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On nonstationary motion of a compressible barotropic viscous fluid bounded by a free surface

100%

Zajączkowski W. M.

Instytut Matematyczny Polskiej Akademii Nauk

We consider the motion of a viscous compressible barotropic fluid in $ℝ^3$ bounded by a free surface which is under constant exterior pressure. For a given initial density, initial domain and initial velocity we prove the existence of local-in-time highly regular solutions. Next assuming that the initial density is sufficiently close to a constant, the initial pressure is sufficiently close to the external pressure, the initial velocity is sufficiently small and the external force vanishes we prove the existence of global-in-time solutions which satisfy, at any moment of time, the properties prescribed at the initial moment.

CONTENTS 1. Introduction.......................................5 2. Global estimates and relations........11 3. Local existence...............................16 4. Global differential inequality............44 5. Korn inequality................................81 6. Global existence.............................89 References.......................................100

Existence and regularity of solutions of some elliptic system in domains with edges

100%

Zajączkowski W. M.

Instytut Matematyczny Polskiej Akademii Nauk

CONTENTS 1. Introduction.......................................................................5 2. Notation and auxiliary results............................................9 3. Statement of the problem (1.1)-(1.3)..............................20 4. The problem (3.14).........................................................22 5. Auxiliary results in $D_ϑ$...............................................34 6. Existence of solutions of (3.14) in $H^k_μ(D_ϑ)$............41 7. Green function................................................................52 8. The problem (3.13) in $L^k_{p,μ}(D_ϑ)$ spaces............59 9. The problem (3.13) in weighted Hölder spaces...............67 10. The problem (1.1)-(1.3) in a bounded domain Ω..........75 Appendix. The distinguished case: μ+2/p∊ℤ......................86 References.........................................................................90

$L_∞$-estimates for solutions of nonlinear parabolic systems with gradient linear growth

92%

Zajączkowski W. M.

nr 1

491-501

Existence of weak solutions and an $L_∞$-estimate are shown for nonlinear nondegenerate parabolic systems with linear growth conditions with respect to the gradient. The $L_∞$-estimate is proved for equations with coefficients continuous with respect to x and t in the general main part, and for diagonal systems with coefficients satisfying the Carathéodory condition.

Preface, Contents

30%

Janeczko S. , Zajączkowski W. M. , Ziemian B.

nr 1

1-8