On the problem of air pollution (Q1373060)

Let \(D\) be a cylindrical region in the space \(\mathbb{R}^3\) with sufficiently smooth boundary. Denote by \(\Omega\) the set \(D\times(0, T)= \{(x,t): x=(x_1, x_2, x_3)\in D\), \(0<t<T<\infty\}\). The process of pollutant transport and diffusion in the atmosphere is described by the following equations \[ LF={\partial F\over\partial t}-\text{div} \nabla F+\text{div }\vec VF+ \sigma F=f,\quad \text{div }\vec V= 0\quad\text{in }\Omega, \] where \(F= F(x,t)\) is the concentration of the pollutant, \(\vec V=(u,v,w)\) is the wind velocity, \(f= f(x,t)\) is the power of the source, \(\lambda=\lambda(x)\) is the diffusion coefficient, and \(\sigma=\sigma(x)\) is the rate of chemical decay transformation, and \(\lambda(x)\) and \(\sigma(x)\) are continuous functions in \(D\), \(0\leq\lambda(x)\leq\lambda_0\); \(0\leq \sigma(x)\leq\sigma_0\); \(\lambda_0\), \(\sigma_0\) being constants. We consider the correctness of the posed problem with boundary conditions \[ F= F_1\quad\text{on }\partial D_1\quad\text{and}\quad V_n\vec V\cdot\vec n<0,\quad{\partial F\over\partial n}= 0\quad\text{on }\partial D_2\quad\text{and} \quad V_n\geq 0, \] \[ {\partial F\over\partial n}+\beta F=0\quad\text{on }\partial D_3\quad\text{and}\quad V_n\geq 0, \] where \(\partial D_1\cup\partial D_2\cup\partial D_3=\partial D\), \(t\in(0, T)\).