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Silu Yin

Publications and source records attributed to Silu Yin.

8 recordsLinked to original sources
arXiv2025

A Morawetz type energy estimate for wave equation in $\mathbb{R}^2$ and application to elastic waves

In this paper, we introduce a modified scaling Morawetz multiplier, which produces a weighted Morawetz type energy (non-negative) estimate for the inhomogeneous wave equation in $\mathbb{R}^2$. With this estimate in hand, an alternative proof of global existence for the Cauchy problem of quasilinear wave equation with small and compactly supported data is given. What is more, such weighted Morawetz type energy estimate also works for certain wave system with multiple speeds, which can be used to prove global existence of some admissible harmonic elastic wave system in $\mathbb{R}^2$.

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arXiv2022

$H^{\frac{11}{4}}(\mathbb{R}^2)$ ill-posedness for 2D Elastic Wave system

In this paper, we prove that for the 2D elastic wave equations, a physical system with multiple wave-speeds, its Cauchy problem fails to be locally well-posed in $H^{\frac{11}{4}}(\mathbb{R}^2)$. The ill-posedness here is driven by instantaneous shock formation. In 2D Smith-Tataru showed that the Cauchy problem for a single quasilinear wave equation is locally well-posed in $H^s$ with $s>\frac{11}{4}$. Hence our $H^{\frac{11}{4}}$ ill-posedness obtained here is a desired result. Our proof relies on combining a geometric method and an algebraic wave-decomposition approach, equipped with detailed analysis of the corresponding hyperbolic system.

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arXiv2021

Low regularity ill-posedness and shock formation for 3D ideal compressible MHD

The study of magnetohydrodynamics (MHD) significantly boosts the understanding and development of solar physics, planetary dynamics and controlled nuclear fusion. Dynamical properties of the MHD system involve nonlinear interactions of waves with multiple travelling speeds (the fast and slow magnetosonic waves, the Alfv\'{e}n wave and the entropy wave). One intriguing topic is the shock phenomena accompanied by the magnetic field, which have been affirmed by astronomical observations. However, permitting the residence of all above multi-speed waves, mathematically, whether one can prove shock formation for three dimensional (3D) MHD is still open. The multiple-speed nature of the MHD system makes it fascinating and challenging. In this paper, we report our recent progress in answering the above question. For 3D ideal compressible MHD, we construct planar symmetric examples of shock formation allowing the presence of all characteristic waves with multiple wave speeds. This also answers a question raised by Majda on conservation law in 1984. Building on our construction, we further prove that the Cauchy problem for 3D ideal MHD is $H^2$ ill-posed. And this is caused by the shock formation. In particular, when the magnetic field is absent, we also provide a desired low-regularity ill-posedness result for the 3D compressible Euler equations, and it is sharp with respect to the regularity of the fluid velocity. Our proof for 3D MHD is based on a coalition of a carefully designed algebraic approach and a geometric approach. To trace the nonlinear interactions of various waves, we algebraically decompose the 3D ideal MHD equations into a $7\times 7$ non-strictly hyperbolic system. Via detailed calculations, we reveal its hidden subtle structures. With them we give a complete description of MHD dynamics up to the earliest singular event, when a shock forms.

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arXiv2020

Low regularity ill-posedness for elastic waves driven by shock formation

In this paper, we construct counterexamples to the local existence of low-regularity solutions to elastic wave equations in three spatial dimensions (3D). Inspired by the recent works of Christodoulou, we generalize Lindblad's classic results on the scalar wave equation by showing that the Cauchy problem for 3D elastic waves, a physical system with multiple wave-speeds, is ill-posed in $H^3(\mathbb{R}^3)$. We further prove that the ill-posedness is caused by instantaneous shock formation, which is characterized by the vanishing of the inverse foliation density. The main difficulties of the 3D case come from the multiple wave-speeds and its associated non-strict hyperbolicity. We obtain the desired results by designing and combining a geometric approach and an algebraic approach, equipped with detailed studies and calculations of the structures and coefficients of the corresponding non-strictly hyperbolic system. Moreover, the ill-posedness we depict also applies to 2D elastic waves, which corresponds to a strictly hyperbolic case.

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arXiv2018

Stability and instability of the standing waves for the Klein-Gordon-Zakharov system in one space dimension

The orbital instability of standing waves for the Klein-Gordon-Zakharov system has been established in two and three space dimensions under radially symmetric condition, see Ohta-Todorova (SIAM J. Math. Anal. 2007). In the one space dimensional case, for the non-degenerate situation, we first check that the Klein-Gordon-Zakharov system satisfies Grillakis-Shatah-Strauss' assumptions on the stability and instability theorems for abstract Hamiltonian systems, see Grillakis-Shatah-Strauss (J. Funct. Anal. 1987). As to the degenerate case that the frequency $|ω|=1/\sqrt{2}$, we follow Wu (ArXiv: 1705.04216, 2017) to describe the instability of the standing waves for the Klein-Gordon-Zakharov system, by using the modulation argument combining with the virial identity. For this purpose, we establish a modified virial identity to overcome several troublesome terms left in the traditional virial identity.

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arXiv2017

Long-time existence of nonlinear inhomogeneous compressible elastic waves

In this paper, we consider the nonlinear inhomogeneous compressible elastic waves in three spatial dimensions when the density is a small disturbance around a constant state. In homogeneous case, the almost global existence was established by Klainerman-Sideris [1996_CPAM], and global existence was built by Agemi [2000_Invent. Math.] and Sideris [1996_Invent. Math., 2000_Ann. Math.] independently. Here we establish the corresponding almost global and global existence theory in the inhomogeneous case.

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arXiv2015

Global existence of radial solutions for general semilinear hyperbolic systems in 3D

We study the well-posedness of radial solutions for general nonlinear hyperbolic systems in three dimensions. We give a proof of the global existence of radial solutions for general semilinear hyperbolic systems in 3D under null condition, with small scaling invariant $\dot{W}^{2,1}(\mathbb{R}^3)$ data. We obtain a bilinear estimate that is effective to the hyperbolic systems which do not have any time decay. It allows us to achieve the boundedness of the weighted BV norm of the radial solution.

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