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On the \(L^p\)–\(L^q\) estimates of the gradient of solutions to the Stokes problem

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Abstract

This paper is concerned with estimates of the gradient of the solutions to the Stokes IBVP both in a bounded and in an exterior domain. More precisely, we look for estimates of the kind \(||\nabla v(t)||_q \le g(t)||\nabla v_0||_p,\;q\ge p>1,\) for all \(t>0\), where function g is independent of v.

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References

  1. K. Abe and Y. Giga, Analyticity of the Stokes semigroup in spaces of bounded functions, Acta Math., 211 (2013) 1–46.

    Article  MathSciNet  MATH  Google Scholar 

  2. K. Abe, Y. Giga, The \(L^\infty \) -Stokes semigroup in exterior domains, J. Evol. Equ. 14 (2014) 1–28.

  3. K. Abe, Y. Giga, M. Hieber, Stokes resolvent estimates in spaces of bounded functions, Ann. Sci. Ec. Norm. Super., 48 (2015) 537–559.

    Article  MathSciNet  MATH  Google Scholar 

  4. M. Bolkart and M. Hieber, Pointwise upper bounds for the solution of the Stokes equation on \(L^{\infty }_\sigma (\Omega )\) and applications, J. of Funct. Analysis, 268 (2015) 1678–1710.

  5. W. Borchers and T. Miyakawa, \(L^2\) decay for the Navier–Stokes flow in half spaces, Math.Ann. 282 (1988) 139–155.

  6. W. Borchers and T. Miyakawa, Algebraic \(L^2\) decay for Navier–Stokes flows in exterior domains, Acta Math., 165 (1990) 189–227.

  7. W. Borchers, H. Sohr, On the semigroup of the Stokes operator for exterior domains in \(L^q\) -spaces, Math. Z. 196 (1987) 415–425.

  8. T. Chang and H. Choe, Maximum modulus estimates for the solution of the Stokes equation, J. Differential Equations 254 (2013) 2682–2704.

    Article  MathSciNet  MATH  Google Scholar 

  9. F. Crispo and Maremonti, An interpolation inequality in exterior domains, Rend. Sem. Mat. Univ. Padova 112 (2004) 11–39.

  10. W. Dan and Y. Shibata, On the \(L^q\)-\(L^r\) estimates of the Stokes semigroup in a two dimensional exterior domain, J. Math. Soc. Japan, bf 51 (1999) 181–207.

  11. W. Dan and Y. Shibata, remark on the \(L^q\)-\(L^\infty \) estimate of the Stokes semigroup in a 2-dimensional exterior domain, Pacifc J. of Mathematics, 189 (1999) 223–239.

  12. W. Desch, M. Hieber and J. Prüss, \(L^p\) -theory of the Stokes equation in a half space, J. Evol. Equ. 1 (2001) 115–142.

  13. H. Engler, Contractive properties for the heat equation in Sobolev spaces, J. of Functional Analysis, 64 (1985) 412–435.

    Article  MathSciNet  MATH  Google Scholar 

  14. H. Engler, B. Kawohl and S. Luckhaus, Gradient estimates for solutions of parabolic equations and systems, J. of Math. Anal. and Appl., 147 (1990) 309–329.

    Article  MathSciNet  MATH  Google Scholar 

  15. R. Farwig, H. Kozono and H. Sohr, An \(L^q\) -approach to Stokes and Navier–Stokes equations in general domains, Acta Math. 195 (2005) 21–53.

  16. R. Farwig and H. Sohr, Generalized resolvent estimates for the Stokes system in bounded and unbounded domains, J. Math. Soc. Japan 46 (1994) 607–643.

    Article  MathSciNet  MATH  Google Scholar 

  17. E. Gagliardo, Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in \(n\) variabili, Rend. Sem. Matem. Univ. Padova, 27 (1957) 284–305.

  18. G.P. Galdi, An Introduction to the Mathematical Theory of the Navier–Stokes Equations, Steady-state Problems, Second Edition, Springer Monographs in Mathematics. Springer. New-York (2011).

    MATH  Google Scholar 

  19. V. Georgiev and K. Taniguchi, Gradient estimates and their optimality for heat equation in an exterior domain, arXiv:1710.00592

  20. Y. Giga, The Stokes operator in \(L^r\) spaces, Proc. Japan Acad. Ser. A Math. Sci. 57 (1981), no. 2, 85–89.

  21. Y. Giga, Analyticity of the semigroup generated by the Stokes operator in \(L^r\) spaces, Math Z. 178 (1981) 297–329.

