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Changxing Miao

Publications and source records attributed to Changxing Miao.

At least 55 records · Page 3Linked to original sources
arXiv2024

On the weak solutions for the MHD systems with controllable total energy and cross helicity

In this paper, we prove the non-uniqueness of three-dimensional magneto-hydrodynamic (MHD) system in $C([0,T];L^2(\mathbb{T}^3))$ for any initial data in $H^{\barβ}(\mathbb{T}^3)$~($\barβ>0$), by exhibiting that the total energy and the cross helicity can be controlled in a given positive time interval. Our results extend the non-uniqueness results of the ideal MHD system to the viscous and resistive MHD system. Different from the ideal MHD system, the dissipative effect in the viscous and resistive MHD system prevents the nonlinear term from balancing the stress error $(\RR_q,\MM_q)$ as doing in \cite{2Beekie}. We introduce the box flows and construct the perturbation consisting in six different kinds of flows in convex integral scheme, which ensures that the iteration works and yields the non-uniqueness.

math.AP↗
arXiv2024

Square function inequality for oscillatory integral operators satisfying homogeneous Carleson-Sjölin type conditions

In this paper, we establish an improved variable coefficient version of square function inequality, by which the local smoothing estimate $L^p_α\rightarrow L^p$ for the Fourier integral operators satisfying cinematic curvature condition is further improved. In particular, we establish almost sharp results for $2<p\leq 3$ and push forward the estimate for the critical point $p=4$. As a consequence, the local smoothing estimate for the wave equation on the manifold is refined. We generalize the results in \cite{LeVa12, Le18P} to its variable coefficient counterpart. The main ingredients in the argument includes multilinear oscillatory integral estimate \cite{BCT06} and decoupling inequality \cite{BelHicSog18P}.

math.AP↗
arXiv2024

The space-time estimates for the Schrödinger equation

In this paper, we studied the space-time estimates for the solution to the Schrödinger equation. By polynomial partitioning, induction arguments, bilinear to linear arguments and broad norm estimates, we set up several maximal estimates for the Schrödinger equation with high-frequency input data. By these maximal estimates, we obtain the sharp global space-time estimate when $n=2$ and improve the known results in the critical cases when $n\geq 3$. The maximal estimate for $n=2$ is also used to extend the results of the local space-time estimates for the solution to the Schrödinger equation.

math.CA↗
arXiv2024

Fourier decay of fractal measures on surfaces of co-dimension two in $\mathbb{R}^5$

Fourier decay of fractal measures on surfaces plays an important role in geometric measure theory and partial differential equations. In this paper, we study the quadratic surfaces of high co-dimensions. Unlike the case of co-dimension 1, quadratic surfaces of high co-dimensions possess some special scaling structures and degenerate characteristics. We will adopt the strategy from Du and Zhang, combined with the broad-narrow analysis with different dimensions as divisions, to obtain a few lower bounds of Fourier decay of fractal measures on quadratic surfaces of co-dimension two in $\mathbb{R}^5$.

math.CA↗
arXiv2023

Local behavior for solutions to anisotropic weighted quasilinear degenerate parabolic equations

This paper aims to study the local behavior of solutions to a class of anisotropic weighted quasilinear degenerate parabolic equations with the weights comprising two power-type weights of different dimensions. We first capture the asymptotic behavior of the solution near the singular or degenerate point of the weights. In particular, we find an explicit upper bound on the decay rate exponent determined by the structures of the equations and weights, which can be achieved under certain condition and meanwhile reflects the damage effect of the weights on the regularity of the solution. Furthermore, we prove the local Hölder regularity of solutions to non-homogeneous parabolic $p$-Laplace equations with single power-type weights.

math.AP↗
arXiv2023

$L^{p}$-estimate of Schrödinger maximal function in higher dimensions

Almost everywhere convergence on the solution of Schrödinger equation is an important problem raised by Carleson in harmonic analysis. In recent years, this problem was essentially solved by building the sharp $L^p$-estimate of Schrödinger maximal function. Du-Guth-Li in \cite{DGL} proved the sharp $L^p$-estimates for all $p \geq 2$ in $\mathbb{R}^{2+1}$. Du-Zhang in \cite{DZ} proved the sharp $L^2$-estimate in $\mathbb{R}^{n+1}$ with $n \geq 3$, but for $p>2$ the sharp $L^p$-estimate of Schrödinger maximal function is still unknown. In this paper, we obtain partial results on this problem by using polynomial partitioning.

math.AP↗
arXiv2023

Maximal estimates for the Weyl sums on $\mathbb{T}^{d}$ (with an appendix by Alex Barron)

In this paper, we obtain the maximal estimate for the Weyl sums on the torus $\mathbb{T}^d$ with $d\geq 2$, which is sharp up to the endpoint. We also consider two variants of this problem which include the maximal estimate along the rational lines and on the generic torus. Applications, which include some new upper bound on the Hausdorff dimension of the sets associated to the large value of the Weyl sums, reflect the compound phenomenon between the square root cancellation and the constructive interference. In the Appendix, an alternate proof of Theorem 1.1 inspired by Baker's argument in [1] is given by Barron, which also improves the $N^ε$ loss in Theorem 1.1, and the Strichartz-type estimates for the Weyl sums with logarithmic losses are obtained by the same argument.

