Local estimates on two linear parabolic equations with singular coefficients

Pacific Journal of Mathematics

Volumn 223, Issue 2, 2006, Pages 367-396

  Zhang, Qi S ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Heat equation; Kato class; Navier Stokes equation; Singular coefficients

Indexed keywords

EID: 32344452439     PISSN: 00308730     EISSN: 00308730     Source Type: Journal    
DOI: 10.2140/pjm.2006.223.367     Document Type: Article

References (31)

1. [Aizenman and Simon 1982] M. Aizenman and B. Simon, "Brownian motion and Harnack inequality for Schrödinger operators", Comm. Pure Appl. Math. 35:2 (1982), 209-273. MR 84a:35062 Zbl 0459.60069
2. [Chen and Song 2003] Z.-Q. Chen and R. Song, "Drift transforms and Green function estimates for discontinuous processes", J. Funct. Anal. 201 (2003), 262-281. MR 2004c:60218 Zbl 1031.60071
3. [Chen and Zhao 1995] Z.-Q. Chen and Z. Zhao, "Diffusion processes and second order elliptic operators with singular coefficients for lower order terms", Math. Ann. 302:2 (1995), 323-357. MR 96e:60136 Zbl 0876.35029
4. [Cranston and Zhao 1987] M. Cranston and Z. Zhao, "Conditional transformation of drift formula and potential theory for 1/2 Δ+b(·)·∇", Comm. Math. Phys. 112:4 (1987), 613-625. MR 89g:35030 Zbl 0647.60071
5. p", Arch. Rational Mech. Anal. 45 (1972), 222-240. MR 47 #5463 Zbl 0254.35097
6. [Frehse and Ruzicka 1994] J. Frehse and M. Ruzicka, "On the regularity of the stationary Navier-Stokes equations", Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 21:1 (1994), 63-95. MR 95e:35158 Zbl 0823.35141
7. [Frehse and Ruzicka 1995] J. Frehse and M. Ruzicka, "Existence of regular solutions to the stationary Navier-Stokes equations", Math. Ann. 302 (1995), 699-717. MR 98b:76024a Zbl 0861.35074
8. [Han and Lin 1997] Q. Han and F. Lin, Elliptic partial differential equations, Courant Lecture Notes in Mathematics 1, New York University, Courant Institute of Mathematical Sciences, New York, 1997. MR 2001d:35035 Zbl 1051.35031
9. 3,∞-solutions of Navier-Stokes equations and backward uniqueness", Uspekhi Mat. Nauk 58:2 (2003), 3-44.
10. In Russian; translation in Russian Math. Surveys 58:2 (2003), 211-250. MR 2004m:35204 Zbl 1064.35134
11. d) generated by Δ + b · ∇", Teor. Veroyatnost. i Primenen. 35:3 (1990), 449-458.
12. In Russian; translation in Theory Probab. Appl. 35 (1990). MR 91m:47070 Zbl 0778.47031
13. [Ladyženskaja et al. 1967] O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural'ceva, Linear and quasilinear equations of parabolic type, vol. 23, Translations of Mathematical Monographs, American Mathematical Society, Providence, R.I., 1967. MR 39 #3159b
14. [Lieberman 1996] G. M. Lieberman, Second order parabolic differential equations, World Scientific, River Edge, NJ, 1996. MR 98k:35003 Zbl 0884.35001
15. [Liskevich and Zhang 2004] V. Liskevich and Q. S. Zhang, "Extra regularity for parabolic equations with drift terms", Manuscripta Math. 113:2 (2004), 191-209. MR 2006a:35127 Zbl 1063.35045
16. [Montgomery-Smith 2005] S. Montgomery-Smith, "Conditions implying regularity of the three dimensional Navier-Stokes equation", Appl. Math. 50:5 (2005), 451-464. MR 2160072
17. [O'Leary 2003] M. O'Leary, "Conditions for the local boundedness of solutions of the Navier-Stokes system in three dimensions", Comm. Partial Differential Equations 28:3-4 (2003), 617-636. MR 2004c:35327 Zbl 1048.35066
18. [Osada 1987] H. Osada, "Diffusion processes with generators of generalized divergence form", J. Math. Kyoto Univ. 27:4 (1987), 597-619. MR 89b:60188 Zbl 0657.35073
19. 2 decay for weak solutions of the Navier-Stokes equations", Arch. Rational Mech. Anal. 88:3 (1985), 209-222. MR 86j:35129 Zbl 0602.76031
20. [Schonbek 1986] M. E. Schonbek, "Large time behaviour of solutions to the Navier-Stokes equations", Comm. Partial Differential Equations 11 (1986), 733-763. MR 87h:35274 Zbl 0607.35071
21. [Semenov 1999] Y. A. Semenov, "Hölder continuity of bounded solutions of parabolic equations", preprint, 1999.
22. [Serrin 1962] J. Serrin, "On the interior regularity of weak solutions of the Navier-Stokes equations", Arch. Rational Mech. Anal. 9 (1962), 187-195. MR 25 #346 Zbl 0106.18302
23. [Simon 1982] B. Simon, "Schrödinger semigroups", Bull. Amer. Math. Soc. (N.S.) 7:3 (1982), 447-526. MR 86b:81001a Zbl 0524.35002
24. [Solonnikov 1964] V. A. Solonnikov, "A priori estimates for solutions of second-order equations of parabolic type", Trudy Mat. Inst. Steklov. 70 (1964), 133-212.
25. In Russian; translated in Amer. Math. Soc. Transl. 75 (1968), 1-116. MR 28 #5267 Zbl 0168.08202
26. [Stampacchia 1965] G. Stampacchia, "Le problème de Dirichlet pour les èquations elliptiques du second ordre à coefficients discontinus", Ann. Inst. Fourier (Grenoble) 15:fasc. 1 (1965), 189-258. MR 33 #404 Zbl 0151.15401
27. [Struwe 1988] M. Struwe, "On partial regularity results for the Navier-Stokes equations", Comm. Pure Appl. Math. 41:4 (1988), 437-458. MR 89h:35270 Zbl 0632.76034
28. [Temam 1984] R. Temam, Navier-Stokes equations, 3rd ed., Studies in Mathematics and its Applications 2, North-Holland, Amsterdam, 1984. Reprinted by the Amer. Math. Soc., Providence, 2001. MR 86m:76003 Zbl 0568.35002
29. t = 0", Manuscripta Math. 93:3 (1997), 381-390. MR 98c:35067 Zbl 0886.35004
30. 2 norms in a new function space", Adv. Differential Equations 9:5-6 (2004), 587-624. MR 2005j:35174
31. [Zhang 2004b] Q. S. Zhang, "A strong regularity result for parabolic equations", Comm. Math. Phys. 244:2 (2004), 245-260. MR 2005b:35116 Zbl 1061.35026