中国科学院数学与系统科学研究院期刊网

15 April 2018, Volume 34 Issue 4
    

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  • Acta Mathematica Sinica. 2018, 34(4): 597-597. https://doi.org/10.1007/s10114-018-7998-5
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  • Hao JIA, Vladimír ŠVERÁK
    Acta Mathematica Sinica. 2018, 34(4): 598-611. https://doi.org/10.1007/s10114-017-7397-3
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    We show that the asymptotics of solutions to stationary Navier Stokes equations in 4, 5 or 6 dimensions in the whole space with a smooth compactly supported forcing are given by the linear Stokes equation. We do not need to assume any smallness condition. The result is in contrast to three dimensions, where the asymptotics for steady states are different from the linear Stokes equation, even for small data, while the large data case presents an open problem. The case of dimension n=2 is still harder.
  • Jun GENG
    Acta Mathematica Sinica. 2018, 34(4): 612-628. https://doi.org/10.1007/s10114-017-7229-5
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    We consider a family of second-order elliptic operators {Lε} in divergence form with rapidly oscillating and periodic coefficients in Lipschitz and convex domains in Rn. We are able to show that the uniform W1,p estimate of second order elliptic systems holds for (2n)/(n+1)-δ < p < (2n)/(n-1)+δ where δ > 0 is independent of ε and the ranges are sharp for n=2, 3. And for elliptic equations in Lipschitz domains, the W1,p estimate is true for (3)/2 -δ < p < 3 + δ if n ≥ 4, similar estimate was extended to convex domains for 1 < p < ∞.
  • Meng Yun LIU, Cheng Bo WANG
    Acta Mathematica Sinica. 2018, 34(4): 629-640. https://doi.org/10.1007/s10114-017-7138-7
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    In this paper, we prove the global existence for some 4-D quasilinear wave equations with small, radial data in H3×H2. The main idea is to exploit local energy estimates with variable coefficients, together with the trace estimates.
  • Jun Yong ZHANG, Ji Qiang ZHENG
    Acta Mathematica Sinica. 2018, 34(4): 641-654. https://doi.org/10.1007/s10114-018-7253-0
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    We study the well-posedness and long-time behavior of solution to both defocusing and focusing nonlinear Schrödinger equations with scaling critical magnetic potentials in dimension two. In the defocusing case, and under the assumption that the initial data is radial, we prove interaction Morawetz-type inequalities and show the scattering holds in the energy space. The magnetic potential considered here is the Aharonov-Bohm potential which decays likely the Coulomb potential|x|-1.
  • Yu Kang CHEN, Zhen LEI, Chang Hua WEI
    Acta Mathematica Sinica. 2018, 34(4): 655-661. https://doi.org/10.1007/s10114-017-7325-6
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    Caffarelli and Silvestre[Comm. Part. Diff. Eqs., 32, 1245-1260 (2007)] characterized the fractional Laplacian (-Δ)s as an operator maps Dirichlet boundary condition to Neumann condition via the harmonic extension problem to the upper half space for 0 < s < 1. In this paper, we extend this result to all s > 0. We also give a new proof to the dissipative a priori estimate of quasi-geostrophic equations in the framework of Lp norm using the Caffarelli-Silvestre's extension technique.
  • Jiang XU
    Acta Mathematica Sinica. 2018, 34(4): 662-680. https://doi.org/10.1007/s10114-017-7344-3
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    In the recent work, we have developed a decay framework in general Lp critical spaces and established optimal time-decay estimates for barotropic compressible Navier-Stokes equations. Those decay rates of Lq-Lr type of the solution and its derivatives are available in the critical regularity framework, which were exactly firstly observed by Matsumura & Nishida, and subsequently generalized by Ponce for solutions with high Sobolev regularity. We would like to mention that our approach is likely to be effective for other hyperbolic/parabolic systems that are encountered in fluid mechanics or mathematical physics. In this paper, a new observation is involved in the high frequency, which enables us to improve decay exponents for the high frequencies of solutions.
  • Jin Gang XIONG
    Acta Mathematica Sinica. 2018, 34(4): 681-690. https://doi.org/10.1007/s10114-018-7309-1
