Higher integrability for weak solutions to a degenerate parabolic system with singular coefficients

• Main Authors: Yan Dong, Guangwei Du, Kelei Zhang
• Format: Article
• Language: English
• Published: SpringerOpen 2019-10-01
• Series: Boundary Value Problems
• Subjects: Degenerate parabolic system; Hörmander’s vector fields; L 2 $L^{2}$ estimates; Higher integrability
• Online Access: http://link.springer.com/article/10.1186/s13661-019-1285-y

Abstract In this paper, we study the degenerate parabolic system uti+Xα∗(aijαβ(z)Xβuj)=gi(z,u,Xu)+Xα∗fiα(z,u,Xu), $$ u_{t}^{i} + X_{\alpha }^{*} \bigl(a_{ij}^{\alpha \beta }(z){X_{\beta }} {u^{j}}\bigr) = {g_{i}}(z,u,Xu) + X_{\alpha }^{*} f_{i}^{\alpha }(z,u,Xu), $$ where X={X1,…,Xm} $X=\{X_{1},\ldots,X_{m} \}$ is a system of smooth real vector fields satisfying Hörmander’s condition and the coefficients aijαβ $a_{ij}^{\alpha \beta }$ are measurable functions and their skew-symmetric part can be unbounded. After proving the L2 $L^{2}$ estimates for the weak solutions, the higher integrability is proved by establishing a reverse Hölder inequality for weak solutions.