ON THE DIMENSION OF THE DIVERGENCE SET OF THE OSTROVSKY EQUATION

doi:10.1007/s10473-022-0418-z

Abstract

Abstract: We investigate the refined Carleson's problem of the free Ostrovsky equation \begin{equation*} \left\{ \begin{aligned} & u_t+\partial_x^3u+\partial_x^{-1}u=0,\\ & u(x,0)=f(x), \end{aligned} \right. \end{equation*} where $(x,t)\in\mathbb{R}\times\mathbb{R}$ and $f\in H^s(\mathbb{R})$. We illustrate the Hausdorff dimension of the divergence set for the Ostrovsky equation \begin{equation*} \alpha_{1,U}(s)=1-2s,\quad \frac{1}{4}\leq s\leq\frac{1}{2}. \end{equation*}

Key words: Free Ostrovsky equation, Hausdorff dimension, divergence set

CLC Number:

35Q53

Yajuan ZHAO, Yongsheng LI, Wei YAN, Xiangqian YAN. ON THE DIMENSION OF THE DIVERGENCE SET OF THE OSTROVSKY EQUATION[J].Acta mathematica scientia,Series B, 2022, 42(4): 1607-1620.

Trendmd

References

[1] Adams D R. Anote on the Choquet integrals with respect to Hausdorff capacity. Function spaces and applications[M]. Berline:Springer-Verlag, 1988
[2] Barceló J A, Bennett J, Carbery A, et al, On the dimension of divergence sets of dispersive equations[J]. Math Ann, 2011, 349:599-622
[3] Bourgain J, On the Schrödinger maximal function in higher dimension[J]. Proc Steklov Inst Math, 2013, 280:46-60
[4] Bourgain J, A note on the Schrödinger maximal function[J]. J Anal Math, 2016, 130:393-396
[5] Carleson L. Some analytical problems related to statistical mechanics//Euclidean Harmonic Analysisi[M]. Berlin:Springer, 1979
[6] Coclite G M, di Ruvo L, On the solutions for an Ostrovsky type equation[J]. Nonlinear Anal Real World Appl, 2020, 55:31 pp
[7] Dahlberg B E, Kenig C E. Anote on the almost everywhere behavior of solutions to the Schrödinger equation[M]. Berlin:Springer, 1981
[8] Ding Y, Niu Y, Global L² estimates for a class of maximal operators associated to general dispersive equations[J]. J Inequal Appl, 2015, 199:20 pp
[9] Ding Y, Niu Y, Maximal estimate for solutions to a class of dispersive equation with radial initial value[J]. Front Math China, 2017, 12:1057-1084
[10] Du X M, Guth L, Li X C, A sharp Schrödinger maximal estimate in $\mathbb{R}$²[J]. Ann Math, 2017, 188:607-640
[11] Du X M, Zhang R X. Sharp L² estimates of the Schrödinger maximal function in higher dimensions[J]. Ann Math, 2019, 189:837-861
[12] Galkin V N, Stepanyants Y A, On the existence of stationary solitary waves in a rotating fluid[J]. J Appl Math Mech, 1991, 55:939-943
[13] Gui G L, Liu Y, On the Cauchy problem for the Ostrovsky equation with positive dispersion[J]. Comm Partial Differential Equations, 2007, 32:1895-1916
[14] Huo Z H, Jia Y L, Low-regularity solutions for the Ostrovsky equation[J]. Proc Edinb Math Soc, 2006, 49:87-100
[15] Isaza P, Mejía J, Cauchy problem for the Ostrovsky equation in spaces of low regularity[J]. J Diff Eqns, 2006, 230:661-681
[16] Isaza P, Mejía J, On the support of solutions to the Ostrovsky equation withpositive dispersion[J]. Nonlinear Anal TMA, 2010, 72:4016-4029
[17] Kenig C E, Ponce G, Vega L, Oscillatory integrals and regularity of dispersive equations[J]. India Uni Math J, 1991, 40:33-69
[18] Lee S. On pointwise convergence of the solutions to Schrödinger equations in $\mathbb{R}$²[J]. Int Math Res Not, 2006, Art ID 32597, 21 pp
[19] Leonov A, The effect of the earth's rotation on the propagation of weak nonlinear surface and internal long oceanic waves[J]. Ann New York Acad Sci, 1981, 373:150-159
[20] Li D, Li J F, On 4-order Schröodinger operator and Beam operator[J]. Front Math China, 2019, 14:1197-1211
[21] Li D, Li J F, Xiao J. A Carleson problem for the Boussinesq operator[J]. arXiv:1912.09636v1[math.CA] 20 Dec 2019
[22] Linares F, Milanés A, Local and global well-posedness for the Ostrovsky equation[J]. J Diff Eqns, 2006, 222:325-340
[23] Lucà R, Rogers K, A note on pointwise convergence for the Schrödinger equation[J]. Math Proc Cambridge Philos Soc, 2019, 166:209-218
[24] Mattila P. Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability[M]. Cambridge:Cambridge University, 1995
[25] Miao C X, Zhang J Y, Zheng J Q, Maximal estimates for Schrödinger equation with inverse-square potential[J]. Pac J Math, 2015, 273:1-19
[26] Ostrovskii L A, Nonlinear internal waves in a rotating ocean[J]. Okeanologiya, 1978, 18:181-191
[27] Sjögren P, Sjölin P, Convergence properties for the time-dependent Schröodinger equation. Ann Acad Sci Fenn Ser A I Math, 1989, 14:13-25
[28] Sjöolin P, Maximal estimates for solutions to the nonelliptic Schröodinger equation[J]. Bull Lond Math Soc, 2007, 39:404-412
[29] Stein E M. Harmonic Analysis:real-variable methods, orthogonality, and oscillatory integrals[M]. Princeton:Princeton University, 1993
[30] Varlamov V, Liu Y, Cauchy problem for the Ostrovsky equation[J]. Discrete Contin Dyn Syst, 2004, 10:731-753
[31] Vega L, Schrödinger equations:pointwise convergence to the initial data[J]. Proc Amer Math Soc, 1988, 102:874-878
[32] Yan W, Li Y S, Huang J H, et al, The Cauchy problem for the Ostrovsky equation with positive dispersion[J]. NoDEA Nonlinear Differential Equations Appl, 2018, 25:37 pp
[33] Yan W, Zhang Q Q, Duan J Q, et al. Pointwise convergence problem of Ostrovsky equation with rough data and random data. arXiv:2006.15981v1[math.AP] 24 Jun 2020
[34] Wang J F, Yan W, The Cauchy problem for quadratic and cubic Ostrovsky equation with negative dispersion[J]. Nonlinear Anal Real World Appl, 2018, 43:283-307
[35] Žubrinić D, Singular sets of Sobolev functions[J]. C R Math Acad Sci Paris, 2002, 334:539-544