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    We study a prescribing functions problem of a conformally invariant integral equation involving Poisson kernel on the unit ball. This integral equation is not the dual of any standard type of PDE. As in Nirenberg problem, there exists a Kazdan-Warner type obstruction to existence of solutions. We prove existence in the antipodal symmetry functions class.
  • Man Ru JIANG, Ren Jin JIANG
    Acta Mathematica Sinica. 2018, 34(4): 691-698. https://doi.org/10.1007/s10114-018-7310-8
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    In this note, we show that, for domains satisfying the separation property, certain weighted Korn inequality is equivalent to the John condition. Our result generalizes previous result from Jiang-Kauranen[Calc. Var. Partial Differential Equations, 56, Art. 109, (2017)] to weighted settings.
  • Su Qing, WU Da Chun, YANG Wen YUAN, Ci Qiang ZHUO
    Acta Mathematica Sinica. 2018, 34(4): 699-748. https://doi.org/10.1007/s10114-018-7311-7
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    This article is devoted to the study of variable 2-microlocal Besov-type and Triebel-Lizorkin-type spaces. These variable function spaces are defined via a Fourier-analytical approach. The authors then characterize these spaces by means of φ-transforms, Peetre maximal functions, smooth atoms, ball means of differences and approximations by analytic functions. As applications, some related Sobolev-type embeddings and trace theorems of these spaces are also established. Moreover, some obtained results, such as characterizations via approximations by analytic functions, are new even for the classical variable Besov and Triebel-Lizorkin spaces.
  • Yuan Yuan NIE, Chun Peng WANG
    Acta Mathematica Sinica. 2018, 34(4): 749-772. https://doi.org/10.1007/s10114-017-7341-6
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    This paper concerns continuous subsonic-sonic potential flows in a two-dimensional convergent nozzle. It is shown that for a given nozzle which is a perturbation of a straight one, a given point on its wall where the curvature is zero, and a given inlet which is a perturbation of an arc centered at the vertex, there exists uniquely a continuous subsonic-sonic flow whose velocity vector is along the normal direction at the inlet and the sonic curve, which satisfies the slip conditions on the nozzle walls and whose sonic curve intersects the upper wall at the given point. Furthermore, the sonic curve of this flow is a free boundary, where the flow is singular in the sense that the speed is only C1/2 Hölder continuous and the acceleration blows up. The perturbation problem is solved in the potential plane, where the flow is governed by a free boundary problem of a degenerate elliptic equation with two free boundaries and two nonlocal boundary conditions, and the equation is degenerate at one free boundary.
  • Hong Liang FENG, Hua WANG, Xiao Hua YAO
    Acta Mathematica Sinica. 2018, 34(4): 773-786. https://doi.org/10.1007/s10114-018-7343-z
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    Based on the endpoint Strichartz estimates for the fourth order Schrödinger equation with potentials for n ≥ 5 by[Feng, H., Soffer, A., Yao, X.:Decay estimates and Strichartz estimates of the fourth-order Schrödinger operator. J. Funct. Anal., 274, 605-658 (2018)], in this paper, the authors further derive Strichartz type estimates with gain of derivatives similar to the one in[Pausader, B.:The cubic fourth-order Schrödinger equation. J. Funct. Anal., 256, 2473-2517 (2009)]. As their applications, we combine the classical Morawetz estimate and the interaction Morawetz estimate to establish scattering theory in the energy space for the defocusing fourth order NLS with potentials and pure power nonlinearity 1 + (8)/n < p < 1 +(8)/n-4 in dimensions n ≥ 7.
  • Liang SONG, Xiao Xiao TIAN, Li Xin YAN
    Acta Mathematica Sinica. 2018, 34(4): 787-800. https://doi.org/10.1007/s10114-018-7368-3
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    Let L be a Schrödinger operator of the form L=-Δ + V acting on L2(Rn) where the nonnegative potential V belongs to the reverse Hölder class Bq for some qn. In this article we will show that a function fL2,λ(Rn), 0 < λ < n, is the trace of the solution of Lu=-utt + Lu=0, u(x, 0)=f(x), where u satisfies a Carleson type condition

    Its proof heavily relies on investigate the intrinsic relationship between the classical Morrey spaces and the new Campanato spaces LL2,λ(Rn) associated to the operator L, i.e.
    LL2,λ (Rn)=L2,λ(Rn).Conversely, this Carleson type condition characterizes all the L-harmonic functions whose traces belong to the space L2,λ(Rn) for all 0 < λ < n.