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 - A, 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, (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. 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. 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. 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. Google Scholar

[6]

G. B. Folland, Subellitic estimates and function space on nilpotent Lie groups,, Ark. Mat., 13 (1975), 161. 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. Google Scholar

[8]

P. Hajlasz and P. Koskela, Sobolev met Poincar $\acutee$,, Mem. Amer. Math. Soc., 145 (2000). 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. 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. 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. Google Scholar

[12]

E. Lanconelli and S. Polidoro, On a class of hypoelliptic evolution operaters,, Rend. Sem. Mat. Univ. Politec. Torino, 52 (1994), 29. 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. 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. Google Scholar

[15]

M. Manfredini, The Dirichlet problem for a class of ultraparabolic equations,, Adv. Diff. Equ., 2 (1997), 831. Google Scholar

[16]

J. Moser, On Harnack's theorem for elliptic differential equations,, Comm. Pure Appl. Math., 14 (1961), 577. doi: 10.1002/cpa.3160140329. Google Scholar

[17]

J. Moser, A Harnack inequality for parabolic differential equations,, Comm. Pure Appl. Math., 17 (1964), 101. Google Scholar

[18]

J. Nash, Continuity of solutions of parabolic and elliptic equations,, Amer. J. Math., 80 (1958), 931. 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. 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. 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. 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. 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

show all references

References:
[1]

A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, "Stratified Lie Groups and Potential Theory for Their Sub-Laplacians,", Springer-Verlag, (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. 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. 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. 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. Google Scholar

[6]

G. B. Folland, Subellitic estimates and function space on nilpotent Lie groups,, Ark. Mat., 13 (1975), 161. 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. Google Scholar

[8]

P. Hajlasz and P. Koskela, Sobolev met Poincar $\acutee$,, Mem. Amer. Math. Soc., 145 (2000). 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. 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. 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. Google Scholar

[12]

E. Lanconelli and S. Polidoro, On a class of hypoelliptic evolution operaters,, Rend. Sem. Mat. Univ. Politec. Torino, 52 (1994), 29. 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. 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. Google Scholar

[15]

M. Manfredini, The Dirichlet problem for a class of ultraparabolic equations,, Adv. Diff. Equ., 2 (1997), 831. Google Scholar

[16]

J. Moser, On Harnack's theorem for elliptic differential equations,, Comm. Pure Appl. Math., 14 (1961), 577. doi: 10.1002/cpa.3160140329. Google Scholar

[17]

J. Moser, A Harnack inequality for parabolic differential equations,, Comm. Pure Appl. Math., 17 (1964), 101. Google Scholar

[18]

J. Nash, Continuity of solutions of parabolic and elliptic equations,, Amer. J. Math., 80 (1958), 931. 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. 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. 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. 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. 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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