arXiv subjects

Chenmin Sun

Publications and source records attributed to Chenmin Sun.

32 records · Page 2Linked to original sources

arXiv2021

Decay for the Kelvin-Voigt damped wave equation: Piecewise smooth damping

We study the energy decay rate of the Kelvin-Voigt damped wave equation with piecewise smooth damping on the multi-dimensional domain. Under suitable geometric assumptions on the support of the damping, we obtain the optimal polynomial decay rate which turns out to be different from the one-dimensional case studied in \cite{LR05}. This optimal decay rate is saturated by high energy quasi-modes localised on geometric optics rays which hit the interface along non orthogonal neither tangential directions. The proof uses semi-classical analysis of boundary value problems.

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arXiv2021

Concerning the pathological set in the context of probabilistic well-posedness

We prove a complementary result to the probabilistic well-posedness for the nonlinear wave equation. More precisely, we show that there is a dense set $S$ of the Sobolev space of super-critical regularity such that (in sharp contrast with the probabilistic well-posedness results) the family of global smooth solutions, generated by the convolution with some approximate identity of the elements of $S$, does not converge in the space of super-critical Sobolev regularity.

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arXiv2021

Observability of Baouendi-Grushin-Type Equations Through Resolvent Estimates

In this article, we study the observability (or, equivalently, the controllability) of some subelliptic evolution equations depending on their step. This sheds light on the speed of propagation of these equations, notably in the "degenerated directions" of the subelliptic structure. First, for any $γ\geq 1$, we establish a resolvent estimate for the Baouendi-Grushin-type operator $Δ_γ=\partial_x^2+|x|^{2γ}\partial_y^2$, which has step $γ+1$. We then derive consequences for the observability of the Schr\''odinger type equation $i\partial_tu-(-Δ_γ)^{s}u=0$ where $s\in N$. We identify three different cases: depending on the value of the ratio $(γ+1)/s$, observability may hold in arbitrarily small time, or only for sufficiently large times, or even fail for any time. As a corollary of our resolvent estimate, we also obtain observability for heat-type equations $\partial_tu+(-Δ_γ)^su=0$ and establish a decay rate for the damped wave equation associated with $Δ_γ$.

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arXiv2021

Refined probabilistic global well-posedness for the weakly dispersive NLS

We continue our study of the cubic fractional NLS with very weak dispersion $α>1$ and data distributed according to the Gibbs measure. We construct the natural strong solutions for $α>α_0=\frac{31-\sqrt{233}}{14}\approx 1.124$ which is strictly smaller than $\frac{8}{7}$, the threshold beyond which the first nontrivial Picard iteration has no longer the Sobolev regularity needed for the deterministic well-posedness theory. This also improves our previous result in Sun-Tzvetkov \cite{Sun-Tz2}. We rely on recent ideas of Bringmann \cite{Bringmann} and Deng-Nahmod-Yue \cite{Deng2}. In particular we adapt to our situation the new resolution ansatz in \cite{Deng2} which captures the most singular frequency interaction parts in the $X^{s,b}$ type space. To overcome the difficulties caused by the weakly dispersive effect, our specific strategy is to benefit from the "almost" transport effect of these singular parts and to exploit their $L^{\infty}$ as well as the Fourier-Lebesgue property in order to inherit the random feature from the linear evolution of high frequency portions.

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arXiv2021

Decays rates for Kelvin-Voigt damped wave equations II: the geometric control condition

We study in this article decay rates for Kelvin-Voigt damped wave equations under a geometric control condition. We prove that when the damping coefficient is sufficiently smooth ($C^1$ vanishing nicely) we show that exponential decay follows from geometric control conditions (see~\cite{BuCh, Te12} for similar results under stronger assumptions on the damping function).

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arXiv2021

Time optimal observability for Grushin Schrödinger equation

We consider two dimensional Grushin Schrödinger equation posed on a finite cylinder $Ω=(-1,1)_x\times \T_y$ with Dirichlet boundary condition. We obtain the sharp observability by any horizontal strip, with the optimal time $T_*>0$ depending on the size of the strip. Consequently, we prove the exact controllability for Grushin Schrödinger equation. By exploiting the concentration of eigenfunctions of harmonic oscillator at $x=0$, we also show that the observability fails for any $T\leq T_*$.

