arXiv subjects

Shunkai Mao

Publications and source records attributed to Shunkai Mao.

3 recordsLinked to original sources

arXiv2025

Non-uniqueness for the nonlinear dynamical Lamé system

We consider the Cauchy problem for the nonlinear dynamical Lamé system with double wave speeds in a $d$-dimensional $(d=2,3)$ periodic domain. Moreover, the equations can be transformed into a linearly degenerate hyperbolic system. We could construct infinitely many continuous solutions in $C^{1,α}$ emanating from the same small initial data for $α<\frac{1}{60}$. The proof relies on the convex integration scheme. We construct a new class of building blocks with compression structure by using the double wave speeds characteristic of the equations.

math.AP↗

arXiv2025

The null condition in elastodynamics leads to non-uniqueness

We consider the Cauchy problem for the system of elastodynamic equations in two dimensions. Specifically, we focus on materials characterized by a null condition imposed on the quadratic part of the nonlinearity. We can construct non-zero weak solutions $u \in C^1([0, T] \times \mathbb{T}^2)$ that emanate from zero initial data. The proof relies on the convex integration scheme. By exploiting the characteristic double wave speeds of the equations, we construct a new class of building blocks. This work extends the application of convex integration techniques to hyperbolic systems with a null condition and reveals the rich solution structure in nonlinear elastodynamics.

math.AP↗

arXiv2023

Non-uniqueness for the compressible Euler-Maxwell equations

We consider the Cauchy problem for the isentropic compressible Euler-Maxwell equations under general pressure laws in a three-dimensional periodic domain. For any smooth initial electron density away from the vacuum and smooth equilibrium-charged ion density, we could construct infinitely many $α$-Hölder continuous entropy solutions emanating from the same initial data for $α<\frac{1}{7}$. Especially, the electromagnetic field belongs to the Hölder class $C^{1,α}$. Furthermore, we provide a continuous entropy solution satisfying the entropy inequality strictly. The proof relies on the convex integration scheme. Due to the constrain of the Maxwell equations, we propose a method of Mikado potential and construct new building blocks.

math.AP↗