A class of solutions to stationary Stokes and Navier-Stokes equations with boundary data in W−1/q,q

Abstract.We develop a theory for a general class of very weak solutions to stationary Stokes and Navier-Stokes equations in a bounded domain Ω with boundary ∂Ω of class C2,1, corresponding to boundary data in the distribution space W−1/q,q(∂Ω), 1<q<∞. These solutions exist and are unique (for small data, in the nonlinear case) in their class of existence, and satisfy a correponding estimate in terms of the data. Moreover, they become regular if the data are regular. To our knowledge, the only existence result for solutions attaining such boundary data is due to Giga, [16], Proposition 2.2, for the Stokes case. However, the methods and the approach used in the present paper are different than Giga’s and cover more general issues, including the nonlinear Navier-Stokes equations and the precise way in which the boundary data are attained by the solutions. We also introduce, in the last section, a further generalization of the solution class.

[1]  Hantaek Bae,et al.  On the Navier-Stokes equations , 2009 .

[2]  C. Simader,et al.  The Dirichlet problem for the Laplacian in bounded and unbounded domains : a new approach to weak, strong and (2+k)-solutions in Sobolev-type spaces , 1996 .

[3]  R. Temam.  Navier-Stokes Equations , 1977 .

[4]  M. Yamazaki.  The Navier-Stokes equations in the weak- $L^n$ space with time-dependent external force , 2000 .

[5]  Hi Jun Choe,et al.  The Stokes problem for Lipschitz domains , 2002 .

[6]  V. A. Solonnikov,et al.  Estimates for solutions of nonstationary Navier-Stokes equations , 1977 .

[7]  C. Foiaș.  Une remarque sur l’unicité des solutions des équations de Navier-Stokes en dimension $n$ , 1961 .

[8]  Yoshikazu Giga,et al.  Analyticity of the semigroup generated by the Stokes operator inLr spaces , 1981 .

[9]  Yoshikazu Giga,et al.  Domains of fractional powers of the Stokes operator in Lr spaces , 1985 .

[10]  M. E. Bogovskii.  Solution of the first boundary value problem for the equation of continuity of an incompressible medium , 1979 .

[11]  H. Amann.  On the Strong Solvability of the Navier—Stokes Equations , 2000 .

[12]  Jason J. Sharples,et al.  Linear and quasilinear parabolic equations in Sobolev space , 2004 .

[13]  G. Grubb.  Nonhomogeneous Dirichlet Navier—Stokes Problems in Low Regularity Lp Sobolev Spaces , 2001 .

[14]  Giovanni P. Galdi,et al.  On the stokes problem in Lipschitz domains , 1994 .

[15]  Wolf von Wahl,et al.  The equations of Navier-Stokes and abstract parabolic equations , 1985 .

[16]  G. Galdi.  An Introduction to the Mathematical Theory of the Navier-Stokes Equations : Volume I: Linearised Steady Problems , 1994 .

[17]  Reinhard Farwig,et al.  Generalized resolvent estimates for the Stokes system in bounded and unbounded domains , 1994 .

[18]  H. Amann.  Nonhomogeneous Navier—Stokes Equations with Integrable Low—Regularity Data , 2002 .

[19]  B. Jones,et al.  The initial value problem for the Navier-Stokes equations with data in Lp , 1972 .

[20]  Takashi Kato,et al.  StrongLp-solutions of the Navier-Stokes equation inRm, with applications to weak solutions , 1984 .

[21]  Y. Giga,et al.  On the Stokes operator in exterior domains , 1988 .

[22]  M. Cannone.  Viscous Flows in Besov Spaces , 2000 .

[23]  J. Serrin.  The initial value problem for the Navier-Stokes equations , 1963 .

[24]  Daisuke Fujiwara,et al.  An L_r-theorem of the Helmholtz decomposition of vector fields , 1977 .

[25]  R. Farwig,et al.  A New Class of Weak Solutions of the Navier–Stokes Equations with Nonhomogeneous Data , 2006 .

[26]  Giovanni P. Galdi,et al.  On the Steady Self‐Propelled Motion of a Body in a Viscous Incompressible Fluid , 1999 .

[27]  Herbert Amann,et al.  Linear and Quasilinear Parabolic Problems , 2019, Monographs in Mathematics.

[28]  Werner Varnhorn,et al.  The Stokes Equations , 1994 .