We introduce a space of L2 vector fields with bounded mean oscillation whose "normal" component to the boundary is well-controlled. We establish its Helmholtz decomposition in the case when the domain is a perturbed C3 half space in Rn (n≥3) with small perturbation.
[1] H. Bahouri, J.-Y. Chemin, R. Danchin, Fourier analysis and nonlinear partial differen- tial equations, Grundlehren der Mathematischen Wissenschaften, 343, Springer, Heidelberg, 2011.
[2] M. Bolkart, Y. Giga, On L∞-BMO estimates for derivatives of the Stokes semigroup, Math. Z., 284 (2016) 1163-1183.
[3] M. Bolkart, Y. Giga, T.-H. Miura, T. Suzuki, Y. Tsutsui, On analyticity of the Lp-Stokes semigroup for some non-Helmholtz domains, Math. Nachr., 290 (2017) 2524-2546.
[4] M. Bolkart, Y. Giga, T. Suzuki, Analyticity of the Stokes semigroup in BMO-type spaces,J. Math. Soc. Japan, 70 (2018) 153-177.
[5] M. Bolkart, Y. Giga, T. Suzuki, Y. Tsutsui, Equivalence of BMO-type norms with appli- cations to the heat and Stokes semigroup, Potential Anal., 49 (2018) 105-130.
[6] L. Brasco, D. G´omez-Castro, J. L. V´azquez, Characterization of homogeneous fractional Sobolev spaces, Calc. Var. 60, 60 (2021) https://doi.org/10.1007/s00526-021-01934-6
[7] R. Farwig, H. Kozono, H. Sohr, An Lq-approach to Stokes and Navier-Stokes equations in general domains, Acta Math., 195 (2005) 21-53.
[8] R. Farwig, H. Kozono, H. Sohr, On the Helmholtz decomposition in general unbounded domains, Arch. Math. (Basel), 88 (2007) 239-248.
[9] C. Fefferman, E. M. Stein, Hp spaces of several variables, Acta Math., 129 (1972) 137-193.
[10] D. Fujiwara, H. Morimoto, An Lr-theorem of the Helmholtz decomposition of vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 24 (1977) 685-700.
[11] G. P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations. Steady-state problems. Second edition, Springer Monographs in Mathematics, Springer, New York, 2011.
[12] Y. Giga, Z. Gu, On the Helmholtz decompositions of vector fields of bounded mean os- cillation and in real Hardy spaces over the half space, Adv. Math. Sci. Appl., 29 (2020) 87-128.
[13] Y. Giga, Z. Gu, Normal trace for a vector field of bounded mean oscillation, Potential Anal., (2022) https://doi.org/10.1007/s11118-021-09973-6
[14] Y. Giga, Z. Gu, The Helmholtz decomposition of a space of vector fields with bounded mean oscillation in a bounded domain, Math. Ann., (2022) https://doi.org/10.1007/s00208-022- 02410-y
[15] Y. Giga, Z. Gu, P.-Y. Hsu, Continuous alignment of vorticity direction prevents the blow-up of the Navier-Stokes flow under the no-slip boundary condition, Nonlinear Anal., 189 (2019) 111579.
[16] Y. Giga, H. Kuroda, M. L- asica, The fourth-order total variation flow in Rn, preprint (2022), arXiv: 2205.07435
[17] I. S. Gradshteyn, I. M. Ryzhik, Table of integrals, series, and products, Seventh edition, Elsevier/Academic Press, Amsterdam, 2007.
[18] L. Grafakos, Classical Fourier Analysis, Third edition, Graduate Texts in Mathematics, 249, Springer-Verlag, New York, 2014.
[19] L. Grafakos, Modern Fourier Analysis, Third edition, Graduate Texts in Mathematics, 250, Springer-Verlag, New York, 2014.
[20] D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, Sec- ond edition, Grundlehren der Mathematischen Wissenschaften, 224, Springer-Verlag, Berlin, 1983.
[21] Z. Gu, Extension theorem for bmo in a domain, preprint (2022).
[22] Q. Han, F. Lin, Elliptic partial differential equations, Second edition, Courant Lecture Notes in Mathematics, 1, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2011.
[23] H. Kozono, H. Wadade, Remarks on Gagliardo-Nirenberg type inequality with critical Sobolev space and BMO, Math. Z., 259 (2008) 935-950.
[24] S. G. Krantz, H. R. Parks, The implicit function theorem. History, theory, and applications, Reprint of the 2003 edition. Modern Birkh¨auser Classics, Birkh¨auser/Springer, New York, 2013.
[25] O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Second English edition, Vol. 2, Gordon and Breach Science Publishers, New York-London-Paris, 1969.
[26] J.-L. Lions, E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, Translated from French by P. Kenneth, Die Grundlehren der mathematischen Wis- senschaften, Band 181, Springer-Verlag, New York-Heidelberg, 1972.
[27] H. Sohr, The Navier-Stokes equations. An elementary functional analytic approach, Birkh¨auser Advanced Texts: Basler Lehrbu¨her, Birkh¨auser Verlag, Basel, 2001.
[28] R. Temam, Navier-Stokes equations. Theory and numerical analysis. Revised edition, With an appendix by F. Thomasset, Studies in Mathematics and its Applications, 2, North- Holland Publishing Co., Amsterdam-New York, 1979.
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