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Yupei Huang

Publications and source records attributed to Yupei Huang.

8 recordsLinked to original sources
arXiv2026

Flexibility and rigidity of steady states of the two-dimensional Euler equations in an infinite channel

We study steady solutions to the two-dimensional incompressible Euler equations in an infinite channel, whose far-field limits are uniformly non-stagnant shear flows. In the smooth category,for a broad class of prescribed far-field shear profiles, non-shear steady states exist via the construction of two-dimensional solutions of the semilinear elliptic equations of stream function by the min--max method. In the analytic category, we establish a comparison principle for the analytic steady states and we show for a dense family of analytic uniformly non-stagnant shear profiles, every analytic steady state with the prescribed far field must itself be a shear flow. In particular, there are far-field shear profiles which exhibit flexibility in the smooth category but rigidity in the analytic category. Furthermore, the dense rigidity is sharp in the sense that there exists analytic shear profile which admits flexibility in the analytic category.

math.AP↗
arXiv2026

On the flexibility of 2D Euler steady states

We consider steady states of the incompressible Euler equation on two-dimensional domains. For non-radial analytic steady states on bounded simply connected domains, it was shown previously that there must be a global functional relationship between the stream function and the vorticity. We show that this does not extend to smooth functions, even under further structural assumptions such as the Morse condition or Arnold's stability criterion. More precisely, we show that a broad class of steady states with multiple critical points can be perturbed to smooth steady states for which the vorticity is not a single-valued function of the stream function. We also establish an analogous flexibility result near the cellular flow on the flat torus, which is a degenerate case. As a consequence of our constructions, there are "branches" of smooth steady states that are isolated from analytic ones. In some cases, the resulting isolated branches can even consist entirely of linearly stable steady states.

math.AP↗
arXiv2024

A Classification Theorem for Steady Euler Flows

Fix a bounded, analytic, and simply connected domain $\Omega\subset\mathbb{R}^2.$ We show that all analytic steady states of the Euler equations with stream function $\psi$ are either radial or solve a semi-linear elliptic equation of the form $\Delta \psi = F(\psi)$ with Dirichlet boundary conditions. In particular, if $\Omega$ is not a ball, then there exists a one to one correspondence between analytic steady states of the Euler equations and analytic solutions of equations of the form $\Delta \psi = F(\psi)$ with Dirichlet boundary conditions.

math.AP↗
arXiv2023

Optimal H{\"o}lder convergence of a class of singular steady states to the Bahouri-Chemin patch

Singular steady states are important objects in obtaining ill-posedness results for 2D incompressible Euler equations. In \cite{elgindi2022regular}, a family of singular steady states near the Bahouri-Chemin patch was introduced. In this paper, we obtain the optimal convergence results for the singular steady states constructed in \cite{elgindi2022regular} to the Bahouri-Chemin patch. We first derive a boundary Harnack principle, and then obtain the optimal convergence results using the singular integral representation based on Green's function.

math.AP↗
arXiv2020

Formation of singularity for the rotating shallow water system

In this paper, we investigate the formation of singularity for general two dimensional and radially symmetric solutions for rotating shallow water system from different aspects. First, the formation of singularity is proved via the study for the associated moments for two dimensional solutions. For the radial symmetric solutions, the formation of singularity is established for the initial data with compact support. Finally, the global existence or formation of singularity for the radial symmetric solutions of the rotating shallow water system are analyzed in detail when the solutions are of the form with separated variables.

math.AP↗