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Volume 10, Issue 1
L2,μ(Q)-estimates for Parabolic Equations and Applications

Hongming Yin

J. Part. Diff. Eq., 10 (1997), pp. 31-44.

Published online: 1997-10

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  • Abstract
In this paper we derive a priori estimates in the Campanato space L^{2,\mu}(Q_T) for solutions of tbe following parabolic equation u_t - \frac{∂}{∂x_i}(a_{ij}(x,t)u_x_j+a_iu) + b_iu_x_i + cu = \frac{∂}{∂_x_i}f_i + f_0 where {a_{ij}(x, t)} are assumed to be measurable and satisfy the ellipticity condition. The proof is based on accurate DeGiorgi-Nash-Moser's estimate and a modified Poincare's inequality. These estimates are very useful in the study of the regularity of solutions for some nonlinear problems. As a concrete example, we obtain the classical solvability for a strongly coupled parabolic system arising from the thermistor problem.
  • Keywords

Parabolic equation a priori estimates in Campanato space DeGiorgi-Nash-Moser's estimate a modified Poincare's inequality

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@Article{JPDE-10-31, author = {}, title = {L2,μ(Q)-estimates for Parabolic Equations and Applications}, journal = {Journal of Partial Differential Equations}, year = {1997}, volume = {10}, number = {1}, pages = {31--44}, abstract = { In this paper we derive a priori estimates in the Campanato space L^{2,\mu}(Q_T) for solutions of tbe following parabolic equation u_t - \frac{∂}{∂x_i}(a_{ij}(x,t)u_x_j+a_iu) + b_iu_x_i + cu = \frac{∂}{∂_x_i}f_i + f_0 where {a_{ij}(x, t)} are assumed to be measurable and satisfy the ellipticity condition. The proof is based on accurate DeGiorgi-Nash-Moser's estimate and a modified Poincare's inequality. These estimates are very useful in the study of the regularity of solutions for some nonlinear problems. As a concrete example, we obtain the classical solvability for a strongly coupled parabolic system arising from the thermistor problem.}, issn = {2079-732X}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jpde/5580.html} }
TY - JOUR T1 - L2,μ(Q)-estimates for Parabolic Equations and Applications JO - Journal of Partial Differential Equations VL - 1 SP - 31 EP - 44 PY - 1997 DA - 1997/10 SN - 10 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jpde/5580.html KW - Parabolic equation KW - a priori estimates in Campanato space KW - DeGiorgi-Nash-Moser's estimate KW - a modified Poincare's inequality AB - In this paper we derive a priori estimates in the Campanato space L^{2,\mu}(Q_T) for solutions of tbe following parabolic equation u_t - \frac{∂}{∂x_i}(a_{ij}(x,t)u_x_j+a_iu) + b_iu_x_i + cu = \frac{∂}{∂_x_i}f_i + f_0 where {a_{ij}(x, t)} are assumed to be measurable and satisfy the ellipticity condition. The proof is based on accurate DeGiorgi-Nash-Moser's estimate and a modified Poincare's inequality. These estimates are very useful in the study of the regularity of solutions for some nonlinear problems. As a concrete example, we obtain the classical solvability for a strongly coupled parabolic system arising from the thermistor problem.
Hongming Yin . (2019). L2,μ(Q)-estimates for Parabolic Equations and Applications. Journal of Partial Differential Equations. 10 (1). 31-44. doi:
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