math.NT↗
arXiv2023

Square function estimates and Local smoothing for Fourier Integral Operators

We prove a variable coefficient version of the square function estimate of Guth--Wang--Zhang. By a classical argument of Mockenhaupt--Seeger--Sogge, it implies the full range of sharp local smoothing estimates for $2+1$ dimensional Fourier integral operators satisfying the cinematic curvature condition. In particular, the local smoothing conjecture for wave equations on compact Riemannian surfaces is completely settled.

math.AP↗
arXiv2023

Improved local smoothing estimate for the wave equation in higher dimensions

In this paper, we establish the sharp $k$-broad estimate for a class of phase functions satisfying the homogeneous convex conditions. As an application, we obtain improved local smoothing estimates for the half-wave operator in dimensions $n\ge3$. As a byproduct, we also generalize the restriction estimates of Ou--Wang to a broader class of phase functions.

math.AP↗
arXiv2022

Local regularity for nonlinear elliptic and parabolic equations with anisotropic weights

The main purpose of this paper is to capture the asymptotic behavior for solutions to a class of nonlinear elliptic and parabolic equations with the anisotropic weights consisting of two power-type weights of different dimensions near the degenerate or singular point, especially covering the weighted $p$-Laplace equations and weighted fast diffusion equations. As a consequence, we also establish the local Hölder estimates for their solutions in the presence of single power-type weights.

math.AP↗
arXiv2022

Global regularity and decay behavior for Leray equations with critical-dissipation and Its Application to Self-similar Solutions

In this paper, we show the global regularity and the optimal decay of weak solutions to the generalized Leray problem with critical dissipation. Our method is based on the maximal smoothing effect, $L^{p}$-type elliptic regularity of linearization, and the action of the heat semigroup generated by the fractional powers of Laplace operator on distributions with Fourier transforms supported in an annulus. As a by-product, we shall construct a self-similar solution to the three-dimensional incompressible Navier-Stokes equations, and more importantly, prove the global regularity and the optimal decay without additional requirement of existing literatures.

math.AP↗
arXiv2022

Construction of solutions to the 3D Euler equations with initial data in $H^β$ for $β>0$

In this paper, we use the method of convex integration to construct infinitely many distributional solutions in $H^β$ for $0<β\ll1$ to the initial value problem for the three-dimensional incompressible Euler equations. We show that if the initial data has any small fractional derivative in $L^2$, then we can construct solutions with some regularity, so that the corresponding $L^2$ energy is continuous in time. This is distinct from the $L^2$ existence result of E. Wiedemann, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 28, No. 5, 727--730 (2011; Zbl 1228.35172), where the energy is discontinuous at $0$.

math.AP↗
arXiv2022

Stress concentration factors for the Lamé system arising from composites

For two neighbouring stiff inclusions, the stress, which is the gradient of a solution to the Lamé system of linear elasticity, may exhibit singular behavior as the distance between these two inclusions becomes arbitrarily small. In this paper, a family of stress concentration factors, which determine whether the stress will blow up or not, are accurately constructed in the presence of the generalized $m$-convex inclusions in all dimensions. We then use these stress concentration factors to establish the optimal upper and lower bounds on the stress blow-up rates in any dimension and meanwhile give a precise asymptotic expression of the stress concentration for interfacial boundaries of inclusions with different principal curvatures in dimension three. Finally, the corresponding results for the perfect conductivity problem are also presented.

math.AP↗
arXiv2022

Singular analysis of the stress concentration in the narrow regions between the inclusions and the matrix boundary

We consider the Lamé system arising from high-contrast composite materials whose inclusions (fibers) are nearly touching the matrix boundary. The stress, which is the gradient of the solution, always concentrates highly in the narrow regions between the inclusions and the external boundary. This paper aims to provide a complete characterization in terms of the singularities of the stress concentration by accurately capturing all the blow-up factor matrices and making clear the dependence on the Lamé constants and the curvature parameters of geometry. Moreover, the precise asymptotic expansions of the stress concentration are also presented in the presence of a strictly convex inclusion close to touching the external boundary for the convenience of application.

math.AP↗
arXiv2022

Soliton resolution for the energy critical wave equation with inverse-square potential in the radial case

In this paper, we establish the soliton resolution for the energy critical wave equation with inverse square potential in the radial case and in all dimensions $N\geq3$. The structure of the radial linear operator $\mathcal{L}_a :=-Δ+\frac{a}{|x|^2}=A^*A$, is essential for the channel of energy, where $A$ is a first order differential operator and $A^*$ is its adjoint operator. Modulation and analysis of the multi-solitons are performed in the function spaces $\dot{H}^1_a(\Bbb R^N)\times L^2(\Bbb R^N)$ associated with $\mathcal{L}_a$.

math.AP↗