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arXiv2020

On the stabilization of a hyperbolic Stokes system under geometric control condition

In this paper, we study the stabilization problem for a hyperbolic type Stokes system posed on a bounded domain. We show that when the damping effects are restricted to a subdomain satisfying the geometrical control condition the system decays exponentially. The result is a consequence of a new quasi-mode estimate for the Stokes system.

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arXiv2020

Internal controllability of non-localized solution for the Kodomtsev-Petviashvili II equation

The internal control problem for the Kadomstev-Petviashvili II equation, known as KP-II, is the object of study in this paper. The controllability in $L^2(T)$ from vertical strip is proved using the Hilbert Unique Method through the techniques of semiclassical and microlocal analysis. Additionally, a negative result for the controllability in $L^2(T)$ from horizontal strip is also showed.

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arXiv2020

Semi-classical propagation of singularity for Stokes system

We study the quasi-mode of Stokes system posed on a smooth bounded domain with Dirichlet boundary condition. We prove that the high energy L2 norm of solutions concentrate on the bi-characteristic of Laplace operator as matrix-valued Radon measure. Moreover, we prove that the support of such measure is invariant under Melrose-Sjöstrand flow.

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arXiv2019

New examples of probabilistic well-posedness for nonlinear wave equations

We consider fractional wave equations with exponential or arbitrary polynomial nonlinearities. We prove the global well-posedness on the support of the corresponding Gibbs measures. We provide ill-posedness constructions showing that the results are truly super-critical in the considered functional setting. We also present a result in the case of a general randomisation in the spirit of the work by N. Burq and the second author.

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arXiv2018

Low regularity blowup solutions for the mass-critical NLS in higher dimensions

In this paper, we study the $H^s$-stability of the log-log blowup regime (which has been completely described in a series of recent works by Merle and Raphael) for the focusing mass-critical nonlinear Schrödinger equations $i\partial_tu+Δu+|u|^\frac4du=0$ in $\mathbb{R}^d$ with $d\geq3$. We aim to extend the result in [Colliander and Raphael, Rough blowup solutions to the $L^2$ critical NLS, Math. Anna., 345(2009), 307-366.] for dimension two to the higher dimensions cases $d\geq3$, where we use the bootstrap argument in the above paper and the commutator estimates in [M. Visan and X. Zhang, On the blowup for the $L^2$-critical focusing nonlinear Schrödinger equation in higher dimensions below the energy class. SIAM J. Math. Anal., 39(2007), 34-56.].

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arXiv2018

Exact Controllability of linear KP-I equation

We prove the exact controllability of linear KP-I equation if the control input is added on a vertical domain. More generally, we have obtained the least dispersion needed to insure observability for fractional linear KP I equation.

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arXiv2017

Scattering below ground state of 3D focusing cubic fractional Schordinger equation with radial data

The aim of this note is to adapt the strategy in [4][See,B.Dodson, J.Murphy, a new proof of scattering below the ground state for the 3D radial focusing cubic NLS, arXiv:1611.04195 ] to prove the scattering of radial solutions below sharp threshold for certain focusing fractional NLS with cubic nonlinearity. The main ingredient is to apply the fractional virial identity proved in [11][See,T.Boulenger, D.Himmelsbach,E.Lenzmann, Blow up for fractional NLS,J.Func.Anal,271(2016),2569-2603] to exclude the concentration of mass near the origin.

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arXiv2015

Probabilistic well-posedness for supercritical wave equation on $\mathbb{T}^3$

In this article, we follow the strategies, listed in \cite{Burq2011} and \cite{OhPo}, in dealing with supercritical cubic and quintic wave equations, we obtain that, the equation \begin{equation*} \left\{ \begin{split} &(\partial^2_t-Δ)u+|u|^{p-1}u=0,\ \ 3<p<5 &\big(u,\partial_tu\big)|_{t=0}=(u_0,u_1)\in H^{s}\times H^{s-1}=:\mathcal{H}^s, \end{split} \right. \end{equation*} is almost surely global well-posed in the sense of Burq and Tzvetkov\cite{Burq2011} for any $s\in (\frac{p-3}{p-1},1)$. The key point here is that $\frac{p-3}{p-1}$ is much smaller than the critical index $\frac{3}{2}-\frac{2}{p-1}$ for $3<p<5$.

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