arXiv subjects

Yaowei Xie

Publications and source records attributed to Yaowei Xie.

7 recordsLinked to original sources

arXiv2026

Norm Inflation for Inviscid and Fully Dissipative Boussinesq Systems in Supercritical Spaces

We prove norm inflation, in the sense of strong ill-posedness, for the two-dimensional Boussinesq system in supercritical Besov spaces. For the inviscid system, norm inflation holds in $\dot B^\beta_{p,q}(\mathbb R^2)\times \dot B^\beta_{p,r}(\mathbb R^2)$ for $\beta\neq0$, $1<p\leq\infty$, $1\leq q,r\leq\infty$, and $-2<\beta-\frac{2}{p}<1$. For the fully dissipative system, the same conclusion holds in the range $-2<\beta-\frac{2}{p}<-1$. In both cases, the results cover almost all supercritical Besov spaces satisfying the local integrability condition. Norm inflation occurs in the density component $\rho$, while the velocity component $u$ remains bounded. In the fully dissipative case, the inflation space is supercritical for $u$, but subcritical for $\rho$ with respect to its own scaling. This is not a contradiction: the density is transported by a velocity field in a supercritical regime, and this transport mechanism is precisely what produces norm inflation in $\rho$.

math.AP↗

arXiv2026

Non-uniform Continuity for the MHD equations with only Magnetic Diffusion

In this paper, we prove the non-uniform continuity of the data-to-solution map for the incompressible magnetohydrodynamic (MHD) equations with only magnetic diffusion in Sobolev spaces $H^s(\mathbb{R}^d)$ for all $s>0$ and $d=2,3$. Our results are first studies on the non-uniform continuity of the data-to-solution map for the resistive MHD equations. Moreover, our results permit the solution perturbation around an arbitrary constant background magnetic fields $\mathbf{B_0} \in \mathbb{R}^d$, which reveal that the strong magnetic background fields may provide the stabilization effect but still preserve the analytical feature of non-uniform continuity of the data-to-solution map.

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arXiv2026

Global Regularity for Non-resistive or Non-viscous MHD System on the Torus

In this paper, we establish the global well-posedness of the incompressible magnetohydrodynamics (MHD) system on $n-$dimensional $(n\geq 2)$ periodic boxes with either no magnetic diffusivity (non-resistive case) or no fluid viscosity (non-viscous case) under assumption that initial magnetic fields are sufficiently close to the background magnetic field ${\bf e}_n=(0,\cdots,0,1)$. In Eulerian coordinates, we develop novel time-weighted energy estimates and commutator estimates involving Riesz transforms in negative Sobolev spaces to handle two distinct dissipation cases under different initial symmetry assumptions. The analysis becomes much more difficult and delicate in three- or higher-dimensional cases. In particular, for the three-dimensional and non-resistive case, compared with the regularity requirement proposed by Pan, Zhou and Zhu {\it [Arch. Ration. Mech. Anal. 2018]}, our result relaxes it from $H^{11}(\mathbb{T}^3)$ to $H^{\frac{9}{2}+}(\mathbb{T}^3)$. And we further establish precise decay rates and growth bounds for both $u(t)$ and $\partial_n(u(t),b(t))$ in Sobolev norms. For the three-dimensional and non-viscous case, we prove the first nonlinear stability result near the background field $\mathbf{e}_3 = (0,0,1)$. This sharply contrasts with the recent blow-up results on the 3D incompressible Euler equations by Elgindi {\it [Ann. Math. 2021]}, Chen-Hou {\it [Commun. Math. Phys. 2021]} and by Chen-Hou {\it [arXiv:2210.07191]}. Our results show that, under certain symmetry assumptions, magnetic fields near the background field provide enhanced dissipations and suppress potential blow-up mechanisms in non-viscous MHD system.

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arXiv2024

Large time behavior of solutions to the 2D damped wave-type magnetohydrodynamic equations

In this paper, we are concerned with the 2D damped wave-type magnetohydrodynamic system (abbreviated as MHD-wave system). The purpose of this paper is to study the large time behavior of solutions to the MHD-wave system, espesically to investigate the influence of the bad term $\gamma \partial_{tt}b$ on the large time behavior. Rates of decay are obtained for both the solutions and higher derivatives in different Sobolev spaces with explicit rates of $\gamma$, which shows that the decay rates closely align with that of the MHD system under the same norm, for any fixed $\gamma>0$. In this sense, these decay rates are optimal.

