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【期刊论文】The energy-critical nonlinear wave equation with an inverse-square potential
Changxing Miao, Jason .Murph, Jiqiang Zheng
Ann. Inst. Henri Poincare-Nonlinear Analysis,2020,37(2):417-456
2020年03月01日
We study the energy-critical nonlinear wave equation in the presence of an inverse-square potential in dimensions three and four. In the defocussingcase, we prove that arbitrary initial data in the energy space lead to global solutions that scatter. In the focusing case, we prove scattering below the ground state threshold.
Nonlinear wave equation, Inverse-square potential, Energy-critical, Scattering, Ground state threshold
调和分析、偏微分方程、自伴算子的谱理论
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【期刊论文】Linear adjoint restriction estimates for paraboloid
Changxing Miao, Junyong Zhang, Jiqiang Zheng
Mathematische Zeitschrift ,2019,292(2):427-451
2019年02月10日
We prove a class of modified paraboloid restriction estimates with a loss of angular derivatives for the full set of paraboloid restriction conjecture indices. This result generalizes the paraboloid restriction estimate in radial case from [Shao, Rev. Mat. Iberoam. 25(2009), 1127-1168], as well as the result from [Miao et al. Proc. AMS 140(2012), 2091-2102]. As an application, we show a local smoothing estimate for a solution of the linear Schr\"odinger equation under the assumption that the initial datum has additional angular regularity.
Linear adjoint restriction estimate,, local restriction estimate,, , Bessel function,, spherical harmonics,, local smoothing.,
调和分析、自伴算子的谱理论、函数空间理论
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【期刊论文】The two-dimensional Euler equation in Yudovich and bmo-type spaces
Qionglei Chen, Changxing Miao, Xiaoxin Zheng
Rev. Mat. Iberoam. ,2019,35(1):195–240
2019年01月01日
We construct global-in-time, unique solutions of the two-dimensional Euler equations in a Yudovich type space and a $\rm bmo$-type space. First, we show the regularity of solutions for the two-dimensional Euler equations in the Spanne space involving an unbounded and non-decaying vorticity. Next, we establish an estimate with a logarithmic loss of regularity for the transport equation in a bmo-type space by developing classical analysis tool such as the John-Nirenberg inequality. We also optimize estimates of solutions to the vorticity-stream formulation of the two-dimensional Euler equations with a bi-Lipschitz vector field in bmo-type space by combining an observation introduced in \cite{Y1} by Yodovich with the so-called ``quasi-conformal property" of the incompressible flow.
Two-dimensional incompressible Euler equations,, Yudovich type data, John– Nirenberg inequality,, global existence and uniqueness of solutions.,
偏微分方程、调和分析、流体力学
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56浏览
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【期刊论文】On the regularity issues of a class of drift-diffusion equations with nonlocal diffusion
Changxing Miao, Liutang Xue
SIAM J. Math. Anal.,2019,51(4): 2927-2970
2019年02月15日
In this paper we address the regularity issues of drift-diffusion equation with nonlocal diffusion, where the diffusion operator is in the realm of stable-type L\'evy operator and the velocity field is defined from the considered quantity by a zero-order pseudo-differential operator. Through using the method of nonlocal maximum principle in a unified way, we prove the eventual regularity result in the supercritical type cases and the global regularity at some logarithmically supercritical cases. The feature of these results is that the time after which the solution is smoothly regular in the supercritical type cases can be evaluated explicitly.
Drift-diffusion equation,, surface quasi-geostrophic equation,, nonlocal maximum principle, Levy operator,, regularity
偏微分方程、调和分析、流体力学
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【期刊论文】Forward self-similar solutions of the fractional Navier-Stokes equations
Baishun Lai, Changxing Miao, Xiaoxin Zheng
Advances in Mathematics,2019,352(2):981–1043
2019年06月28日
We study forward self-similar solutions to the 3-D Navier-Stokes equations with the fractional diffusion (−Δ)^α. First, we construct a global-time forward self-similar solutions to the fractional Navier-Stokes equations with 5/6 <α ≤1for arbitrarily large self-similar initial data by making use of the so called blow-up argument. Moreover, we prove that this so-lution is smooth in R^3×(0, +∞). In particular, when α =1, we prove that the solution constructed by Korobkov and Tsai (2016) [16]satisfies the decay estimate by establishing regu-larity of solution for the corresponding elliptic system, which implies this solution has the same properties as a solution which was constructed in Jia and Šverák (2014) [13].
Self-similar solution,, Nonlocal smoothing effect,, Blowup argument,, The weighted estimate
偏微分方程、调和分析、流体力学
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