Global regular motions for compressible barotropic viscous fluids: Stability

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Abstract: We consider viscous compressible barotropic motions in a bounded domain

O m e g a s u b s e t m a t h b b R^{3}

with the Dirichlet boundary conditions for velocity. We assume the existence of some special sufficiently regular solutions

v_{s}

(velocity),

v a r r h o_{s}

(density) of the problem. By the special solutions we can choose spherically symmetric solutions. Let

v

,

v a r r h o

be a~solution to our problem. Then we are looking for differences

u = v - v_{s}

,

e t a = v a r r h o - v a r r h o_{s}

. We prove existence of

u

,

e t a

such that

u, e t a i n L_{i} n f t y (k T, (k + 1) T; H^{2} (O m e g a))

,

u_{t}, e t a_{t} i n L_{i} n f t y (k T, (k + 1) T; H^{1} (O m e g a))

,

u i n L_{2} (k T, (k + 1) T; H^{3} (O m e g a))

,

u_{t} i n L_{2} (k T, (k + 1) T; H^{2} (O m e g a))

, where

T > 0

is fixed and

k i n m a t h b b N c u p 0

. Moreover,

u

,

e t a

are sufficiently small in the above norms. This also means that stability of the special solutions

v_{s}

,

v a r r h o_{s}

is proved. Finally, we proved existence of solutions such that

v = v_{s} + u

,

v a r r h o = v a r r h o_{s} + e t a

.