Robin Ming Chen
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View article: Finite-time self-similar implosion of hollow vortices
Finite-time self-similar implosion of hollow vortices Open
In this paper, we consider the finite-time blowup of hollow vortices. These are solutions of the two-dimensional Euler equations for which the fluid domain is the complement of finitely many Jordan curves $Γ_1, \ldots, Γ_M$, and such that …
View article: Nonlinear stability of compressible vortex sheets in three-dimensional elastodynamics
Nonlinear stability of compressible vortex sheets in three-dimensional elastodynamics Open
We investigate the nonlinear stability of compressible vortex sheet solutions for three-dimensional (3D) isentropic elastic flows. Building upon previous results on the weakly linear stability of elastic vortex sheets [19], we perform a de…
View article: Global bifurcation for monotone fronts of elliptic equations
Global bifurcation for monotone fronts of elliptic equations Open
We present two results on global continuation of monotone front-type solutions to elliptic PDEs posed on infinite cylinders. This is done under quite general assumptions, and in particular applies even to fully nonlinear equations as well …
View article: Vortex-carrying solitary gravity waves of large amplitude
Vortex-carrying solitary gravity waves of large amplitude Open
In this paper, we study two-dimensional traveling waves in finite-depth water that are acted upon solely by gravity. We prove that, for any supercritical Froude number (non-dimensionalized wave speed), there exists a continuous one-paramet…
View article: Spectral analysis of periodic $b$-KP equation under transverse perturbation
Spectral analysis of periodic $b$-KP equation under transverse perturbation Open
The $b$-family-Kadomtsev-Petviashvili equation ($b$-KP) is a two dimensional generalization of the $b$-family equation. In this paper, we study the spectral stability of the one-dimensional small-amplitude periodic traveling waves with res…
View article: Desingularization and global continuation for hollow vortices
Desingularization and global continuation for hollow vortices Open
A hollow vortex is a region of constant pressure bounded by a vortex sheet and suspended inside a perfect fluid; it can therefore be interpreted as a spinning bubble of air in water. This paper gives a general method for desingularizing no…
View article: Rigidity of three-dimensional internal waves with constant vorticity
Rigidity of three-dimensional internal waves with constant vorticity Open
This paper studies the structural implications of constant vorticity for steady three-dimensional internal water waves. It is known that in many physical regimes, water waves beneath vacuum that have constant vorticity are necessarily two …
View article: Center manifolds without a phase space for quasilinear problems in elasticity, biology, and hydrodynamics
Center manifolds without a phase space for quasilinear problems in elasticity, biology, and hydrodynamics Open
In this paper, we present a center manifold reduction theorem for quasilinear elliptic equations posed on infinite cylinders that is done without a phase space in the sense that we avoid explicitly reformulating the PDE as an evolution pro…
View article: On the regularity of the flow map for the gravity-capillary equations
On the regularity of the flow map for the gravity-capillary equations Open
We prove via explicitly constructed initial data that solutions to the gravity-capillary wave system in R3R3 representing a 2d air–water interface immediately fail to be C3C3 with respect to the initial data if the initial data (h0,ψ0)∈Hs+…
View article: Transverse instability of the CH-KP-I equation
Transverse instability of the CH-KP-I equation Open
The Camassa-Holm-Kadomtsev-Petviashvili-I equation (CH-KP-I) is a two dimensional generalization of the Camassa-Holm equation (CH). In this paper, we prove transverse instability of the line solitary waves under periodic transverse perturb…
View article: $W^{1,\infty}$ instability of $H^1$-stable peakons in the Novikov equation
$W^{1,\infty}$ instability of $H^1$-stable peakons in the Novikov equation Open
It is known from the previous works that the peakon solutions of the Novikov equation are orbitally and asymptotically stable in $H^1$. We prove, via the method of characteristics, that these peakon solutions are unstable under $W^{1,\inft…
View article: On the stability of two-dimensional nonisentropic elastic vortex sheets
On the stability of two-dimensional nonisentropic elastic vortex sheets Open
We are concerned with the stability of vortex sheet solutions for the two-dimensional nonisentropic compressible flows in elastodynamics. This is a nonlinear free boundary hyperbolic problem with characteristic discontinuities, which has e…