math.AP↗

arXiv2024

Mild ill-posedness in $W^{1,\infty}$ for the incompressible porous media equation

In this paper, we establish the mild ill-posedness of 2D IPM equation in the critical Sobolev space $W^{1,\infty}$ when the initial data are small perturbations of stable profile $g(x_2).$ Consequently, instability can be inferred. Notably, our results are valid for arbitrary vertically stratified density profiles $g(x_2)$ without imposing any restrictions on the sign of $g'(x_2).$ From a physical perspective, since gravity acts downward, density profiles satisfying $g'(x_2) < 0$ typically correspond to stable configurations, whereas those with $g '(x_2) > 0$ are generally expected to be unstable. Surprisingly, our analysis uncovers an unexpected instability even when $g'(x_2) < 0$ and $g'(x_2)\in W^{2,\infty}(\mathbb{R})$. To the best of our knowledge, this work provides the first rigorous demonstration of IPM instability for vertically nonlinear density profiles, marking a significant departure from conventional physical expectations.

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arXiv2024

Sharp decay estimates and asymptotic stability for incompressible MHD equations without viscosity or magnetic diffusion

Whether the global existence and uniqueness of strong solutions of $n$-dimensional incompressible magnetohydrodynamic (MHD for short) equations with only kinematic viscosity or magnetic diffusion holds true or not remains an outstanding open problem. In recent years, more attention has been paid to the case when the magnetic field close to an equilibrium state (the background magnetic field for short). Specifically, when the background magnetic field satisfies the Diophantine condition (see (1.2) for details), Chen, Zhang and Zhou [Sci. China Math. 41 (2022), pp.1-10] first studied the perturbation system and established the decay estimates and stability of its solutions in 3D periodic domain $\mathbb{T}^3$, which was then improved to $H^{(3+2β)r+5+(α+2β)}(\mathbb{T}^2)$ for 2D periodic domain $\mathbb{T}^2$ and any $α>0$, $β>0$ by Zhai [J. Differ. Equ. 374 (2023), pp.267-278]. In this paper, we seek to find the optimal decay estimates and improve the space where the global stability is taking place. Through deeply exploring and fully utilizing the structure of perturbation system, we discover a new dissipative mechanism, which enables us to establish the decay estimates in Sobolev space with much lower regularity. Based on the above discovery, we greatly reduce the initial regularity requirement of aforementioned two works from $H^{4r+7}(\mathbb{T}^3)$ and $H^{(3+2β)r+5+(α+2β)}(\mathbb{T}^2)$ to $H^{(3r+3)^+}(\mathbb{T}^n)$ for $r>n-1$ when $n=3$ and $n=2$ respectively. Additionally, we first present the linear stability result via the method of spectral analysis in this paper. From which, the decay estimates obtained for the nonlinear system can be seen as sharp in the sense that they are in line with those for the linearized system.

math.AP↗

arXiv2024

Asymptotic stability for $n$-dimensional isentropic compressible MHD equations without magnetic diffusion

Whether the global well-posedness of strong solutions of $n$-dimensional compressible isentropic magnetohydrodynamic (MHD for short) equations without magnetic diffusion holds true or not remains an challenging open problem, even for the small initial data. In recent years, stared from the pioneer work by Wu and Wu [Adv. Math. 310 (2017), 759--888], much more attention has been paid to the system when the magnetic field near an equilibrium state (the background magnetic field for short). In particular, when the background magnetic field satisfies the Diophantine condition (see (1.3) for details), Wu and Zhai [Math. Models Methods Appl. Sci. 33 (2023), no. 13, 2629--2656] established the decay estimates and asymptotic stability for smooth solutions of the 3D compressible isentropic MHD system without magnetic diffusion in $H^{4r+7}(\mathbb{T}^3)$ with $r>2$ by exploiting a wave structure. In this paper, a new dissipative mechanism is found out and applied so that we can improve the spaces where the decay estimates and asymptotic stability of solutions are taking place by Wu and Zhai. More precisely, we establish the decay estimates of solutions in $H^{r+1}(\mathbb{T}^n)$ and asymptotic stability result in $H^{\left(3r+3\right)^+}(\mathbb{T}^n)$ for any dimensional periodic domain $\mathbb{T}^n$ with $n\geq 2$ and $r>n-1$. Our results provide an approach for establishing the decay estimates and asymptotic stability in the Sobolev spaces with much lower regularity and uniform dimension, which can be used to study many other related models such as the compressible non-isentropic MHD system without magnetic diffusion and so on.

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