Download PDF by S.N. Antontsev, A.V. Kazhiktov, V.N. Monakhov: Boundary Value Problems in Mechanics of Nonhomogeneous

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

ISBN-10: 0080875432

ISBN-13: 9780080875439

ISBN-10: 0444883827

ISBN-13: 9780444883827

The target of this ebook is to record the result of investigations made via the authors into sure hydrodynamical types with nonlinear platforms of partial differential equations.

The investigations contain the implications pertaining to Navier-Stokes equations of viscous heat-conductive fuel, incompressible nonhomogeneous fluid and filtration of multi-phase blend in a porous medium. The correctness of the preliminary boundary-value difficulties and the qualitative houses of suggestions also are thought of. The ebook is written if you happen to have an interest within the concept of nonlinear partial differential equations and their purposes in mechanics.

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Extra info for Boundary Value Problems in Mechanics of Nonhomogeneous Fluids

Sample text

The Banach theorem. e. then t h e r e e x i s t s a s i n g l e f i x e d p o i n t u = nu on H. The Shauder theorem. If A i s a completly continuous o p e r a t o r and maps a l i m i t e d closed convex s e t K onto i t s e l f , then t h e r e e x i s t s a t l e a s t one f i x e d point u E K. It should be r e c a l l e d t h a t a q u i t e continuous operator i s a continuous operator which maps any bounded closed s e t i n t o a compact one. The Tikhonov Shauder theorem. If K i s a compact convex closed s e t of a Banach apace B and t h e o p e r a t o r A self-maps K continuously i n the norm of B then t h e r e i s a f i x e d point on K.

The s c a l a r product and the norm i n i z ( Q , P o ) can be 24 Chapter I defined by t h e formulas In conclusion l e t us note t h a t t h e functions of the c l a s s have the property of c o n t i n u i t y on t h e average: p Z 1 - f(x>llp,a -A-0 0 6 f ( A > = Ilf(x + A > where f ( x + A > 0 at x + A Lp(Q), # 52- Besides, t h e existence of the generalized d e r i v a t i v e s i s r e l a t e d Namely, if t o the modulus of continuity of t h e function 8 f ( A ) . 8f(A) I c A then f ( x ) E i V i ( a ) and f o r any s t r i c t l y i n t e r n a l subdomain 9' c G?

In this case valid is the presentation 3O. Yvl(d), cp(x> E Wl(Q) , + in which case the tangen- tial components of the vector Y on the boundary r = dR equal t o zero. + The two addents in the right-hand part r o t Y and V q are orthoa$ce, according to the Gauss-Ostrogradsky gonal in L2(Q) formula, for smooth Y and c p : ( r o t Y, q'p),,? = J g ( r o t Y 4 r 3 - n)m. 49) below). In this connection let us cite the formulas of calculating the differential operators V, div, rot in an arbitrary curvi-linear system of coordinates.

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Boundary Value Problems in Mechanics of Nonhomogeneous Fluids by S.N. Antontsev, A.V. Kazhiktov, V.N. Monakhov

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