  22. Y. Giga and H. Sohr, On the Stokes operator in exterior domains, J. Fac. Sci. Univ. Tokyo, 36 (1988) 103–130.

    MathSciNet  Google Scholar 

  23. J. Heywood, On nonstationary Stokes flow past a obstacle, Indiana Univ. Math. J., 24 (1974) 271–284.

    Article  MathSciNet  MATH  Google Scholar 

  24. J. Heywood, The Navier–Stokes equations: on the existence, regularity and decay of solutions, Indiana Univ. Math. J., 29 (1980) 639–681.

    Article  MathSciNet  MATH  Google Scholar 

  25. M. Hieber and P. Maremonti, Bounded analyticity of the Stokes semigroup on spaces of bounded functions, Recent Advances in Fluid Mechanics, Birkhuser, Basel, 2014.

  26. M. Higaki, Navier wall law for nonstationary viscous incompressible flows, J. Differential Equations, 260 (2016) 7358–7396, https://doi.org/10.1016/j.jde.2016.01.028.

    Article  MathSciNet  MATH  Google Scholar 

  27. H. Iwashita, \( L^q\)–\(L^r\) estimates for solutions of the nonstationary Stokes equations in an exterior domain and the Navier–Stokes initial value problems in Lq spaces, Math. Ann., 285 (1989) 265–289.

  28. P. Maremonti, On the Stokes problem in exterior domains: the maximum modulus theorems, Discrete Contin. Dyn. Syst. 34 (2014) 2135–2171.

    Article  MathSciNet  MATH  Google Scholar 

  29. P. Maremonti and S. Shimizu, Global existence of solutions to 2-D Navier–Stokes flow with non-decaying initial data in exterior domains, J. Math. Fluid Mech., 20 (2018) 899–927, https://doi.org/10.1007/s00021-017-0348-z.

    Article  MathSciNet  MATH  Google Scholar 

  30. P. Maremonti and S. Shimizu, Global existence of solutions to 2-D Navier–Stokes flow with non-decaying initial data in half-plane, J. Diff. Equations, 265 (2018) 5352–5383.

    Article  MathSciNet  MATH  Google Scholar 

  31. P. Maremonti and V.A. Solonnikov, An estimate for the solutions of Stokes system in exterior domains, Zap. Nauch. Sem. LOMI, 180 (1990) 105–120, trasl. in J. of Math. Sciences (1994) 229–239.

  32. P. Maremonti and V.A. Solonnikov, On nonstationary Stokes problem in exterior domains, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 24 (1997), no. 3, 395–449.

  33. V.A. Solonnikov, Estimates of the solutions of the nonstationary Navier–Stokes system. (Russian) Boundary value problems of mathematical physics and related questions in the theory of functions, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 38 (1973), 153–231, (e.t.) J. Soviet Math., 8 (1977), 467–529.

  34. V.A. Solonnikov, On the estimates of the solution of the evolution Stokes problem in weighted Hölder norms, Annali dell’Univ. di Ferrara, (Sez. VII, Sci. Mat.), 52 (2006) 137–172.

  35. V.A. Solonnikov, On nonstationary Stokes problem and Navier–Stokes problem in a half-space with initial data nondecreasing at infinity, Function theory and applications, J. Math. Sci. (N. Y.) 114 (2003), no. 5, 1726–1740.

  36. S. Ukai, A solution formula for the Stokes equation in \(\mathbb{R}^n_+\), Theory of nonlinear evolution equations and its applications (Japanese). Sūrikaisekikenkyūsho Kōkyūroku No. 604 (1987), 124–138.

  37. M. Yamazaki, The Navier–Stokes equations in the weak- \(L^n\) space with time-dependent external force, Math. Ann., 317 (2000) 635–675.

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Acknowledgements

The author is grateful to some anonymous referees for the careful reading of the paper. The referee’s comments have made the paper more readable.

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  1. Dipartimento di Matematica e Fisica, Università degli Studi della Campania “L. Vanvitelli”, via Vivaldi 43, 81100, Caserta, Italy

    Paolo Maremonti

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Correspondence to Paolo Maremonti.

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The research is partially supported by GNFM (INdAM) and by MIUR via the PRIN 2017 “Hyperbolic Systems of Conservation Laws and Fluid Dynamics: Analysis and Applications”.

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Maremonti, P. On the \(L^p\)–\(L^q\) estimates of the gradient of solutions to the Stokes problem. J. Evol. Equ. 19, 645–676 (2019). https://doi.org/10.1007/s00028-019-00490-z

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  • DOI: https://doi.org/10.1007/s00028-019-00490-z

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