# American Institute of Mathematical Sciences

• Previous Article
$L^\infty$-estimates for parabolic systems with VMO-coefficients
• DCDS-S Home
• This Issue
• Next Article
Local existence theorems with unbounded set of input data and unboundedness of stable invariant manifolds for 3D Navier-Stokes equations
June  2010, 3(2): 291-297. doi: 10.3934/dcdss.2010.3.291

## Remarks on the $L^p$-approach to the Stokes equation on unbounded domains

 1 Technische Universität Darmstadt, Fachbereich Mathematik, Schlossgartenstr. 7, D-64289 Darmstadt, Germany, Germany, Germany

Received  April 2009 Revised  October 2009 Published  April 2010

Consider a domain $\Omega \subset \R^n$ with uniform $C^3$-boundary and assume that the Helmholtz projection $P$ exists on $L^p(\Omega)$ for some $1 < p < \infty$. Of concern are recent results on the Stokes operator in $L^p(\Omega)$ generating an analytic semigroup on $L^p(\Omega)$ and admitting maximal $L^p$-$L^q$-regularity.
Citation: Matthias Geissert, Horst Heck, Matthias Hieber, Okihiro Sawada. Remarks on the $L^p$-approach to the Stokes equation on unbounded domains. Discrete and Continuous Dynamical Systems - S, 2010, 3 (2) : 291-297. doi: 10.3934/dcdss.2010.3.291
 [1] Lianzhang Bao, Wenxian Shen. Logistic type attraction-repulsion chemotaxis systems with a free boundary or unbounded boundary. I. Asymptotic dynamics in fixed unbounded domain. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 1107-1130. doi: 10.3934/dcds.2020072 [2] Donatella Donatelli, Eduard Feireisl, Antonín Novotný. On incompressible limits for the Navier-Stokes system on unbounded domains under slip boundary conditions. Discrete and Continuous Dynamical Systems - B, 2010, 13 (4) : 783-798. doi: 10.3934/dcdsb.2010.13.783 [3] Carlos Fresneda-Portillo. A new family of boundary-domain integral equations for the diffusion equation with variable coefficient in unbounded domains. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5097-5114. doi: 10.3934/cpaa.2020228 [4] Helmut Abels. Nonstationary Stokes system with variable viscosity in bounded and unbounded domains. Discrete and Continuous Dynamical Systems - S, 2010, 3 (2) : 141-157. doi: 10.3934/dcdss.2010.3.141 [5] Xiaoming He, Xin Zhao, Wenming Zou. Maximum principles for a fully nonlinear nonlocal equation on unbounded domains. Communications on Pure and Applied Analysis, 2020, 19 (9) : 4387-4399. doi: 10.3934/cpaa.2020200 [6] Reinhard Farwig, Paul Felix Riechwald. Regularity criteria for weak solutions of the Navier-Stokes system in general unbounded domains. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 157-172. doi: 10.3934/dcdss.2016.9.157 [7] Reinhard Farwig, Yasushi Taniuchi. Uniqueness of backward asymptotically almost periodic-in-time solutions to Navier-Stokes equations in unbounded domains. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1215-1224. doi: 10.3934/dcdss.2013.6.1215 [8] Sylvie Monniaux. Various boundary conditions for Navier-Stokes equations in bounded Lipschitz domains. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1355-1369. doi: 10.3934/dcdss.2013.6.1355 [9] Zhong-Qing Wang, Jing-Xia Wu. Generalized Jacobi rational spectral methods with essential imposition of Neumann boundary conditions in unbounded domains. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 325-346. doi: 10.3934/dcdsb.2012.17.325 [10] Peter I. Kogut. On approximation of an optimal boundary control problem for linear elliptic equation with unbounded coefficients. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2105-2133. doi: 10.3934/dcds.2014.34.2105 [11] Giorgio Fusco, Francesco Leonetti, Cristina Pignotti. On the asymptotic behavior of symmetric solutions of the Allen-Cahn equation in unbounded domains in $\mathbb{R}^2$. Discrete and Continuous Dynamical Systems, 2017, 37 (2) : 725-742. doi: 10.3934/dcds.2017030 [12] Esa V. Vesalainen. Rellich type theorems for unbounded domains. Inverse Problems and Imaging, 2014, 8 (3) : 865-883. doi: 10.3934/ipi.2014.8.865 [13] Paulo Cesar Carrião, Olimpio Hiroshi Miyagaki. On a class of variational systems in unbounded domains. Conference Publications, 2001, 2001 (Special) : 74-79. doi: 10.3934/proc.2001.2001.74 [14] Arnaud Heibig, Mohand Moussaoui. Exact controllability of the wave equation for domains with slits and for mixed boundary conditions. Discrete and Continuous Dynamical Systems, 1996, 2 (3) : 367-386. doi: 10.3934/dcds.1996.2.367 [15] Marc Briant. Perturbative theory for the Boltzmann equation in bounded domains with different boundary conditions. Kinetic and Related Models, 2017, 10 (2) : 329-371. doi: 10.3934/krm.2017014 [16] Michal Beneš. Mixed initial-boundary value problem for the three-dimensional Navier-Stokes equations in polyhedral domains. Conference Publications, 2011, 2011 (Special) : 135-144. doi: 10.3934/proc.2011.2011.135 [17] Alessio Falocchi, Filippo Gazzola. Regularity for the 3D evolution Navier-Stokes equations under Navier boundary conditions in some Lipschitz domains. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1185-1200. doi: 10.3934/dcds.2021151 [18] Bixiang Wang, Xiaoling Gao. Random attractors for wave equations on unbounded domains. Conference Publications, 2009, 2009 (Special) : 800-809. doi: 10.3934/proc.2009.2009.800 [19] Martino Prizzi. A remark on reaction-diffusion equations in unbounded domains. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 281-286. doi: 10.3934/dcds.2003.9.281 [20] Lingwei Ma, Zhenqiu Zhang. Monotonicity for fractional Laplacian systems in unbounded Lipschitz domains. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 537-552. doi: 10.3934/dcds.2020268

2021 Impact Factor: 1.865