MR
Reinhard FARWIG, Yoshikazu GIGA and Pen-Yuan HSU
Reinhard FARWIG, Yoshikazu GIGA and Pen-Yuan HSU
Initial Values for the Navier-Stokes Equations in Spaces with Weights in Time
Funkcialaj Ekvacioj. Serio Internacia
59
2016
199--216
http://fe.math.kobe-u.ac.jp/FE/FullPapers/59-2/59_199.pdf
http://www.ams.org/mathscinet-getitem?mr=MR
We consider the nonstationary Navier-Stokes system in a smooth bounded domain $\Omega\subset\mathbf{R}^3$ with initial value $u_0\in L^2_\sigma(\Omega)$. It is an important question to determine the optimal initial value condition in order to prove the existence of a unique local strong solution satisfying Serrin's condition. In this paper, we introduce a weighted Serrin condition that yields a necessary and sufficient initial value condition to guarantee the existence of local strong solutions $u(\cdot)$ contained in the weighted Serrin class $\int_0^T(\tau^\alpha\|u(\tau)\|_q)^s d\tau<\infty$ with $2/s+3/q=1-2\alpha$, $0<\alpha<1/2$. Moreover, we prove a restricted weak-strong uniqueness theorem in this Serrin class.
Instationary Navier-Stokes system, Initial values, Local strong solutions, Weighted Serrin condition, Well-chosen weak solutions, Restricted Serrin's uniqueness theorem.
35Q30, 76D05.
59-199
2016
Initial Values for the Navier-Stokes Equations in Spaces with Weights in Time
Reinhard FARWIG, Yoshikazu GIGA and Pen-Yuan HSU
Reinhard FARWIG, Yoshikazu GIGA and Pen-Yuan HSU
1
Amann, H.
Linear and Quasilinear Parabolic Problems, Vol. I
Monographs in Mathematics, 89. Birkhäuser Boston, Inc., Boston, MA
1995
MR1345385
2
Amann, H.
On the strong solvability of the Navier-Stokes equations
J. Math. Fluid Mech.
2
2000
16-98
MR1755865
3
Amann, H.
Nonhomogeneous Navier-Stokes equations with integrable low-regularity data
Int. Math. Ser. (N. Y.), 2, Kluwer/Plenum, New York
2002
pp. 1-28
MR1971987
4
Beirão da Veiga, H.
On the construction of suitable weak solutions to the Navier-Stokes equations via a general approximation theorem
J. Math. Pures Appl. (9)
64
1985
321-334
MR0823407
5
Farwig, R.
On regularity of weak solutions to the instationary Navier-Stokes system - a review on recent results
Ann. Univ. Ferrara Sez. VII Sci. Mat.
60
2014
91-122
MR3208788
6
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
7
Farwig, R.; Sohr, H.
Generalized resolvent estimates for the Stokes system in bounded and unbounded domains
J. Math. Soc. Japan
46
1994
607-643
MR1291109
8
Farwig, R.; Sohr, H.
Optimal initial value conditions for the existence of local strong solutions of the Navier-Stokes equations
Math. Ann.
345
2009
631-642
MR2534111
9
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
10
Fujita, H.; Kato, T.
On the Navier-Stokes initial value problem
Arch. Rational Mech. Anal.
16
1964
269-315
MR0166499
11
Galdi, G. P.
An Introduction to the Mathematical Theory of the Navier-Stokes Equations; Nonlinear Steady Problems
Springer Tracts in Natural Philosophy, 39, Springer-Verlag, New York
1994
MR1284206
12
Giga, Y.
Analyticity of the semigroup generated by the Stokes operator in $L^r$-spaces
Math. Z.
178
1981
297-329
MR0635201
13
Giga, Y.
Solution for semilinear parabolic equations in $L^p$ and regularity of weak solutions for the Navier-Stokes system
J. Differential Equations
62
1986
186-212
MR0833416
14
Giga, Y.; Miyakawa, T.
Solutions in $L_r$ of the Navier-Stokes initial value problem
Arch. Rational Mech. Anal.
89
1985
267-281
MR0786550
15
Giga, Y.; Sohr, H.
Abstract $L^q$-estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains
J. Funct. Anal.
102
1991
72-94
MR1138838
16
Guermond, J.-L.
Faedo-Galerkin weak solutions of the Navier-Stokes equations with Dirichlet boundary conditions are suitable
J. Math. Pures Appl. (9)
88
2007
87-106
MR2334774
17
Haak, B. H.; Kunstmann, P. C.
On Kato's method for Navier-Stokes equations
J. Math. Fluid Mech.
11
2009
492-535
MR2574794
18
Heywood, J. G.
The Navier-Stokes equations: on the existence, regularity and decay of solutions
Indiana Univ. Math. J.
29
1980
639-681
MR0589434
19
Hopf, E.
Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen
Math. Nachr.
4
1951
213-231
MR0050423
20
Kato, T.
Strong $L^p$-solutions of the Navier-Stokes equation in $\mathbf{R}^m$, with applications to weak solutions
Math. Z.
187
1984
471-480
MR0760047
21
Kiselev, A. A.; Ladyzhenskaya, O. A.
On the existence and uniqueness of solutions of the non-stationary problems for flows of non-compressible fluids
Amer. Math. Soc. Transl. (2)
24
1963
79-106
MR0100448
22
Kozono, H.; Yamazaki, M.
Local and global unique solvability of the Navier-Stokes exterior problem with Cauchy data in the space $L^{n,\infty}$
Houston J. Math.
21
1995
755-799
MR1368344
23
Kunstmann, P. C.
Navier-Stokes equations on unbounded domains with rough initial data
Czechoslovak Math. J.
60(135)
2010
297-313
MR2657950
24
Leray, J.
Sur le mouvement d'un liquide visqueux emplissant l'espace
Acta Math.
63
1934
193-248
MR1555394
25
Miyakawa, T.
On the initial value problem for the Navier-Stokes equations in $L^p$-spaces
Hiroshima Math. J.
11
1981
9-20
MR0606832
26
Sohr, H.
The Navier-Stokes Equations. An Elementary Functional Analytic Approach
Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel
2001
MR1928881
27
Solonnikov, V. A.
Estimates for solutions of nonstationary Navier-Stokes equations
J. Soviet Math.
8
1977
467-529
MR0415097
28
Stein, E. M.; Weiss, G.
Fractional integrals on $n$-dimensional Euclidean space
J. Math. Mech.
7
1958
503-514
MR0098285
29
Strichartz, R. S.
$L^p$ estimates for integral transforms
Trans. Amer. Math. Soc.
136
1969
33-50
MR0234321
30
Triebel, H.
Interpolation Theory, Function Spaces, Differential Operators
North-Holland Mathematical Library, 18. North-Holland Publishing Co., Amsterdam-New York
1978
MR0503903
31
Varnhorn, W.
The Stokes Equations
Mathematical Research, 76. Akademie-Verlag, Berlin
1994
MR1282728