144
Views
5
CrossRef citations to date
0
Altmetric
Original Articles

Necessary and sufficient conditions on local strong solvability of the Navier–Stokes system

, &
Pages 47-58 | Received 31 Jan 2010, Accepted 21 Feb 2010, Published online: 29 Jun 2010

References

  • Sohr , H . 2001 . The Navier–Stokes Equations. An Elementary Functional Analytic Approach , Basel : Birkhäuser Advanced Texts, Birkhäuser Verlag .
  • Leray , J . 1934 . Sur le mouvement d'un liquide visqueux emplissant l'espace . Acta Math. , 63 : 193 – 248 .
  • Hopf , E . 1950/51 . Ber die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen . Math. Nachr. , 4 : 213 – 231 .
  • Kiselev , AA and Ladyzenskaya , OA . 1963 . On the existence of uniqueness of solutions of the non-stationary problems for flows of non-compressible fluids . Amer. Math. Soc. Transl. Ser. 2 , 24 : 79 – 106 .
  • Fujita , H and Kato , T . 1964 . On the Navier–Stokes initial value problem . Arch. Ration. Mech. Anal. , 16 : 269 – 315 .
  • Kato , T . 1984 . Strong L p -solutions of the Navier–Stokes equation in ℝ m , with applications to weak solutions . Math. Z. , 187 : 471 – 480 .
  • Giga , Y . 1986 . Solution for semilinear parabolic equations in L p and regularity of weak solutions for the Navier–Stokes system . J. Differ. Eqns , 61 : 186 – 212 .
  • Heywood , JG . 1980 . The Navier–Stokes equations: On the existence, regularity and decay of solutions . Indiana Univ. Math. J. , 29 : 639 – 681 .
  • Kozono , H and Yamazaki , M . 1995 . Local and global unique solvability of the Navier–Stokes exterior problem with Cauchy data in the space L n,∞ . Houston J. Math. , 21 : 755 – 799 .
  • Solonnikov , VA . 1977 . Estimates for solutions of nonstationary Navier–Stokes equations . J. Soviet Math. , 8 : 467 – 529 .
  • Amann , H . 2000 . On the strong solvability of the Navier–Stokes equations . J. Math. Fluid Mech. , 2 : 16 – 98 .
  • Amann , H . 2002 . “ Nonhomogeneous Navier–Stokes Equations with Integrable Low-regularity Data ” . In International Mathematics Series , 1 – 26 . New York : Kluwer Academic/Plenum .
  • Triebel , H . 1978 . Interpolation Theory, Function Spaces, Differential Operators , Amsterdam : North-Holland .
  • Farwig , R and Sohr , H . 2009 . Optimal initial value conditions for the existence of local strong solutions of the Navier–Stokes equations . Math. Ann. , 345 : 631 – 642 .
  • Galdi , GP . 1998 . An Introduction to the Mathematical Theory of the Navier–Stokes Equations: Nonlinear Steady Problems , New York : Springer Tracts in Natural Philosophy, Springer-Verlag .
  • Giga , Y and Sohr , H . 1991 . Abstract L q -estimates for the Cauchy problem with applications to the Navier–Stokes equations in exterior domains . J. Funct. Anal. , 102 : 72 – 94 .
  • Miyakawa , T . 1981 . On the initial value problem for the Navier–Stokes equations in L r -spaces . Math. Z. , 178 : 9 – 20 .
  • Varnhorn , W . 1994 . The Stokes Equations , Berlin : Akademie Verlag .
  • Farwig , R , Galdi , GP and Sohr , H . 2006 . A new class of weak solutions of the Navier–Stokes equations with nonhomogeneous data . J. Math. Fluid Mech. , 8 : 423 – 444 .
  • Giga , Y . 1981 . Analyticity of the semigroup generated by the Stokes operator in L r -spaces . Math. Z. , 178 : 297 – 329 .
  • Farwig , R , Sohr , H and Varnhorn , W . On optimal initial value conditions for local strong solutions of the Navier–Stokes equations, Ann. Univ. Ferrara 55 (2009), pp. 89–110
  • Sohr , H . 2001 . A regularity class for the Navier–Stokes equations in Lorentz spaces . J. Evol. Equ. , 1 : 441 – 467 .

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.