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Cauchy problems for parabolic equations in Sobolev–Slobodeckii and Hölder spaces on uniformly regular Riemannian manifolds

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Abstract

In this paper, we establish optimal solvability results—maximal regularity theorems—for the Cauchy problem for linear parabolic differential equations of arbitrary order acting on sections of tensor bundles over boundaryless complete Riemannian manifolds \(({M, g})\) with bounded geometry. We employ an anisotropic extension of the Fourier multiplier theorem for arbitrary Besov spaces introduced in Amann (Math Nachr 186:5–56, 1997). This allows for a unified treatment of Sobolev–Slobodeckii and little Hölder spaces. In the flat case \({(M, g=(\mathbb{R}^{m},|dx|^{2})}\) , we recover classical results for Petrowskii-parabolic Cauchy problems.

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References

  1. Agranovich M.S., Vishik M.I.: Elliptic problems with a parameter and parabolic problems of general type. Russ. Math. Surveys, 19, 53–157 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  2. H. Amann. Ordinary Differential Equations.W. de Gruyter & Co., Berlin, 1990.

  3. H. Amann. Linear and Quasilinear Parabolic Problems, Volume I: Abstract Linear Theory. Birkhäuser, Basel, 1995.

  4. Amann H..: Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications. Math. Nachr., 186, 5–56 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  5. Amann H.: Elliptic operators with infinite-dimensional state spaces. J. Evol. Equ., 1, 143–188 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. H. Amann. Anisotropic Function Spaces and Maximal Regularity for Parabolic Problems. Part 1: Function Spaces. Jindřich Nečas Center for Mathematical Modeling, Lecture Notes, 6, Prague, 2009.

  7. H. Amann. Anisotropic function spaces on singular manifolds, 2012. arXiv:1204.0606.

  8. Amann H.: Function spaces on singular manifolds. Math. Nachr., 286, 436–475 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. H. Amann. Uniformly regular and singular Riemannian manifolds. In Elliptic and parabolic equations, volume 119 of Springer Proc. Math. Stat., pages 1–43. Springer, Cham, 2015.

  10. H. Amann. Parabolic equations on uniformly regular Riemannian manifolds and degenerate initial boundary value problems. In Recent Developments of Mathematical Fluid Mechanics, H. Amann, Y. Giga, H. Kozono, H. Okamoto, M. Yamazaki (Eds.), pages 43–77. Birkhäuser, Basel, 2016.

  11. H. Amann. Linear and Quasilinear Parabolic Problems, Volume II: Function Spaces and Linear Differential Operators. 2017. In preparation.

  12. Amann H., Hieber M., Simonett G.: Bounded \({{H}_\infty}\) -calculus for elliptic operators. Diff. Int. Equ., 7, 613–653 (1994)

    MathSciNet  MATH  Google Scholar 

  13. Angenent S.B.: Nonlinear analytic semiflows. Proc. Royal Soc. Edinburgh, 115A, 91–107 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  14. Da Prato G., Grisvard P.: Sommes d’opérateurs linéaires et équations différentielles opérationelles. J. Math. Pures Appl., 54, 305–387 (1975)

    MathSciNet  MATH  Google Scholar 

  15. Davies E.B.: Heat Kernels and Spectral Theory. Cambridge Univ. Press, Cambridge (1989)

    Book  MATH  Google Scholar 

  16. S.P. Degtyarev. On Fourier multipliers in function spaces with partial Hölder condition and their application to the linearized Cahn–Hilliard equation with dynamic boundary conditions. volume 4 of Evolution Equations and Control Theory, pages 391–429. 2015.

  17. R. Denk, M. Hieber, J. Prüss. \({\mathcal{R}}\) -boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem. Amer. Math. Soc., 166(788) (2003).

  18. R. Denk, T. Seger. Inhomogeneous boundary value problems in spaces of higher regularity. In Recent Developments of Mathematical Fluid Mechanics, H. Amann, Y. Giga, H. Kozono, H. Okamoto, M. Yamazaki (Eds.), pages 157–173. Birkhäuser, Basel, 2016.

  19. J. Dieudonné. Eléments d’Analyse III. Gauthier-Villars, Paris, 1974.

  20. Disconzi M., Shao Y., Simonett G.: Remarks on uniformly regular Riemannian manifolds. Math. Nachr., 289, 232–242 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. G. Dore. \({L^{p}}\) regularity for abstract differential equations. In Functional analysis and related topics, 1991 (Kyoto), Lecture Notes in Math. 1540, pages 25–38. Springer-Verlag, Berlin, 1993.

  22. Dore G., Venni A.: On the closedness of the sum of two closed operators. Math. Z., 196, 189–201 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  23. S.D. Eidelman, N.V. Zhitarashu. Parabolic Boundary Value Problems. Birkhäuser Verlag, Basel, 1998.

