace differential operator, $\lambda,\alpha,\beta$ are positive constants depending on characteristics of the watery soil. The explicit solution to the Cauchy problem for the above linear partial differential equation is obtained in the space $L_p(R^2)$, $1$, by means of reducing the considered filtration problem to the abstract Cauchy problem in a Banach space. Solution of the corresponding homogeneous equation with respect to the temporary variable $t$ satisfies the semi-group property. The resulting estimation of the solution to the Cauchy problem in the space $L_p(R^2)$, $1$, entails that the solution is continuously dependent on the initial data in any finite time interval.

Détail

Keywords: free surface of the filtered fluid, strongly continuous semi-groups of operators.