# American Institute of Mathematical Sciences

## BMO type space associated with Neumann operator and application to a class of parabolic equations

 1 School of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 310018, China 2 Department of Mathematics, Jiangxi University of Finance and Economics, Nanchang 330032, China

*Corresponding author: yangmh7@mail2.sysu.edu.cn

Received  November 2020 Revised  January 2021 Published  March 2021

Fund Project: Zhang Chao's work was supported by National Natural Science Foundation of China(Grant Nos. 11971431, 11401525), the Natural Science Foundation of Zhejiang Province (Grant No. LY18A010006) and the first Class Discipline of Zhejiang-A(Zhejiang Gongshang University- Statistics). Minghua Yang's work was supported by the National Natural Science Foundation of China (Grant No. 11801236), Postdoctoral Science Foundation of China (Grant Nos. 2020T130265, 2018M632593), Natural Science Foundation of Jiangxi Province (Grant No.20204BCJL23056), the Postdoctoral Science Foundation of Jiangxi Province (Grant No. 2017KY23) and Educational Commission Science Programm of Jiangxi Province (Grant No. GJJ190272)

Let
 ${\rm BMO}_{\Delta_{N}}(\mathbb{R}^{n})$
denote a BMO space on
 $\mathbb{R}^{n}$
associated to a Neumann operator
 $\mathcal{L}: = -\Delta_{N}$
 $f\in {\rm BMO}_{\Delta_{N}}(\mathbb{R}^{n})$
is the trace of the solution of
 ${\mathbb L}u = u_{t}+ \mathcal{L} u = 0, u(x, 0) = f(x),$
where
 $u$
satisfies a Carleson-type condition
 $\begin{eqnarray*} \sup\limits_{x_B, r_B} r_B^{-n}\int_0^{r_B^2}\int_{B(x_B, r_B)} \left\{t| \partial_t u(x, t) |^2+ | \nabla_x u(x, t) |^2 \right\}{dx dt } \leq C <\infty, \end{eqnarray*}$
for some constant
 $C>0$
. Conversely, this Carleson condition characterizes all the
 ${\mathbb L}$
-carolic functions whose traces belong to the space
 ${\rm BMO}_{\Delta_{N}}(\mathbb{R}^{n})$
. Furthermore, based on the characterization of
 ${\rm BMO}_{\Delta_{N}}(\mathbb{R}^{n})$
mentioned above, we prove the global well-posedness for parabolic equations of Navier-Stokes type with the Neumann boundary condition under smallness condition on the intial data
 $u_{0}\in {{\rm BMO}_{\Delta_{N}}^{-1}(\mathbb{R}^{n})}$
.
Citation: Zhang Chao, Minghua Yang. BMO type space associated with Neumann operator and application to a class of parabolic equations. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021104
##### References:
 [1] P. Auscher and D. Frey, On the Well-Posedness of parabolic equations of Navier-Stokes type with ${\rm {BMO}}^{-1}$ data, J. Inst. Math. Jussieu., 16 (2017), 947-985.  doi: 10.1017/S1474748015000158.  Google Scholar [2] P. Auscher, S. Monniaux and P. Portal, The maximal regularity operator on tent spaces, Commun. Pure Appl. Anal., 11 (2012), 2213-2219.  doi: 10.3934/cpaa.2012.11.2213.  Google Scholar [3] P. Auscher and P. Tchamitchian., Square Root Problem for Divergence Operators and Related Topics, Ast$\acute{e}$risqueno., 249, 1998.  Google Scholar [4] D. Deng, X. T. Duong, A. Sikora and L. Yan, Comparison of the classical BMO with the BMO spaces associated with operators and applications, Rev. Mat. Iberoamericana., 24 (2008), 267-296.  doi: 10.4171/RMI/536.  Google Scholar [5] M. Dindos, C. Kenig and J. Pipher, BMO solvability and the $A_{\infty}$ condition for elliptic operators, J. Geom. Anal., 21 (2011), 78-95. doi: 10.1007/s12220-010-9142-3.  