  24. Escher J., Prüss J., Simonett G.: Analytic solutions for a Stefan problem with Gibbs-Thomson correction. J. Reine Angew. Math., 563, 1–52 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  25. A. Friedman. Partial differential equations of parabolic type. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964.

  26. A. Grigor’yan. Heat Kernel and Analysis on Manifolds. Amer. Math. Soc., Providence, RI, 2009.

  27. Grubb G.: Parameter-elliptic and parabolic pseudodifferential boundary problems in global \({{L}_p}\) Sobolev spaces. Math. Z., 218, 43–90 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  28. G. Grubb. Functional Calculus of Pseudodifferential Boundary Problems. Birkhäuser, Boston, MA, 1996.

  29. Grubb G., Kokholm N.J.: A global calculus of parameter-dependent pseudodifferential boundary problems in \({{L}_p}\) Sobolev spaces. Acta Math., 171, 165–229 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  30. M. Hieber, J. Prüss. Heat kernels and maximal \({L^{p}}\) -\({L^{q}}\) estimates for parabolic evolution equations. Comm. Partial Differential Equations, 22(9-10) (1997), 1647–1669.

  31. L. Hörmander. The Analysis of Linear Partial Differential Operators I. Springer Verlag, Berlin, 1983.

  32. Kato T.: Perturbation theory for linear operators. Springer, Berlin (1995)

    MATH  Google Scholar 

  33. Ladyzhenskaya O.A.: A multiplier theorem in nonhomogeneous Hölder spaces and some of its applications. J. Math. Sci. (N.Y.), 115, 2792–2802 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  34. O.A. Ladyzhenskaya, V.A. Solonnikov, N.N. Ural’ceva. Linear and Quasilinear Equations of Parabolic Type. Amer. Math. Soc., Transl. Math. Monographs, Providence, R.I., 1968.

  35. LeCrone J., Simonett G.: On the flow of non-axisymmetric perturbations of cylinders via surface diffusion. 260, 5510–5531 (2016)

  36. A. Lunardi. Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhäuser, Basel, 1995.

  37. Mazzucato A.L., Nistor V.: Mapping properties of heat kernels, maximal regularity, and semi-linear parabolic equations on noncompact manifolds. J. Hyperbolic Differ. Equ., 3(4), 599–629 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  38. L.I. Nicolaescu. Lectures on the geometry of manifolds. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007.

  39. J. Prüss, Simonett G. Moving Interfaces and Quasilinear Parabolic Evolution Equations, volume 105 of Monographs in Mathematics. Birkhäuser, Basel, 2016.

  40. Y. Shao. Real analytic solutions for the Willmore flow. In Proceedings of the Ninth MSU-UAB Conference on Differential Equations and Computational Simulations, volume 20 of Electron. J. Differ. Equ. Conf., pages 151–164, 2013.

  41. Y. Shao. Continuous maximal regularity on singular manifolds and its applications, 2014. arXiv:1410.1082.

  42. Shao Y.: A family of parameter-dependent diffeomorphisms acting on function spaces over a Riemannian manifold and applications to geometric flows. NoDEA Nonlinear Differential Equations Appl., 22(1), 45–85 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  43. Shao Y., Simonett G.: Continuous maximal regularity on uniformly regular Riemannian manifolds. J. Evol. Equ., 14(1), 211–248 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  44. Solonnikov V.A.: On boundary value problems for linear parabolic systems of differential equations of general form. Proc. Steklov Inst. Math., 83, 1–184 (1965)

    MathSciNet  MATH  Google Scholar 

  45. V.A. Solonnikov. Maximal regularity and weak solutions of linear parabolic equations, 2015. Talk given at the conference Mathematical Fluid Mechanics: Old Problems, New Trends—a week for W. Zajączkowski, Bedlewo, 30 August 2015–5 September 2015.

  46. M.E. Taylor. Partial differential equations II. Qualitative studies of linear equations, volume 116 of Applied Mathematical Sciences. Springer, New York, 2011.

  47. H. Triebel. Interpolation Theory, Function Spaces, Differential Operators. North Holland, Amsterdam, 1978.

  48. Triebel H.: Theory of Function Spaces. Birkhäuser, Basel (1983)

    Book  MATH  Google Scholar 

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  1. Math. Institut, Universität Zürich, Winterthurerstr. 190, 8057, Zürich, Switzerland

    Herbert Amann

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  1. Herbert Amann
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Correspondence to Herbert Amann.

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Dedicated to Professor Jan Prüss on the occasion of his retirement.

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Amann, H. Cauchy problems for parabolic equations in Sobolev–Slobodeckii and Hölder spaces on uniformly regular Riemannian manifolds. J. Evol. Equ. 17, 51–100 (2017). https://doi.org/10.1007/s00028-016-0347-1

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