World Scientific
Skip main navigation

Cookies Notification

We use cookies on this site to enhance your user experience. By continuing to browse the site, you consent to the use of our cookies. Learn More
Our website is made possible by displaying certain online content using javascript.
In order to view the full content, please disable your ad blocker or whitelist our website

System Upgrade on Mon, Jun 21st, 2021 at 1am (EDT)

During this period, the E-commerce and registration of new users may not be available for up to 6 hours.
For online purchase, please visit us again. Contact us at [email protected] for any enquiries.

Besov regularity of parabolic and hyperbolic PDEs

    This paper is concerned with the regularity of solutions to linear and nonlinear evolution equations on nonsmooth domains. In particular, we study the smoothness in the specific scale Bτ,τr,1τ=rd+1p of Besov spaces. The regularity in these spaces determines the approximation order that can be achieved by adaptive and other nonlinear approximation schemes. We show that for all cases under consideration the Besov regularity is high enough to justify the use of adaptive algorithms.

    AMSC: 35B65, 35K55, 46E35, 35L15, 35A02, 35K05, 65M12


    • 1. S. Agmon, A. Douglis and L. Nierenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I, Comm. Pure Appl. Math. 12 (1959) 623–727. Crossref, ISIGoogle Scholar
    • 2. H. Aimar and I. Gómez, Parabolic Besov regularity for the heat equation, Constr. Approx. 36 (2012) 145–159. CrossrefGoogle Scholar
    • 3. H. Aimar, I. Gómez and B. Iaffei, Parabolic mean values and maximal estimates for gradients of temperatures, J. Funct. Anal. 255 (2008) 1939–1956. CrossrefGoogle Scholar
    • 4. H. Aimar, I. Gómez and B. Iaffei, On Besov regularity of temperatures, J. Fourier Anal. Appl. 16 (2010) 1007–1020. CrossrefGoogle Scholar
    • 5. T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Oxford Lecture Series in Mathematics and Its Applications (Clarendon Press, 1998). Google Scholar
    • 6. P. Cioica, Besov regularity of stochastic partial differential equations on bounded Lipschitz domains, Ph.D. thesis, Philipps-Universität Marburg (2013). Google Scholar
    • 7. P. Cioica, K.-H. Kim, K. Lee and F. Lindner, On the Lq(Lp)-regularity and Besov smoothness of stochastic parabolic equations on bounded Lipschitz domains, Electron. J. Probab. 18(82) (2013) 1–41. Google Scholar
    • 8. A. Cohen, W. Dahmen and R. DeVore, Adaptive wavelet methods for elliptic operator equation: Convergence rates, Math. Comp. 70 (2001) 27–75. Crossref, ISIGoogle Scholar
    • 9. S. Dahlke, Besov regularity for elliptic boundary value problems with variable coefficients, Manuscripta Math. 95 (1998) 59–77. CrossrefGoogle Scholar
    • 10. S. Dahlke, Besov regularity for interface problems, Z. Angew. Math. Mech. 79(6) (1999) 383–388. CrossrefGoogle Scholar
    • 11. S. Dahlke, Besov regularity for elliptic boundary value problems on polygonal domains, Appl. Math. Lett. 12(6) (1999) 31–38. CrossrefGoogle Scholar
    • 12. S. Dahlke, Besov regularity of edge singularities for the Poisson equation in polyhedral domains, Numer. Linear Algebra Appl. 9(6–7) (2002) 457–466. CrossrefGoogle Scholar
    • 13. S. Dahlke, W. Dahmen and R. DeVore, Nonlinear approximation and adaptive techniques for solving elliptic operator equations, in Multiscale Wavelet Methods for Partial Differential Equations, eds. W. DahmenA. J. KurdilaP. Oswald, Wavelet Analysis and Applications, Vol. 6 (Academic Press, San Diego, 1997), pp. 237–283. CrossrefGoogle Scholar
    • 14. S. Dahlke and R. DeVore (1997). Besov regularity for elliptic boundary value problems, Comm. Partial Differential Equations 22(1–2) (1997) 1–16. CrossrefGoogle Scholar
    • 15. S. Dahlke, L. Diening, C. Hartmann, B. Scharf and M. Weimar, Besov regularity of solutions to the p-Poisson equation, Nonlinear Anal. 130 (2016) 298–329. CrossrefGoogle Scholar
    • 16. S. Dahlke, M. Hansen, C. Schneider and W. Sickel, Properties of Kondratiev spaces, preprint, Reihe Philipps University Marburg, Bericht Mathematik No. 2018-06 (2018). Google Scholar
    • 17. S. Dahlke and C. Schneider, Describing the singular behaviour of parabolic equations on cones in fractional Sobolev spaces, Int. J. Geomath. 9(2) (2018) 293–315. CrossrefGoogle Scholar
    • 18. S. Dahlke and C. Schneider, Besov regularity of parabolic and hyperbolic PDEs, preprint (2018), arXiv:1811.09428 [math.AP]. Google Scholar
    • 19. R. DeVore, Nonlinear approximation, Acta Numer. 7 (1998) 51–150. CrossrefGoogle Scholar
    • 20. R. DeVore, B. Jawerth and V. Popov, Compression of wavelet decompositions, Amer. J. Math. 114 (1992) 737–785. Crossref, ISIGoogle Scholar
    • 21. R. DeVore and B. Lucier, High order regularity for conservation laws, Indiana Math. J. 39(2) (1990) 413–430. Crossref, ISIGoogle Scholar
