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Hao-Guang Li

Publications and source records attributed to Hao-Guang Li.

10 recordsLinked to original sources
arXiv2026

Local well-posedness for the Boltzmann equation with hard potentials

We consider the spatially inhomogeneous non-cutoff Boltzmann equation with hard potentials in the non-perturbative setting. For initial data with polynomial decay in the velocity variable, we establish the local-in-time existence and uniqueness of weak solutions, conditional to pointwise bounds on the hydrodynamic quantities (mass, energy, and entropy). Compared to the soft potential case, the key challenge for full-range hard potentials lies in the more severe loss of velocity moments. The proof combines a hypoelliptic estimate with interpolation inequalities to handle the moment-loss terms.

math.AP↗
arXiv2022

Analytic Gelfand-Shilov smoothing effect of the spatially homogeneous Landau equation

In this work, we study the nonlinear spatially homogeneous Landau equation with hard potential in a close-to-equilibrium framework, we show that the solution to the Cauchy problem with $L^2$ initial datum enjoys a analytic Gelfand-Shilov regularizing effect in the class $S^1_1(\mathbb{R}^3)$, meaning that the solution of the Cauchy problem and its Fourier transformation are analytic for any positive time, the evolution of analytic radius is similar to the heat equation.

math.AP↗
arXiv2019

The Cauchy problem for the inhomogeneous non-cutoff Kac equation in critical Besov space

In this work, we investigate the Cauchy problem for the spatially inhomogeneous non-cutoff Kac equation. If the initial datum belongs to the spatially critical Besov space, we can prove the well-posedness of weak solution under a perturbation framework. Furthermore, it is shown that the solution enjoys Gelfand-Shilov regularizing properties with respect to the velocity variable and Gevrey regularizing properties with respect to the position variable. In comparison with the recent result in [18], the Gelfand-Shilov regularity index is improved to be optimal. To the best of our knowledge, our work is the first one that exhibits smoothing effect for the kinetic equation in Besov spaces.

math.AP↗
arXiv2017

Cauchy problem for the spatially homogeneous landau equation with shubin class initial datum and gelfand-shilov smoothing effect

In this work, we study the nonlinear spatially homogeneous Lan-dau equation with Maxwellian molecules, by using the spectral analysis, we show that the non linear Landau operators is almost linear, and we prove the existence of weak solution for the Cauchy problem with the initial datum belonging to Shubin space of negative index which conatins the probability measures. Based on this spectral decomposition, we prove also that the Cauchy problem enjoys Gelfand-Shilov smoothing effect, meaning that the weak solution of the Cauchy problem with Shubin class initial datum is ultra-analytics and exponential decay for any positive time.

math.AP↗
arXiv2017

The cauchy problem for radially symmetric homogeneous boltzmann equation with shubin class initial datum and gelfand-shilov smoothing effect

In this paper, we study the Cauchy problem for radially symmetric homogeneous non-cutoff Boltzmann equation with Maxwellian molecules, the initial datum belongs to Shubin space of the negative index which can be characterized by spectral decomposition of the harmonic oscillators. The Shubin space of the negative index contains the measure functions. Based on this spectral decomposition, we construct the weak solution with Shubin class initial datum, we also prove that the Cauchy problem enjoys Gelfand-Shilov smoothing effect, meaning that the smoothing properties are the same as the Cauchy problem defined by the evolution equation associated to a fractional harmonic oscillator.

math.AP↗
arXiv2016

Shubin regularity for the radially symmetric spatially homogeneous Boltzmann equation with Debye-Yukawa potential

In this work, we study the Cauchy problem for the radially symmetric spatially homogeneous Boltzmann equation with Debye-Yukawa potential. We prove that this Cauchy problem enjoys the same smoothing effect as the Cauchy problem defined by the evolution equation associated to a fractional logarithmic harmonic oscillator. To be specific, we can prove the solution of the Cauchy problem belongs to Shubin spaces.

math.AP↗
arXiv2015

Sharp regularity and Cauchy problem of the spatially homogeneous Boltzmann equation with Debye-Yukawa potential

In this paper, we study the Cauchy problem for the linear spatially homogeneous Boltzamnn equation with Debye-Yukawa potential. Using the spectral decomposition of the linear operator, we prove that, for an initial datum in the sense of distribution which contains the dual of the Sobolev spaces, there exists a unique solution which belongs to a more regular Sobolev space for any positive time. We also study the sharp regularity of the solution.

math.AP↗
arXiv2015

Sharp regularity properties for the non-cutoff spatially homogeneous Boltzmann equation

In this work, we study the Cauchy problem for the spatially homogeneous non-cutoff Boltzamnn equation with Maxwellian molecules. We prove that this Cauchy problem enjoys Gelfand-Shilov regularizing effect, that means the smoothing properties is same as the Cauchy problem defined by the evolution equation associated to a fractional harmonic oscillator. The power of this fractional is exactly the singular index of non-cutoff collisional kernel of Boltzmann operator. So that we get the regularity of solution in the Gevery class with the sharp power and the optimal exponential decay of solutions. We also give a method to construct the solution of the nonlinear Boltzmann equation by solving an infinite ``triangular'' systems of ordinary differential equations.The key tools is the spectral decomposition of linear and non-linear Boltzmann operators.

math.AP↗