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Regularity estimates for stationary Stokes problem in some generalized function spaces

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Abstract

We investigate the global regularity for weak solutions to a generalized stationary Stokes problem with BMO coefficients in a non-smooth domains. The results in this paper fall into two categories to get estimates for both velocity gradient and its associated pressure, corresponding to the two classes of generalized function spaces. Our first outcome is the generalized Lorentz spaces \(\Lambda _{\nu ,\omega }^{s,t}(\Omega )\) estimate with Muckenhoupt weights. The second concerns a global bound in \(\psi \)-generalized Morrey spaces \(\mathcal {M}^{s,\psi }(\Omega )\). The proof technique in our study is an adaption of innovative method launched in Acerbi and Mingione (Duke Math J 136(2):285–320, 2007), and distinctive feature is that we employ the so-called fractional maximal distribution functions.

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Acknowledgements

M.-P. Tran gratefully acknowledges support of this work under the funding of Ministry of Education and Training, Vietnam. Project entitled: “Gradient estimates for some classes of partial differential equations in generalized Lebesgue spaces” (Nghien cuu danh gia gradient cho mot so lop phuong trinh dao ham rieng trong cac khong gian Lebesgue tong quat).

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Ministry of Education and Training, Vietnam.

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Authors and Affiliations

  1. Group of Analysis and Applied Mathematics, Department of Mathematics, Ho Chi Minh City University of Education, Ho Chi Minh City, Vietnam

    Thanh-Nhan Nguyen

  2. Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Minh-Phuong Tran

  3. Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    N.-T.-Nhu Tran

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  1. Thanh-Nhan Nguyen
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  2. Minh-Phuong Tran
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  3. N.-T.-Nhu Tran
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Contributions

T.-N. Nguyen and M.-P. Tran wrote the main parts of this paper. T.-N. Nguyen proved the first result concerning the regularity in generalized Lorentz spaces with Muckenhoupt weights, M.-P. Tran proved the second result and completed the last version of this paper. Nhu Tran, a student, who wrote the raw version of the content and very first proofs of preparatory lemmas. All the authors reviewed the manuscript.

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Correspondence to Minh-Phuong Tran.

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Nguyen, TN., Tran, MP. & Tran, NTN. Regularity estimates for stationary Stokes problem in some generalized function spaces. Z. Angew. Math. Phys. 74, 13 (2023). https://doi.org/10.1007/s00033-022-01901-x

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