Very weak solutions of the stationary Navier–Stokes equations for an incompressible fluid past obstacles
We consider the stationary motion of an incompressible Navier–Stokes fluid past obstacles in R3, subject to the given boundary velocity vb, external force f=divF and nonzero constant vector ke1 at infinity. Our main result is the existence of at least one very weak solution v in ke1+L3(Ω) for arbitr...
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| Published in | Nonlinear analysis Vol. 147; pp. 145 - 168 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Elmsford
Elsevier Ltd
01.12.2016
Elsevier BV |
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| Online Access | Get full text |
| ISSN | 0362-546X 1873-5215 |
| DOI | 10.1016/j.na.2016.08.017 |
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| Abstract | We consider the stationary motion of an incompressible Navier–Stokes fluid past obstacles in R3, subject to the given boundary velocity vb, external force f=divF and nonzero constant vector ke1 at infinity. Our main result is the existence of at least one very weak solution v in ke1+L3(Ω) for arbitrary large F∈L3/2(Ω)+L12/7(Ω) provided that the flux of vb−ke1 on the boundary of each body is sufficiently small with respect to the viscosity ν. The uniqueness of very weak solutions is proved by assuming that F and vb−ke1 are suitably small. Moreover, we establish weak and strong regularity results for very weak solutions. In particular, our existence and regularity results enable us to prove the existence of a weak solution v satisfying ∇v∈L3/2(Ω). |
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| AbstractList | We consider the stationary motion of an incompressible Navier-Stokes fluid past obstacles in R3, subject to the given boundary velocity vb, external force f=divF and nonzero constant vector ke1 at infinity. Our main result is the existence of at least one very weak solution v in ke1+L3(Ω) for arbitrary large F∈L3/2(Ω)+L12/7(Ω) provided that the flux of vb-ke1 on the boundary of each body is sufficiently small with respect to the viscosity ν. The uniqueness of very weak solutions is proved by assuming that F and vb-ke1 are suitably small. Moreover, we establish weak and strong regularity results for very weak solutions. In particular, our existence and regularity results enable us to prove the existence of a weak solution v satisfying ∇v∈L3/2(Ω). We consider the stationary motion of an incompressible Navier–Stokes fluid past obstacles in R3, subject to the given boundary velocity vb, external force f=divF and nonzero constant vector ke1 at infinity. Our main result is the existence of at least one very weak solution v in ke1+L3(Ω) for arbitrary large F∈L3/2(Ω)+L12/7(Ω) provided that the flux of vb−ke1 on the boundary of each body is sufficiently small with respect to the viscosity ν. The uniqueness of very weak solutions is proved by assuming that F and vb−ke1 are suitably small. Moreover, we establish weak and strong regularity results for very weak solutions. In particular, our existence and regularity results enable us to prove the existence of a weak solution v satisfying ∇v∈L3/2(Ω). |
| Author | Kim, Hyunseok Kim, Dugyu |
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| Cites_doi | 10.14492/hokmj/1381517172 10.1016/j.jde.2014.02.021 10.1007/s002459911018 10.1007/s00208-004-0573-7 10.1142/S0219530506000735 10.1007/s00021-005-0182-6 10.1016/j.jmaa.2012.05.039 10.1512/iumj.1991.40.40001 10.1007/s00205-010-0340-8 10.1016/j.na.2011.12.032 10.2969/jmsj/04420307 10.1007/BF00253485 10.1007/s002080050149 10.1007/BF02384076 10.1007/s00205-008-0168-7 10.1007/s00208-012-0861-6 |
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| SubjectTerms | Barriers Exterior domains Fluid dynamics Fluid flow Incompressible flow Infinity Navier-Stokes equations Nonlinear equations Regularity Uniqueness Very weak solutions Weak solutions |
| Title | Very weak solutions of the stationary Navier–Stokes equations for an incompressible fluid past obstacles |
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