MR2730622
R. FARWIG, H. KOZONO and H. SOHR
R. FARWIG, H. KOZONO and H. SOHR
Global Weak Solutions of the Navier-Stokes System with Nonzero Boundary Conditions
Funkcialaj Ekvacioj. Serio Internacia
53
2010
231--247
http://fe.math.kobe-u.ac.jp/FE/FullPapers/53-2/53_231.pdf
http://www.ams.org/mathscinet-getitem?mr=MR2730622
Consider the Navier-Stokes equations in a smooth bounded domain $\Omega\subset \mathbf{R}^3$ and a time interval $[0,T)$, $0<T\leq\infty$. It is well-known that there exists at least one global weak solution $u$ with vanishing boundary values $u|_{\partial\Omega}=0$ for any given initial value $u_0\in L_\sigma^2(\Omega)$, external force $f=\div F$, $F\in L^2(0,T;L^2(\Omega))$, and satisfying the strong energy inequality. Our aim is to extend this existence result to a much larger class of global in time “'Leray-Hopf type” weak solutions $u$ with nonzero boundary values $u|_{\partial\Omega}=g\in W^{1/2,2}(\partial\Omega)$. As for usual weak solutions we do not need any smallness condition on $g$; indeed, our generalized weak solutions $u$ exist globally in time. The solutions will satisfy an energy estimate with exponentially increasing terms in time, but for simply connected domains the energy increases at most linearly in time.
Navier-Stokes equations, Weak solution, Nonhomogeneous boundary values, Strong energy inequality.
76D05, 35Q30, 35J65.
53-231
2010
Global Weak Solutions of the Navier-Stokes System with Nonzero Boundary Conditions
R. FARWIG, H. KOZONO and H. SOHR
R. FARWIG, H. KOZONO and H. SOHR
1
Amann, H.
On the strong solvability of the Navier-Stokes equations
J. Math. Fluid Mech.
2
2000
16-98
MR1755865
2
Amann, H.
Navier-Stokes equations with nonhomogeneous Dirichlet data
J. Nonlinear Math. Phys.
10 Suppl. 1
2003
1-11
MR2063541
3
Farwig, R.; Galdi, G. P.; Sohr, H.
A new class of weak solutions of the Navier-Stokes equations with nonhomogeneous data
J. Math. Fluid Mech.
8
2006
423-444
MR2258419
4
Farwig, R.; Kozono, H.; Sohr, H.
Very weak, weak and strong solutions to the instationary Navier-Stokes system
Topics on partial differential equations, ed. by P. Kaplický, Š. Nečasová. J. Nečas Center Math. Model., Lecture Notes, Vol. 2, Praha
2007
1-54
MR
5
Farwig, R.; Kozono, H.; Sohr, H., Global Leray-Hopf weak wolutions of the Navier-Stokes equations with nonzero time-dependent boundary values. Manuscript 2010
6
Farwig, R.; Sohr, H.; Varnhorn, W.
On optimal initial value conditions for local strong solutions of the Navier-Stokes equations
Ann. Univ. Ferrara Sez. VII Sci. Mat.
55
2009
89-110
MR2506065
7
Fursikov, A.V.; Gunzburger, M. D.; Hou, L. S.
Inhomogeneous boundary value problems for the three-dimensional evolutionary Navier-Stokes equations
J. Math. Fluid Mech.
4
2002
45-75
MR1891074
8
Galdi, G. P.
An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I
Springer Verlag, New York
1994
MR1284205
9
Galdi, G. P.
An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. II
Springer Verlag, New York
1994
MR1284206
10
Grubb, G.
Nonhomogeneous Dirichlet Navier-Stokes problems in low regularity $L_p$ Sobolev spaces
J. Math. Fluid Mech.
3
2001
57-81
MR1830655
11
Heywood, J. G.
The Navier-Stokes Equations: On the existence, regularity and decay of solutions
Indiana Univ. Math. J.
29
1980
639-681
MR0589434
12
Hopf, E.
Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen
Math. Nachr.
4
1950
213-231
MR0050423
13
Ladyzhenskaya, O. A.
The Mathematical Theory of Viscous Incompressible Flow
Gordon and Breach, New York
1969
MR0254401
14
Leray, J.
Sur le mouvement d'un liquide visqueux emplissant l' espace
Acta Math.
63
1934
193-248
MR1555394
15
Raymond, J. P.
Stokes and Navier-Stokes equations with nonhomogeneous boundary conditions
Ann. Inst. H. Poincaré Anal. Non Linéaire
24
2007
921-951
MR2371113
16
Sohr, H.
The Navier-Stokes Equations
Birkhäuser Verlag, Basel
2001
MR1928881
17
Solonnikov, V. A.
Estimates for solutions of nonstationary Navier-Stokes equations
J. Soviet Math.
8
1977
467-529
MR0415097