arXiv subjects

Mingjuan Chen

Publications and source records attributed to Mingjuan Chen.

5 recordsLinked to original sources

arXiv2024

Global Well-posedness for the Fourth-order Nonlinear Schrodinger Equation

The local and global well-posedness for the one dimensional fourth-order nonlinear Schrödinger equation are established in the modulation space $M^{s}_{2,q}$ for $s\geq \frac12$ and $2\leq q <\infty$. The local result is based on the $U^p-V^p$ spaces and crucial bilinear estimates. The key ingredient to obtain the global well-posedness is that we achieve a-priori estimates of the solution in modulation spaces by utilizing the power series expansion of the perturbation determinant introduced by Killip-Visan-Zhang for completely integrable PDEs.

math.AP↗

arXiv2019

Long-time Asymptotic Behavior of the Fifth-order Modified KdV Equation in Low Regularity Spaces

Based on the nonlinear steepest descent method of Deift and Zhou for oscillatory Riemann--Hilbert problems and the Dbar approach, the long-time asymptotic behavior of solutions to the fifth-order modified Korteweg-de Vries equation on the line is studied in the case of initial conditions that belong to some weighted Sobolev spaces. Using techniques in Fourier analysis and the idea of $I$-method, we give its global well-posedness in lower regularity Sobolev spaces, and then obtain the asymptotic behavior in these spaces with weights.

math.AP↗

arXiv2019

Uniform local well-posedness and inviscid limit for the Benjamin-Ono-Burgers equation

In this paper, we study the Cauchy problem for the Benjamin-Ono-Burgers equation $\partial_t u-ε\partial_x^2 u+\mathcal{H}\partial_x^2u+u u_x=0$, where $\mathcal{H}$ denotes the Hilbert transform. We obtain that it is uniformly locally well-posed for small data in the refined Sobolev space $\widetilde{H}^σ(\mathbb{R})$($σ\geq 0$), whose low-frequency part is scaling critical and high-frequency part is equal to Sobolev space $H^σ$($σ\geq 0$). Furthermore, we also obtain its inviscid limit behavior in $\widetilde{H}^σ(\mathbb{R})$($σ\geq 0$).

math.AP↗

arXiv2018

Local Well and Ill Posedness for the Modified KdV Equations in Subcritical Modulation Spaces

We consider the Cauchy problem of the modified KdV equation (mKdV). Local well-posedness of this problem is obtained in modulation spaces $M^{1/4}_{2,q}(\mathbb{R})$ $(2\leq q\leq\infty)$. Moreover, we show that the data-to-solution map fails to be $C^3$ continuous in $M^{s}_{2,q}(\mathbb{R})$ when $s<1/4$. It is well-known that $H^{1/4}$ is a critical Sobolev space of mKdV so that it is well-posedness in $H^s$ for $s\geq 1/4$ and ill-posed (in the sense of uniform continuity) in $H^{s'}$ with $s'<1/4$. Noticing that $M^{1/4}_{2,q} \subset B^{1/q-1/4}_{2,q}$ is a sharp embedding and $H^{-1/4}\subset B^{-1/4}_{2,\infty}$, our results contains all of the subcritical data in $M^{1/4}_{2,q}$, which contains a class of functions in $H^{-1/4}\setminus H^{1/4}$.

math.AP↗

arXiv2017

On Dissipative Nonlinear Evolutional Pseudo-Differential Equations

First, using the uniform decomposition in both physical and frequency spaces, we obtain an equivalent norm on modulation spaces. Secondly, we consider the Cauchy problem for the dissipative evolutionary pseudo-differential equation \partial_t u + A(x,D) u = F\big((\partial^α_x u)_{|α|\leq κ}\big), \ \ u(0,x)= u_0(x), where $A(x,D)$ is a dissipative pseudo-differential operator and $F(z)$ is a multi-polynomial. We will develop the uniform decomposition techniques in both physical and frequency spaces to study its local well posedness in modulation spaces $M^s_{p,q}$ and in Sobolev spaces $H^s$. Moreover, the local solution can be extended to a global one in $L^2$ and in $H^s$ ($s>κ+d/2$) for certain nonlinearities.

math.AP↗