Volume 33, Issue 4
A Nash Type Result for Divergence Parabolic Equation Related to Hörmander's Vector Fields

J. Part. Diff. Eq., 33 (2020), pp. 341-376.

Published online: 2020-08

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• Abstract

In this paper we consider the divergence parabolic equation with bounded and measurable coefficients related to Hörmander's vector fields and establish a Nash type result, i.e., the local Hölder regularity for weak solutions. After deriving the parabolic Sobolev inequality, (1,1) type Poincaré inequality of Hörmander's vector fields and a De Giorgi type Lemma, the Hölder regularity of weak solutions to the equation is proved based on the estimates of oscillations of solutions and the  isomorphism between parabolic Campanato space and parabolic Hölder space. As a consequence, we give the Harnack inequality of weak solutions by showing an extension property of positivity for functions in the De Giorgi class.

• Keywords

Hörmander's vector fields divergence parabolic equation weak solution Hölder regularity Harnack inequality.

35K10, 35D30, 35B65, 35B45 35K10, 35D30, 35B65, 35B45

hll67800@163.com (Lingling Hou)

pengchengniu@nwpu.edu.cn (Pengcheng Niu)

• BibTex
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@Article{JPDE-33-341, author = {Hou , Lingling and Niu , Pengcheng}, title = {A Nash Type Result for Divergence Parabolic Equation Related to Hörmander's Vector Fields}, journal = {Journal of Partial Differential Equations}, year = {2020}, volume = {33}, number = {4}, pages = {341--376}, abstract = {

In this paper we consider the divergence parabolic equation with bounded and measurable coefficients related to Hörmander's vector fields and establish a Nash type result, i.e., the local Hölder regularity for weak solutions. After deriving the parabolic Sobolev inequality, (1,1) type Poincaré inequality of Hörmander's vector fields and a De Giorgi type Lemma, the Hölder regularity of weak solutions to the equation is proved based on the estimates of oscillations of solutions and the  isomorphism between parabolic Campanato space and parabolic Hölder space. As a consequence, we give the Harnack inequality of weak solutions by showing an extension property of positivity for functions in the De Giorgi class.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v33.n4.3}, url = {http://global-sci.org/intro/article_detail/jpde/17863.html} }
TY - JOUR T1 - A Nash Type Result for Divergence Parabolic Equation Related to Hörmander's Vector Fields AU - Hou , Lingling AU - Niu , Pengcheng JO - Journal of Partial Differential Equations VL - 4 SP - 341 EP - 376 PY - 2020 DA - 2020/08 SN - 33 DO - http://doi.org/10.4208/jpde.v33.n4.3 UR - https://global-sci.org/intro/article_detail/jpde/17863.html KW - Hörmander's vector fields KW - divergence parabolic equation KW - weak solution KW - Hölder regularity KW - Harnack inequality. AB -

In this paper we consider the divergence parabolic equation with bounded and measurable coefficients related to Hörmander's vector fields and establish a Nash type result, i.e., the local Hölder regularity for weak solutions. After deriving the parabolic Sobolev inequality, (1,1) type Poincaré inequality of Hörmander's vector fields and a De Giorgi type Lemma, the Hölder regularity of weak solutions to the equation is proved based on the estimates of oscillations of solutions and the  isomorphism between parabolic Campanato space and parabolic Hölder space. As a consequence, we give the Harnack inequality of weak solutions by showing an extension property of positivity for functions in the De Giorgi class.

Lingling Hou & Pengcheng Niu. (2020). A Nash Type Result for Divergence Parabolic Equation Related to Hörmander's Vector Fields. Journal of Partial Differential Equations. 33 (4). 341-376. doi:10.4208/jpde.v33.n4.3
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