By S.N. Antontsev, A.V. Kazhiktov, V.N. Monakhov

The target of this booklet is to document the result of investigations made via the authors into definite hydrodynamical versions with nonlinear structures of partial differential equations.

The investigations contain the implications relating Navier-Stokes equations of viscous heat-conductive gasoline, incompressible nonhomogeneous fluid and filtration of multi-phase combination in a porous medium. The correctness of the preliminary boundary-value difficulties and the qualitative homes of strategies also are thought of. The ebook is written should you have an interest within the thought of nonlinear partial differential equations and their functions in mechanics.

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**Extra resources for Boundary Value Problems in Mechanics of Nonhomogeneous Fluids**

**Example text**

V. N. Antontsev [7, 131 for planeparallel flows. At the next stage it was proved [20 to 22, 1561 that in the general three-dimensional case asw well, the Masket-Leverette's equations are reduced to the elliptic-parabolic system, if the saturation s ( x , t) and some mean pressure p(x, t) are taken as the unknown functions. Owing to this circumstance, the existence of the generalised solutions to the boundary-value problems was proved and some significant qualitative properties of these solutions were established: the maximum principle for the saturation s(x, t) ensuring the satisfiability of the inequalities 0 I s I 1 the smoothness properties depending upon the Smoothness of the data of the problem, the uniqueness of the solutions in the nonsingular case, hen 0 < S 5 S(X, t) 5 1 S etc.

11. If t h e sequence {u,} of a H i l b e r t ' s space H weakly then un + u i n t h e converges t o u E H and l i m Ilu,llH = IIuIIH n-+m norm of H This statement can be e a s i l y deduced from t h e R i s s theorem on pres e n t a t i o n of the f u n c t i o n a l s over a H i l b e r t space: any l i n e a r continuous f u n c t i o n a l g on H i s r e a l i z e d a s a s c a l a r product . < 65, u > = (v, UIH, where v E H i s uniquely defined with t h e f u n c t i o n a l g. 3". Theorems on t h e f i x e d p o i n t s of operators.

Cllflla - l 5 a <- 1. Exclusive - k(1- -n/pa)/q,i s ak / non-negative integer, where l/r = k/n + a(l/p i s the case when t h e number 1 and 1 < p < M. Then a < 1. It should be noted t h a t here t h e bounded, i n p a r t i c u l a r , (2 = H* Let UB e s p e c i a l l y s i n g l e out t h e t h e f u n c t i o n s of c l a s s W i ( Q ) , m . 3. Let I U ( x ) h = 0. 21) f o r 5 1. 22) where a = (l/r l/Ci)(l/r (n m)/mn)"in which case i f m < n then q E [ r , mn/(n m)] a t r I mn/(n m) while a t r > mn/(n m> we can t a k e q E [mn/(n m), r].