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Global Lorentz estimates for nonlinear parabolic equations on nonsmooth domains

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Abstract

Consider the nonlinear parabolic equation in the form

$$\begin{aligned} u_t-\mathrm{div}{\mathbf {a}}(D u,x,t)=\mathrm{div}\,(|F|^{p-2}F) \quad \text {in} \quad \Omega \times (0,T), \end{aligned}$$

where \(T>0\) and \(\Omega \) is a Reifenberg domain. We suppose that the nonlinearity \({\mathbf {a}}(\xi ,x,t)\) has a small BMO norm with respect to x and is merely measurable and bounded with respect to the time variable t. In this paper, we prove the global Calderón-Zygmund estimates for the weak solution to this parabolic problem in the setting of Lorentz spaces which includes the estimates in Lebesgue spaces. Our global Calderón-Zygmund estimates extend certain previous results to equations with less regularity assumptions on the nonlinearity \({\mathbf {a}}(\xi ,x,t)\) and to more general setting of Lorentz spaces.

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Acknowledgements

The authors would like to thank the referee for useful comments and suggestions to improve the paper. The first named author was supported by the research Grant ARC DP140100649 from the Australian Research Council and Vietnam’s National Foundation for Science and Technology Development (NAFOSTED) under Project 101.02–2016.25. The second named author was supported by the research Grant ARC DP140100649.

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  1. Department of Mathematics, Macquarie University, Sydney, NSW, 2109, Australia

    The Anh Bui & Xuan Thinh Duong

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  1. The Anh Bui
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  2. Xuan Thinh Duong
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Correspondence to Xuan Thinh Duong.

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Communicated by N. Trudinger.

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Bui, T.A., Duong, X.T. Global Lorentz estimates for nonlinear parabolic equations on nonsmooth domains. Calc. Var. 56, 47 (2017). https://doi.org/10.1007/s00526-017-1130-z

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