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Russian participants
Research of problems of the forecast of an
ecological condition of an atmosphere and
ocean on the basis of methods of mathematical modelling is
reduced to the solution initial - boundary value problems
for the equations describing distribution of an impurity.
The specified boundary value problems contain a number of hydrodynamical
parameters,
and also the functions describing density of pollutant sources.
The solution of problems of protection
of an environment from emissions of harmful impurity
results in necessity of the decision mathematical methods
of problems of detection of unknown pollutant sources and
identification of their parameters.
On the statement the specified problems concern to a class of inverse
problems.
In the strict mathematical formulation they consist in a finding of
parameters of
a unknown pollutant source under the measured information on the field of
concentration created by this source in some area,
and also under the certain information on a source.
Besides, the important role in appendices is played with
extremal problems of the theory of distribution of an impurity.
In these problems it is entered determined cost functional
and it is required to minimize it, for example, due to a choice of density
of pollutant sources. It is necessary to note,
that the solution of inverse problems can be shown to the solution of
extremal
problems at the appropriate choice cost functionals.
The purpose of the present work is the theoretical analysis of
extremal problems for the mass transfer equations in
the viscous fluid, considered in the limited area with Lipschetz boundary.
The specified problems are formulated as a problem of minimization certain
cost functionals on weak solutions of an origin boundary value problem.
As required parameters the distributed density of polutant
sources and distribution of
concentration to parts of bordary.
Theorems of existence and uniqueness of solutions of
extremal problems are proved, optimality systems as
for general nonlinear, and linear mass tranfer models
are deduced and analyzed. In the latter case
the simple explicit formula for a
finding of a minimum linear concerning
concentration cost functional is deduced.
The work was supported by the Russian Found of the Fundamental Researches (a code of the project 99-01-00214).
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