# American Institute of Mathematical Sciences

July  2011, 29(3): 1261-1275. doi: 10.3934/dcds.2011.29.1261

## The $C^{\alpha}$ regularity of weak solutions of ultraparabolic equations

 1 Institute of Mathematics, AMSS, Academia Sinica, Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences, Beijing, 100190, China, China

Received  November 2009 Revised  May 2010 Published  November 2010

We obtained the $C^{\alpha}$ continuity of weak solutions for a class of ultraparabolic equations with measurable coefficients of the form

$\partial_t \ u= \sum_{i,j=1}^{m_0}X_i(a_{ij}(x,t)X_j\ u )+X_0 u.$

By choosing a new cut-off function, we simplified and generalized the earlier arguments and proved the $C^{\alpha}$ regularity of weak solutions for more general ultraparabolic equations.

Citation: Wendong Wang, Liqun Zhang. The $C^{\alpha}$ regularity of weak solutions of ultraparabolic equations. Discrete & Continuous Dynamical Systems, 2011, 29 (3) : 1261-1275. doi: 10.3934/dcds.2011.29.1261
##### References:
 [1] A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, "Stratified Lie Groups and Potential Theory for Their Sub-Laplacians," Springer-Verlag, Berlin, Heidelberg, 2007.  Google Scholar [2] M. Bramanti, M. C. Cerutti and M. Manfredini, $L^p$ estimates for some ultraparabolic equations, J. Math. Anal. Appl., 200 (1996), 332-354. doi: 10.1006/jmaa.1996.0209.  Google Scholar [3] C. Cinti, A. Pascucci and S. Polidoro, Pointwise estimates for solutions to a class of non-homogenous Kolmogorov equations, Math. Annal., 340 (2008), 237-264. doi: 10.1007/s00208-007-0147-6.  Google Scholar [4] C. Cinti and S. Polidoro, Pointwise local estimates and Gaussian upper bounds for a class of uniformly subelliptic ultraparabolic operators, J. Math. Anal. Appl., 338 (2008), 946-969. doi: 10.1016/j.jmaa.2007.05.059.  Google Scholar [5] E. De Giorgi, Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat., 3 (1957), 25-43.  Google Scholar [6] G. B. Folland, Subellitic estimates and function space on nilpotent Lie groups, Ark. Mat., 13 (1975), 161-207. doi: 10.1007/BF02386204.  Google Scholar [7] M. Di Francesco and S. Polidoro, Harnack inequality for a class of degenerate parabolic equations of Kolmogorov type, Adv. Diff. Equ., 11 (2006), 1261-1320.  Google Scholar [8] P. Hajlasz and P. Koskela, Sobolev met Poincar $\acutee$, Mem. Amer. Math. Soc., 145 (2000), x+101.  Google Scholar [9] A. E. Kogoj and E. Lanconelli, An invariant Harnack inequality for a class of hypoelliptic ultraparabolic equations, Mediterr. J. Math., 1 (2004), 51-80. doi: 10.1007/s00009-004-0004-8.  Google Scholar [10] S. N. Kruzhkov, A priori bounds and some properties of solutions of elliptic and parabolic equations, Math. Sb. (N.S.), 65 (1964), 522-570.  Google Scholar [11] S. N. Kruzhkov, A priori bounds for generalized solutions of second-order elliptic and parabolic equations, (Russian) Dokl. Akad. Nauk SSSR, 150 (1963), 748-751.  Google Scholar [12] E. Lanconelli and S. Polidoro, On a class of hypoelliptic evolution operaters, Rend. Sem. Mat. Univ. Politec. Torino, 52 (1994), 29-63.  Google Scholar [13] A. Lunardi, Schauder estimates for a class of degenerate elliptic and parabolic operators with unbounded coefficients in $R^N$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 24 (1997), 133-164.  Google Scholar [14] M. Manfredini and S. Polidoro, Interior regularity for weak solutions of ultraparabolic equations in the divergence form with discontinuous coefficients, Boll Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 1 (1998), 651-675.  Google Scholar [15] M. Manfredini, The Dirichlet problem for a class of ultraparabolic equations, Adv. Diff. Equ., 2 (1997), 831-866.  Google Scholar [16] J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math., 14 (1961), 577-591. doi: 10.1002/cpa.3160140329.  Google Scholar [17] J. Moser, A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math., 17 (1964), 101-134.  Google Scholar [18] J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math., 80 (1958), 931-954. doi: 10.2307/2372841.  Google Scholar [19] A. Pascucci and S. Polidoro, The Moser's iterative method for a class of ultraparabolic equations, Commun. Contemp. Math., 6 (2004), 395-417. doi: 10.1142/S0219199704001355.  Google Scholar [20] S. Polidoro and M. A. Ragusa, Hölder regularity for solutions of ultraparabolic equations in divergence form, Potential Anal., 14 (2001), 341-350. doi: 10.1023/A:1011261019736.  Google Scholar [21] W. Wang and L. Zhang, The $C^{\alpha}$ regularity of a class of non-homogeneous ultraparabolic equations, Science in China Series A: Math., 52 (2009), 1589-1605. doi: 10.1007/s11425-009-0158-8.  Google Scholar [22] Z. P. Xin and L. Zhang, On the global existence of solutions to the Prandtl's system, Adv. in Math., 181 (2004), 88-133. doi: 10.1016/S0001-8708(03)00046-X.  Google Scholar [23] Z. P. Xin, L. Zhang and J. N. Zhao, Global well-posedness for the two dimensional Prandtl's boundary layer equations,, to appear., ().   Google Scholar [24] L. Zhang, The $C^\alpha$ reglarity of a class of ultraparabolic equations,, preprint, ().   Google Scholar