View article: A Kato-type criterion for vanishing viscosity near the Onsager's critical regularity
A Kato-type criterion for vanishing viscosity near the Onsager's critical regularity Open
We consider a vanishing viscosity sequence of weak solutions of the three-dimensional Navier--Stokes equations on a bounded domain. In a seminal paper [25] Kato showed that for sufficiently regular solutions, the vanishing viscosity limit …
View article: Global bifurcation for monotone fronts of elliptic equations
Global bifurcation for monotone fronts of elliptic equations Open
In this paper, we present two results on global continuation of monotone front-type solutions to elliptic PDEs posed on infinite cylinders. This is done under quite general assumptions, and in particular applies even to fully nonlinear equ…
View article: $W^{1,\\infty}$ instability of $H^1$-stable peakons in the Novikov\n equation
$W^{1,\\infty}$ instability of $H^1$-stable peakons in the Novikov\n equation Open
It is known from the previous works that the peakon solutions of the Novikov\nequation are orbitally and asymptotically stable in $H^1$. We prove, via the\nmethod of characteristics, that these peakon solutions are unstable under\n$W^{1,\\…
View article: Center manifolds without a phase space for quasilinear problems in elasticity, biology, and hydrodynamics
Center manifolds without a phase space for quasilinear problems in elasticity, biology, and hydrodynamics Open
In this paper, we present a novel center manifold reduction theorem for quasilinear elliptic equations posed on infinite cylinders. This is done without a phase space in the sense that we avoid explicitly reformulating the PDE as an evolut…
View article: Energy equality in compressible fluids with physical boundaries
Energy equality in compressible fluids with physical boundaries Open
We study the energy balance for weak solutions of the three-dimensional compressible Navier--Stokes equations in a bounded domain. We establish an $L^p$-$L^q$ regularity conditions on the velocity field for the energy equality to hold, pro…
View article: Linear Stability of Compressible Vortex Sheets in 2D Elastodynamics: Variable Coefficients
Linear Stability of Compressible Vortex Sheets in 2D Elastodynamics: Variable Coefficients Open
The linear stability with variable coefficients of the vortex sheets for the two-dimensional compressible elastic flows is studied. As in our earlier work on the linear stability with constant coefficients, the problem has a free boundary …
View article: Martingale solutions for the three-dimensional stochastic nonhomogeneous incompressible Navier-Stokes equations driven by Levy processes
Martingale solutions for the three-dimensional stochastic nonhomogeneous incompressible Navier-Stokes equations driven by Levy processes Open
In this paper, the three-dimensional stochastic nonhomogeneous incompressible Navier-Stokes equations driven by Lévy process consisting of the Brownian motion, the compensated Poisson random measure and the Poisson random measure are consi…
View article: On a shallow-water approximation to the Green-Naghdi equations with the Coriolis effect
On a shallow-water approximation to the Green-Naghdi equations with the Coriolis effect Open
We consider an asymptotic 1D (in space) rotation-Camassa-Holm (R-CH) model, which could be used to describe the propagation of long-crested shallow-water waves in the equatorial ocean regions with allowance for the weak Coriolis effect due…
View article: Existence and uniqueness of the global conservative weak solutions for the integrable Novikov equation
Existence and uniqueness of the global conservative weak solutions for the integrable Novikov equation Open
The integrable Novikov equation can be regarded as one of the Camassa-Holm-type equations with cubic nonlinearity. In this paper, we prove the global existence and uniqueness of the Hölder continuous energy conservative solutions for the C…
View article: Linear Stability of Compressible Vortex Sheets in Two-Dimensional Elastodynamics
Linear Stability of Compressible Vortex Sheets in Two-Dimensional Elastodynamics Open
The linear stability of rectilinear compressible vortex sheets is studied for two-dimensional isentropic elastic flows. This problem has a free boundary and the boundary is characteristic. A necessary and sufficient condition is obtained f…
View article: Reconstruction of stratified steady water waves from pressure readings on the ocean bed
Reconstruction of stratified steady water waves from pressure readings on the ocean bed Open
Consider a two-dimensional stratified solitary wave propagating through a body of water that is bounded below by an impermeable ocean bed. In this work, we study how such a wave can be reconstructed from data consisting of the wave speed, …