No Access

Besov regularity of parabolic and hyperbolic PDEs

Stephan Dahlke

FB12 Mathematik und Informatik, Philipps-Universität Marburg, Hans-Meerwein Straße, Lahnberge, 35032 Marburg, Germany

E-mail Address: [email protected]

Search for more papers by this author

and

Cornelia Schneider

Department Mathematik, AM3, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstr. 11, 91058 Erlangen, Germany

E-mail Address: [email protected]

Search for more papers by this author

https://doi.org/10.1142/S0219530518500306Cited by:6

Abstract

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 $B_{τ, τ}^{r}, \frac{1}{τ} = \frac{r}{d} + \frac{1}{p}$ 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.

Keywords:

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

References

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, ISI, Google Scholar
2. H. Aimar and I. Gómez, Parabolic Besov regularity for the heat equation, Constr. Approx. 36 (2012) 145–159. Crossref, Google 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. Crossref, Google Scholar
4. H. Aimar, I. Gómez and B. Iaffei, On Besov regularity of temperatures, J. Fourier Anal. Appl. 16 (2010) 1007–1020. Crossref, Google 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 $L_{q} (L_{p})$ -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, ISI, Google Scholar
9. S. Dahlke, Besov regularity for elliptic boundary value problems with variable coefficients, Manuscripta Math. 95 (1998) 59–77. Crossref, Google Scholar
10. S. Dahlke, Besov regularity for interface problems, Z. Angew. Math. Mech. 79(6) (1999) 383–388. Crossref, Google Scholar
11. S. Dahlke, Besov regularity for elliptic boundary value problems on polygonal domains, Appl. Math. Lett. 12(6) (1999) 31–38. Crossref, Google 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. Crossref, Google 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. Crossref, Google 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. Crossref, Google 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. Crossref, Google 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. Crossref, Google 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. Crossref, Google Scholar
20. R. DeVore, B. Jawerth and V. Popov, Compression of wavelet decompositions, Amer. J. Math. 114 (1992) 737–785. Crossref, ISI, Google Scholar
21. R. DeVore and B. Lucier, High order regularity for conservation laws, Indiana Math. J. 39(2) (1990) 413–430. Crossref, ISI, Google Scholar
22. F. D. Gaspoz and P. Morin, Convergence rates for adaptive finite elements, IMA J. Numer. Anal. 29(4) (2009) 917–936. Crossref, Google 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). Crossref, Google Scholar
26. M. Hansen, Nonlinear approximation rates and Besov regularity for elliptic PDEs on polyhedral domains, Found. Comput. Math. 15 (2015) 561–589. Crossref, Google Scholar
27. D. Jerison and C. E. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal. 130 (1995) 161–219. Crossref, ISI, Google Scholar
28. K.-H. Kim, On stochastic partial differential equations with variable coefficients in $C^{1}$ domains, Stochastic Process. Appl. 112(2) (2004) 261–283. Crossref, Google Scholar
29. K.-H. Kim, A $W_{n}^{p}$ -theory of parabolic equations with unbounded leading coefficients on non-smooth domains, J. Math. Anal. Appl. 350 (2009) 294–305. Crossref, Google 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. Crossref, Google Scholar
31. K.-H. Kim and N. V. Krylov, On the Sobolev space theory of parabolic and elliptic equations in $C^{1}$ domains, SIAM J. Math. Anal. 36(2) (2004) 618–642. Crossref, Google 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. Crossref, Google 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. Crossref, Google 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). Crossref, Google 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. Crossref, Google 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). Crossref, Google 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). Crossref, Google 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. Crossref, Google Scholar
42. R. Stevenson and C. Schwab, Space-time adaptive wavelet methods for parabolic evolution problems, Math. Comput. 78 (2009) 1293–1318. Crossref, Google 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). Crossref, Google Scholar
45. H. Triebel, Function Spaces and Wavelets on Domains, EMS Tracts on Mathematics, Vol. 7 (EMS Publishing House, Zürich, 2008). Crossref, Google Scholar
46. I. Wood, Maximal $L^{p}$ -regularity for the Laplacian on Lipschitz domains, Math. Z. 255(4) (2007) 855–875. Crossref, Google 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

Vol. 17, No. 02

Metrics

Downloaded 33 times

History

Received 6 January 2018

Accepted 3 December 2018

Keywords

PDF download

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

Besov regularity of parabolic and hyperbolic PDEs

Abstract

References

Recommended