Geodesic
Stokes and Navier–Stokes problems in the half-space: existence and uniqueness of solutions non converging to a limit at infinity
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 39, Tome 362 (2008), pp. 176-240 
P. Maremonti. Stokes and Navier–Stokes problems in the half-space: existence and uniqueness of solutions non converging to a limit at infinity. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 39, Tome 362 (2008), pp. 176-240. http://geodesic.mathdoc.fr/item/ZNSL_2008_362_a7/
@article{ZNSL_2008_362_a7,
     author = {P. Maremonti},
     title = {Stokes and {Navier{\textendash}Stokes} problems in the half-space: existence and uniqueness of solutions non converging to a~limit at infinity},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {176--240},
     year = {2008},
     volume = {362},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2008_362_a7/}
}
TY  - JOUR
AU  - P. Maremonti
TI  - Stokes and Navier–Stokes problems in the half-space: existence and uniqueness of solutions non converging to a limit at infinity
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2008
SP  - 176
EP  - 240
VL  - 362
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2008_362_a7/
LA  - en
ID  - ZNSL_2008_362_a7
ER  - 
%0 Journal Article
%A P. Maremonti
%T Stokes and Navier–Stokes problems in the half-space: existence and uniqueness of solutions non converging to a limit at infinity
%J Zapiski Nauchnykh Seminarov POMI
%D 2008
%P 176-240
%V 362
%U http://geodesic.mathdoc.fr/item/ZNSL_2008_362_a7/
%G en
%F ZNSL_2008_362_a7

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

We study the Cauchy problem and the initial boundary value problem in the half-space of the Stokes and Navier–Stokes equations. We furnish existence and uniqueness of classical solutions $(u,\pi)$ (meaning at least $C^2\times C^1$ smooth with respect to the space variable and $C^1\times C^0$ smooth with respect to the time variable) without requiring of convergence at infinity. A priori the fields $u$ and $\pi$ are nondecreasing at infinity. In the case of the Stokes problem we prove the existence, for any $t>0$, and uniqueness of solutions with kinetic field $u=O([1+t^\frac\beta2][1+|x|^\beta])$ and pressure field $\pi=O([1+t^\frac\beta2][1+|x|^\beta]|x|^\gamma)$, for some $\beta\in(0,1)$ and $\gamma\in(0,1-\beta)$. In the case of the Navier–Stokes equations we prove the existence (local in time) and the uniqueness of classical solutions to the Navier–Stokes equations assuming an initial data only continuous and bounded, proving that, for any $t\in(0,T)$, the kinetic field $u(x,t)$ is bounded and, for any $\gamma\in(0,1)$, the pressure field $\pi(x,t)=O(1+|x|^\gamma)$. Bibl. – 20 titles.

[1] Zap. Nauch. Sem. POMI, 318, 2004, 147–202 | DOI | MR | Zbl

[2] W. Desch, M. Hieber, J. Prüss, “$L^p$-Theory of the Stokes equation in a half space”, J. Evol. Equation, 1 (2001), 115–142 | DOI | MR | Zbl

[3] G. P. Galdi, P. Maremonti, “A uniquenesss theorem for viscous fluid motions in exterior domains”, Arch. Ratioanl Mech. Anal., 91 (1986), 375–384 | DOI | MR | Zbl

[4] G. P. Galdi, S. Rionero, “On the best conditions on the gradient of pressure for uniqueness of viscous flows in the whole space”, Pac. J. Math., 104 (1983), 77–83 | MR | Zbl

[5] Y. Giga, K. Inui, Sh. Matsui, “On the Cauchy problem for the Navier–Stokes equations with nondecaying initial data”, Quad. Mat., 4 (1999), 27–68 | MR | Zbl

[6] J. Kato, “The uniqueness of nondecaying solutions for the Navier–Stokes equations”, Arch. Rational Mech. Anal., 169 (2003), 159–175 | DOI | MR | Zbl

[7] M. Hieber, A. Rhandi, O. Sawada, “The Navier-Stokes flow for globally Lipschitz continuous initial data” (to appear)

[8] M. Hieber, O. Sawada, “The Navier–Stokes equations in $\mathbb R^n$ with linearly growing initial data”, Arch. Rational Mech. Anal., 175 (2005), 269–285 | DOI | MR | Zbl

[9] G. H. Knightly, “A note on the Cauchy problem for the Navier–Stokes equations”, SIAM J. Appl. Math., 18 (1970), 641–644 | DOI | MR | Zbl

[10] G. H. Knightly, Stability of uniform solutions of the Navier–Stokes equations in $n$-dimensions, Math. Research Center Technical Summary Report # 1085, Wisconsin University, 1971

[11] H. Koch, V. A. Solonnikov, “$L_q$-estimtes of the first-order derivatives of solutions to the nonstationary Stokes problem”, Nonlinear problems of mathematical physics and related topics, V. I, International Math. Series, 1, Kluwer Academic, Plenum Publisher, New York, 2002, 203–218 | MR | Zbl

[12] O. A. Ladyzhenskaya, N. N. Uralceva, Linear and quasilinear elliptic equations, Academic Press, New York–London, 1968 | MR | Zbl

[13] P. Maremonti, G. Starita, “On the nonstationary Stokes equations in half-space with continuous initial data”, Zap. Nauch. Sem. POMI, 295, 2003, 118–167 | MR | Zbl

[14] P. Maremonti, “On the uniqueness of bounded weak solutions to the Navier–Stokes Cauchy problem” (to appear)

[15] P. Muratori, “Su un problema esterno relativo alle equazioni di Navier–Stokes”, Rend. Sem. Mat. Univ. Padova, 46 (1971), 53–100 | MR

[16] O. Sawada, T. Usui, “The Navier–Stokes equations for linearly growing velocity with nondecaying initial disturbance” (to appear)

[17] V. A. Solonnikov, “Estimates for solutions to the nonstationary Navier–Stokes equations”, Zap. Nauch. Sem. LOMI, 38, 1973, 153–231 | MR | Zbl

[18] V. A. Solonnikov, “Estimates for solutions of the nonstationary Stokes problem in anisotropic Sobolev spaces and estimates for the resolvent of the Stokes operator”, Usp. Mat. Nauk, 58:2 (2003), 123–156 | MR | Zbl

[19] V. A. Solonnikov, “On nonstationary Stokes problem and Navier–Stokes problem in a half-space with initial data nondecreasing at infinity”, J. Math. Sci., 114 (2003), 1726–1740 | DOI | MR | Zbl

[20] V. A. Solonnikov, “Weighted Schauder estimates for evolution Stokes problem”, Ann. Univ. Ferrara Sez. VII Sci. Mat., 52 (2006), 137–172 | DOI | MR | Zbl