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##### References:
 [1] A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, "Stratified Lie Groups and Potential Theory for Their Sub-Laplacians," Springer-Verlag, Berlin, Heidelberg, 2007.  Google Scholar [2] M. Bramanti, M. C. Cerutti and M. Manfredini, $L^p$ estimates for some ultraparabolic equations, J. Math. Anal. Appl., 200 (1996), 332-354. doi: 10.1006/jmaa.1996.0209.  Google Scholar [3] C. Cinti, A. Pascucci and S. Polidoro, Pointwise estimates for solutions to a class of non-homogenous Kolmogorov equations, Math. Annal., 340 (2008), 237-264. doi: 10.1007/s00208-007-0147-6.  Google Scholar [4] C. Cinti and S. Polidoro, Pointwise local estimates and Gaussian upper bounds for a class of uniformly subelliptic ultraparabolic operators, J. Math. Anal. Appl., 338 (2008), 946-969. doi: 10.1016/j.jmaa.2007.05.059.  Google Scholar [5] E. De Giorgi, Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat., 3 (1957), 25-43.  Google Scholar [6] G. B. Folland, Subellitic estimates and function space on nilpotent Lie groups, Ark. Mat., 13 (1975), 161-207. doi: 10.1007/BF02386204.  Google Scholar [7] M. Di Francesco and S. Polidoro, Harnack inequality for a class of degenerate parabolic equations of Kolmogorov type, Adv. Diff. Equ., 11 (2006), 1261-1320.  Google Scholar [8] P. Hajlasz and P. Koskela, Sobolev met Poincar $\acutee$, Mem. Amer. Math. Soc., 145 (2000), x+101.  Google Scholar [9] A. E. Kogoj and E. Lanconelli, An invariant Harnack inequality for a class of hypoelliptic ultraparabolic equations, Mediterr. J. Math., 1 (2004), 51-80. doi: 10.1007/s00009-004-0004-8.  Google Scholar [10] S. N. Kruzhkov, A priori bounds and some properties of solutions of elliptic and parabolic equations, Math. Sb. (N.S.), 65 (1964), 522-570.  Google Scholar [11] S. N. Kruzhkov, A priori bounds for generalized solutions of second-order elliptic and parabolic equations, (Russian) Dokl. Akad. Nauk SSSR, 150 (1963), 748-751.  Google Scholar [12] E. Lanconelli and S. Polidoro, On a class of hypoelliptic evolution operaters, Rend. Sem. Mat. Univ. Politec. Torino, 52 (1994), 29-63.  Google Scholar [13] A. Lunardi, Schauder estimates for a class of degenerate elliptic and parabolic operators with unbounded coefficients in $R^N$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 24 (1997), 133-164.  Google Scholar [14] M. Manfredini and S. Polidoro, Interior regularity for weak solutions of ultraparabolic equations in the divergence form with discontinuous coefficients, Boll Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 1 (1998), 651-675.  Google Scholar [15] M. Manfredini, The Dirichlet problem for a class of ultraparabolic equations, Adv. Diff. Equ., 2 (1997), 831-866.  Google Scholar [16] J. Moser, On Harnack's theorem for elliptic differential equations, Comm. Pure Appl. Math., 14 (1961), 577-591. doi: 10.1002/cpa.3160140329.  Google Scholar [17] J. Moser, A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math., 17 (1964), 101-134.  Google Scholar [18] J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math., 80 (1958), 931-954. doi: 10.2307/2372841.  Google Scholar [19] A. Pascucci and S. Polidoro, The Moser's iterative method for a class of ultraparabolic equations, Commun. Contemp. Math., 6 (2004), 395-417. doi: 10.1142/S0219199704001355.  Google Scholar [20] S. Polidoro and M. A. Ragusa, Hölder regularity for solutions of ultraparabolic equations in divergence form, Potential Anal., 14 (2001), 341-350. doi: 10.1023/A:1011261019736.  Google Scholar [21] W. Wang and L. Zhang, The $C^{\alpha}$ regularity of a class of non-homogeneous ultraparabolic equations, Science in China Series A: Math., 52 (2009), 1589-1605. doi: 10.1007/s11425-009-0158-8.  Google Scholar [22] Z. P. Xin and L. Zhang, On the global existence of solutions to the Prandtl's system, Adv. in Math., 181 (2004), 88-133. doi: 10.1016/S0001-8708(03)00046-X.  Google Scholar [23] Z. P. Xin, L. Zhang and J. N. Zhao, Global well-posedness for the two dimensional Prandtl's boundary layer equations,, to appear., ().   Google Scholar [24] L. Zhang, The $C^\alpha$ reglarity of a class of ultraparabolic equations,, preprint, ().   Google Scholar
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