    • 22. F. D. Gaspoz and P. Morin, Convergence rates for adaptive finite elements, IMA J. Numer. Anal. 29(4) (2009) 917–936. CrossrefGoogle Scholar
    • 23. P. Grisvard, Singularities in Boundary Value Problems, Recherches en Mathématiques Appliquées, Vol. 22 (Masson, Paris; Springer-Verlag, Berlin, 1992). Google Scholar
    • 24. P. Grisvard, Elliptic Problems in Nonsmooth Domains, Classics in Applied Mathematics, Vol. 69 (SIAM, Philadelphia, 2011). Reprint of the 1985 original. Google Scholar
    • 25. W. Hackbusch, Elliptic Differential Equations: Theory and Numerical Treatment (Springer, Berlin, 1992). CrossrefGoogle Scholar
    • 26. M. Hansen, Nonlinear approximation rates and Besov regularity for elliptic PDEs on polyhedral domains, Found. Comput. Math. 15 (2015) 561–589. CrossrefGoogle Scholar
    • 27. D. Jerison and C. E. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal. 130 (1995) 161–219. Crossref, ISIGoogle Scholar
    • 28. K.-H. Kim, On stochastic partial differential equations with variable coefficients in C1 domains, Stochastic Process. Appl. 112(2) (2004) 261–283. CrossrefGoogle Scholar
    • 29. K.-H. Kim, A Wnp-theory of parabolic equations with unbounded leading coefficients on non-smooth domains, J. Math. Anal. Appl. 350 (2009) 294–305. CrossrefGoogle Scholar
    • 30. K.-H. Kim, A weighted Sobolev space theory of parabolic stochastic PDEs on non-smooth domains, J. Theoret. Probab. 27(1) (2012) 107–136. CrossrefGoogle Scholar
    • 31. K.-H. Kim and N. V. Krylov, On the Sobolev space theory of parabolic and elliptic equations in C1 domains, SIAM J. Math. Anal. 36(2) (2004) 618–642. CrossrefGoogle Scholar
    • 32. V. A. Kondratiev and A. O. Oleinik, Boundary value problems for partial differential equations in non-smooth domains, Russian Math. Surveys 8 (1983) 1–86. CrossrefGoogle Scholar
    • 33. V. A. Kozlov, On the spectrum of the pencil generated by the Dirichlet problem for an elliptic equation in an angle, Siberian Math. J. 32(2) (1991) 238–251. CrossrefGoogle Scholar
    • 34. V. A. Kozlov, V. G. Mazya and J. Rossman, Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations, Mathematical Surveys and Monographs, Vol. 85 (American Mathematical Society, Providence, RI, 2001). Google Scholar
    • 35. J. Lang, Adaptive Multilevel Solution of Nonlinear Parabolic PDE Systems: Theory, Algorithm, and Applications, Lecture Notes in Computational Science and Engineering, Vol. 16 (Springer-Verlag, Berlin, 2001). CrossrefGoogle Scholar
    • 36. S. V. Lototsky, Sobolev spaces with weights in domains and boundary value problems for degenerate elliptic equations, Methods Appl. Anal. 7(1) (2000) 195–204. Google Scholar
    • 37. V. T. Luong and D. V. Loi, The first initial-boundary value problem for parabolic equations in a cone with edges, Vestn. St.-Petersbg. Univ. Ser. 1. Mat. Mekh. Asron. 2 60(3) (2015) 394–404. Google Scholar
    • 38. V. T. Luong and N. T. Tung, The Dirichlet–Cauchy problem for nonlinear hyperbolic equations in a domain with edges, Nonlinear Anal. 125 (2015) 457–467. CrossrefGoogle Scholar
    • 39. V. Maz’ya and J. Rossmann, Elliptic Equations in Polyhedral Domains, Mathematical Surveys and Monographs, Vol. 162 (American Mathematical Society, Providence, RI, 2010). CrossrefGoogle Scholar
    • 40. T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, De Gruyter Series in Nonlinear Analysis and Applications (Walter de Gruyter, 1996). CrossrefGoogle Scholar
    • 41. R. Stevenson, Adaptive wavelet methods for solving operator equations: An overview, in Multiscale, Nonlinear and Adaptive Approximation, eds. R. DeVoreet al., (Springer, Berlin, 2009), pp. 543–597; Dedicated to Wolfgang Dahmen on the occasion of his 60th birthday. CrossrefGoogle Scholar
    • 42. R. Stevenson and C. Schwab, Space-time adaptive wavelet methods for parabolic evolution problems, Math. Comput. 78 (2009) 1293–1318. CrossrefGoogle Scholar
    • 43. V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, 2nd edn., Springer Series in Computational Mathematics, Vol. 25 (Springer-Verlag, Berlin, 2006). Google Scholar
    • 44. H. Triebel, Theory of Function Spaces, Monographs in Mathematics, Vol. 78 (Birkhäuser Verlag, Basel, 1983). CrossrefGoogle Scholar
    • 45. H. Triebel, Function Spaces and Wavelets on Domains, EMS Tracts on Mathematics, Vol. 7 (EMS Publishing House, Zürich, 2008). CrossrefGoogle Scholar
    • 46. I. Wood, Maximal Lp-regularity for the Laplacian on Lipschitz domains, Math. Z. 255(4) (2007) 855–875. CrossrefGoogle Scholar
    Published: 21 January 2019
    Remember to check out the Most Cited Articles!

    Check out our Differential Equations and Mathematical Analysis books in our Mathematics 2021 catalogue
    Featuring authors such as Ronen Peretz, Antonio Martínez-Abejón & Martin Schechter