Google Scholar [6] Y. Du and K. Wang, Regularity of the solutions to the liquid crystal equations with small rough data, J. Differential Equations, 256 (2014), 65-81. doi: 10.1016/j.jde.2013.07.066.  Google Scholar [7] X. T. Duong, L. Yan and C. Zhang, On characterization of Poisson integrals of Schrödinger operators with BMO traces, J. Funct. Anal., 266 (2014), 2053-2085.  doi: 10.1016/j.jfa.2013.09.008.  Google Scholar [8] X. T. Duong and L. Yan, New function spaces of ${\rm{BMO}}$ type, the John-Nirenberg inequality, interpolation and applications, Comm. Pure Appl. Math., 58 (2005), 1375-1420.  doi: 10.1002/cpa.20080.  Google Scholar [9] X. T. Duong and L. Yan, Duality of Hardy and BMO spaces associated with operators with heat kernel bounds, J. Amer. Math. Soc., 18 (2005), 943-973.  doi: 10.1090/S0894-0347-05-00496-0.  Google Scholar [10] E. B. Fabes, R. L. Johnson and U. Neri, Spaces of harmonic functions representable by Poisson integrals of functions in BMO and $L_{p, \lambda}$, Indiana Univ. Math. J., 25 (1976), 159-170. doi: 10.1512/iumj.1976.25.25012.  Google Scholar [11] E. B. Fabes and U. Neri, Characterization of temperatures with initial data in BMO, Duke Math. J., 42 (1975), 725-734. doi: 10.1215/S0012-7094-75-04260-X.  Google Scholar [12] E. B. Fabes and U. Neri, Dirichlet problem in Lipschitz domains with BMO data, Proc. Amer. Math. Soc., 78 (1980), 33-39. doi: 10.1090/S0002-9939-1980-0548079-8.  Google Scholar [13] C. Fefferman and E. M. Stein, $H^p$ spaces of several variables, Acta Math., 129 (1972), 137-193.  doi: 10.1007/BF02392215.  Google Scholar [14] S. Hofmann, G. Lu, D. Mitrea, M. Mitrea and L. Yan, Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates, Memoirs of the Amer. Math. Soc., 214 (2011), no. 1007. doi: 10.1090/S0065-9266-2011-00624-6.  Google Scholar [15] R. Jiang, J. Xiao and D. Yang, Towards spaces of harmonic functions with traces in square Campanato spaces and their scaling invariants, Anal. Appl. (Singap.), 14 (2016), 679-703. doi: 10.1142/S0219530515500190.  Google Scholar [16] H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35.  doi: 10.1006/aima.2000.1937.  Google Scholar [17] J. Li and B. D. Wick, Characterizations of $\rm H^{1}_{\Delta_{N}}(\mathbb{R}^{n})$ and $\rm BMO_{\Delta_{N}}(\mathbb{R}^{n})$ via weak factorizations and commutators, J. Funct. Anal., 272 (2017), 5384-5416.  doi: 10.1016/j.jfa.2017.03.007.  Google Scholar [18] L. Song, X. X. Tian and L. X. Yan, On characterization of Poisson integrals of Schrödinger operators with Morry traces, Acta Math. Sin. (Engl. Ser.), 34 (2018), 787-800.  doi: 10.1007/s10114-018-7368-3.  Google Scholar [19] E. M. Stein, Topics in Harmonic Analysis Related to the Littlewood-Paley Theory, Annals of Mathematics Studies, 63, Princeton Univ. Press, Princeton, N.J., 1970.  Google Scholar [20] M. Yang and C. Zhang, Characterization of temperatures associated to Schrödinger operators with initial data in BMO spaces, to appear in Math. Nachr. arXiv: 1710.01160 Google Scholar [21] W. Yuan, W. Sickel and D. Yang, Morrey and Campanato meet Besov, Lizorkin and Triebel, Lecture Notes in Mathematics, 2005. Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-14606-0.  Google Scholar

