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Ufa Mathematical Journal, 2011, Volume 3, Issue 2, Pages 79–84 (Mi ufa95)  

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

Kh. G. Umarov

Chechen State University, Groznyi, Russia
References:
Abstract: 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 $L_p(R^2)$, $1<p<+\infty$, 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 $L_p(R^2)$, $1<p<+\infty$, entails that the solution is continuously dependent on the initial data in any finite time interval.
Keywords: free surface of the filtered fluid, strongly continuous semi-groups of operators.
Received: 11.01.2011
Bibliographic databases:
Document Type: Article
UDC: 517.95+517.986.7
Language: English
Original paper language: Russian
Citation: Kh. G. Umarov, “Explicit solution of the Cauchy problem to the equation for groundwater motion with a free surface”, Ufa Math. J., 3:2 (2011), 79–84
Citation in format AMSBIB
\Bibitem{Uma11}
\by Kh.~G.~Umarov
\paper Explicit solution of the Cauchy problem to the equation for groundwater motion with a~free surface
\jour Ufa Math. J.
\yr 2011
\vol 3
\issue 2
\pages 79--84
\mathnet{http://mi.mathnet.ru//eng/ufa95}
\zmath{https://zbmath.org/?q=an:1249.76009}
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    Abstract page:397
    Russian version PDF:130
    English version PDF:27
    References:73
    First page:2
     
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