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##### References:
 [1] P. Auscher and D. Frey, On the Well-Posedness of parabolic equations of Navier-Stokes type with ${\rm {BMO}}^{-1}$ data, J. Inst. Math. Jussieu., 16 (2017), 947-985.  doi: 10.1017/S1474748015000158.  Google Scholar [2] P. Auscher, S. Monniaux and P. Portal, The maximal regularity operator on tent spaces, Commun. Pure Appl. Anal., 11 (2012), 2213-2219.  doi: 10.3934/cpaa.2012.11.2213.  Google Scholar [3] P. Auscher and P. Tchamitchian., Square Root Problem for Divergence Operators and Related Topics, Ast$\acute{e}$risqueno., 249, 1998.  Google Scholar [4] D. Deng, X. T. Duong, A. Sikora and L. Yan, Comparison of the classical BMO with the BMO spaces associated with operators and applications, Rev. Mat. Iberoamericana., 24 (2008), 267-296.  doi: 10.4171/RMI/536.  Google Scholar [5] M. Dindos, C. Kenig and J. Pipher, BMO solvability and the $A_{\infty}$ condition for elliptic operators, J. Geom. Anal., 21 (2011), 78-95. doi: 10.1007/s12220-010-9142-3.  Google Scholar [6] Y. Du and K. Wang, Regularity of the solutions to the liquid crystal equations with small rough data, J. Differential Equations, 256 (2014), 65-81. doi: 10.1016/j.jde.2013.07.066.  Google Scholar [7] X. T. Duong, L. Yan and C. Zhang, On characterization of Poisson integrals of Schrödinger operators with BMO traces, J. Funct. Anal., 266 (2014), 2053-2085.  doi: 10.1016/j.jfa.2013.09.008.  Google Scholar [8] X. T. Duong and L. Yan, New function spaces of ${\rm{BMO}}$ type, the John-Nirenberg inequality, interpolation and applications, Comm. Pure Appl. Math., 58 (2005), 1375-1420.  doi: 10.1002/cpa.20080.  Google Scholar [9] X. T. Duong and L. Yan, Duality of Hardy and BMO spaces associated with operators with heat kernel bounds, J. Amer. Math. Soc., 18 (2005), 943-973.  doi: 10.1090/S0894-0347-05-00496-0.  Google Scholar [10] E. B. Fabes, R. L. Johnson and U. Neri, Spaces of harmonic functions representable by Poisson integrals of functions in BMO and $L_{p, \lambda}$, Indiana Univ. Math. J., 25 (1976), 159-170. doi: 10.1512/iumj.1976.25.25012.  Google Scholar [11] E. B. Fabes and U. Neri, Characterization of temperatures with initial data in BMO, Duke Math. J., 42 (1975), 725-734. doi: 10.1215/S0012-7094-75-04260-X.  Google Scholar [12] E. B. Fabes and U. Neri, Dirichlet problem in Lipschitz domains with BMO data, Proc. Amer. Math. Soc., 78 (1980), 33-39. doi: 10.1090/S0002-9939-1980-0548079-8.  Google Scholar [13] C. Fefferman and E. M. Stein, $H^p$ spaces of several variables, Acta Math., 129 (1972), 137-193.  doi: 10.1007/BF02392215.  Google Scholar [14] S. Hofmann, G. Lu, D. Mitrea, M. Mitrea and L. Yan, Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates, Memoirs of the Amer. Math. Soc., 214 (2011), no. 1007. doi: 10.1090/S0065-9266-2011-00624-6.  Google Scholar [15] R. Jiang, J. Xiao and D. Yang, Towards spaces of harmonic functions with traces in square Campanato spaces and their scaling invariants, Anal. Appl. (Singap.), 14 (2016), 679-703. doi: 10.1142/S0219530515500190.  Google Scholar [16] H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35.  doi: 10.1006/aima.2000.1937.  Google Scholar [17] J. Li and B. D. Wick, Characterizations of $\rm H^{1}_{\Delta_{N}}(\mathbb{R}^{n})$ and $\rm BMO_{\Delta_{N}}(\mathbb{R}^{n})$ via weak factorizations and commutators, J. Funct. Anal., 272 (2017), 5384-5416.  doi: 10.1016/j.jfa.2017.03.007.  Google Scholar [18] L. Song, X. X. Tian and L. X. Yan, On characterization of Poisson integrals of Schrödinger operators with Morry traces, Acta Math. Sin. (Engl. Ser.), 34 (2018), 787-800.  doi: 10.1007/s10114-018-7368-3.  Google Scholar [19] E. M. Stein, Topics in Harmonic Analysis Related to the Littlewood-Paley Theory, Annals of Mathematics Studies, 63, Princeton Univ. Press, Princeton, N.J., 1970.  Google Scholar [20] M. Yang and C. Zhang, Characterization of temperatures associated to Schrödinger operators with initial data in BMO spaces, to appear in Math. Nachr. arXiv: 1710.01160 Google Scholar [21] W. Yuan, W. Sickel and D. Yang, Morrey and Campanato meet Besov, Lizorkin and Triebel, Lecture Notes in Mathematics, 2005. Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-14606-0.  Google Scholar
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