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Communications on Pure & Applied Analysis

July & August 2021 , Volume 20 , Issue 7&8

Special issue in honor of Professor Shuxing Chen on the occasion of his 80th birthday

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Preface: Special issue in honor of Professor Shuxing Chen on the occasion of his 80th birthday
Gui-Qiang G. Chen, Daoyuan Fang, Zhen Lei, Ping Zhang and Yongqian Zhang
2021, 20(7&8): i-iii doi: 10.3934/cpaa.2021089 +[Abstract](352) +[HTML](143) +[PDF](113.45KB)
Abstract:
Three-dimensional supersonic flows of Euler-Poisson system for potential flow
Myoungjean Bae and Hyangdong Park
2021, 20(7&8): 2421-2440 doi: 10.3934/cpaa.2021079 +[Abstract](513) +[HTML](198) +[PDF](466.05KB)
Abstract:

We prove the unique existence of supersonic solutions of the Euler-Poisson system for potential flow in a three-dimensional rectangular cylinder when prescribing the velocity and the strength of electric field at the entrance. Overall, the main framework is similar to [1], but there are several technical differences to be taken care of vary carefully. And, it is our main goal to treat all the technical differences occurring when one considers a three dimensional supersonic solution of the steady Euler-Poisson system.

Nonlinear stability of rarefaction waves for a hyperbolic system with Cattaneo's law
Yinsong Bai, Lin He and Huijiang Zhao
2021, 20(7&8): 2441-2474 doi: 10.3934/cpaa.2021049 +[Abstract](506) +[HTML](238) +[PDF](594.48KB)
Abstract:

This paper is concerned with the time-asymptotically nonlinear stability of rarefaction waves to the Cauchy problem and the initial-boundary value problem in the half space with impermeable wall boundary condition for a scalar conservation laws with an artificial heat flux satisfying Cattaneo's law. In our results, although the \begin{document}$ L^2\cap L^\infty- $\end{document}norm of the initial perturbation is assumed to be small, the \begin{document}$ H^1- $\end{document}norm of the first order derivative of the initial perturbation with respect to the spatial variable can indeed be large. Moreover the far fields of the artificial heat flux can be different. Our analysis is based on the \begin{document}$ L^2 $\end{document} energy method.

Global solutions of a two-dimensional Riemann problem for the pressure gradient system
Gui-Qiang G. Chen, Qin Wang and Shengguo Zhu
2021, 20(7&8): 2475-2503 doi: 10.3934/cpaa.2021014 +[Abstract](655) +[HTML](333) +[PDF](670.18KB)
Abstract:

We are concerned with a two-dimensional Riemann problem for the pressure gradient system that is a hyperbolic system of conservation laws. The Riemann initial data consist of four constant states in four sectorial regions such that two shocks and two vortex sheets are generated between the adjacent states. The solutions keep the four constant states and four planar waves outside the outer sonic circle in the self-similar coordinates, while the two shocks keep planar until meeting the outer sonic circle at two different points and then generate a diffracted shock to connect these points, whose location is apriori unknown. Then the problem can be formulated as a free boundary problem, in which the diffracted transonic shock is the one-phase free boundary to connect the two points, while the other part of the sonic circle forms a fixed boundary. We establish the global existence of a solution and the optimal Lipschitz regularity of both the diffracted shock across the two points and the solution across the outer sonic boundary. Then this Riemann problem is solved globally, whose solution contains two vortex sheets and one global shock containing the two originally separated shocks generated by the Riemann data.

Multiple solutions for nonlinear cone degenerate elliptic equations
Hua Chen and Yawei Wei
2021, 20(7&8): 2505-2518 doi: 10.3934/cpaa.2020272 +[Abstract](959) +[HTML](418) +[PDF](397.0KB)
Abstract:

The present paper is concerned with the Dirichlet boundary value problem for nonlinear cone degenerate elliptic equations. First we introduce the weighted Sobolev spaces, inequalities and the property of compactness. After the appropriate energy functional established, we obtain the existence of infinitely many solutions in the weighted Sobolev spaces by applying the variational methods.

On the stability of two-dimensional nonisentropic elastic vortex sheets
Robin Ming Chen, Feimin Huang, Dehua Wang and Difan Yuan
2021, 20(7&8): 2519-2533 doi: 10.3934/cpaa.2021083 +[Abstract](414) +[HTML](185) +[PDF](404.88KB)
Abstract:

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 extra difficulties when considering the effect of entropy. The addition of the thermal effect to the system makes the analysis of the Lopatinski\begin{document}$ \breve{{\mathrm{i}}} $\end{document} determinant extremely complicated. Our results are twofold. First, through a qualitative analysis of the roots of the Lopatinski\begin{document}$ \breve{{\mathrm{i}}} $\end{document} determinant for the linearized problem, we find that the vortex sheets are weakly stable in some supersonic and subsonic regions. Second, under the small perturbation of entropy, the nonlinear stability can be adapted from the previous two-dimensional isentropic elastic vortex sheets [6] by applying the Nash-Moser iteration. The two results confirm the strong elastic stabilization of the vortex sheets. In particular, our conditions for the linear stability (1) ensure that a stable supersonic regime as well as a stable subsonic one always persist for any given nonisentropic configuration, and (2) show how the stability condition changes with the thermal fluctuation. The existence of the stable subsonic bubble, a phenomenon not observed in the Euler flow, is specially due to elasticity.

Uniqueness of steady 1-D shock solutions in a finite nozzle via vanishing viscosity aguments
Beixiang Fang and Qin Zhao
2021, 20(7&8): 2535-2553 doi: 10.3934/cpaa.2021066 +[Abstract](434) +[HTML](205) +[PDF](426.96KB)
Abstract:

This paper studies the uniqueness of steady 1-D shock solutions in a finite flat nozzle via vanishing viscosity arguments. It is proved that, for both barotropic gases and non-isentropic gases, the steady viscous shock solutions converge under the \begin{document}$ \mathcal{L}^{1} $\end{document} norm. Hence only one shock solution of the inviscid Euler system could be the limit as the viscosity coefficient goes to \begin{document}$ 0 $\end{document}, which shows the uniqueness of the steady 1-D shock solution in a finite flat nozzle. Moreover, the position of the shock front for the limit shock solution is also obtained.

A degenerate elliptic problem from subsonic-sonic flows in convergent nozzles
Guanming Gai, Yuanyuan Nie and Chunpeng Wang
2021, 20(7&8): 2555-2577 doi: 10.3934/cpaa.2021070 +[Abstract](443) +[HTML](196) +[PDF](475.38KB)
Abstract:

This paper concerns continuous subsonic-sonic potential flows in a two dimensional convergent nozzle, which is governed by a free boundary problem of a quasilinear degenerate elliptic equation. It is shown that for a given nozzle which is a perturbation of an straight one, and a given mass flux, there exists uniquely a continuous subsonic-sonic flow whose velocity vector is along the normal direction at the inlet and the sonic curve. Furthermore, the sonic curve of this flow is a free boundary, where the flow is singular in the sense that the speed is only \begin{document}$ C^{1/2} $\end{document} Hölder continuous and the acceleration blows up at the sonic state.

Kuramoto order parameters and phase concentration for the Kuramoto-Sakaguchi equation with frustration
Seung-Yeal Ha, Javier Morales and Yinglong Zhang
2021, 20(7&8): 2579-2612 doi: 10.3934/cpaa.2021013 +[Abstract](673) +[HTML](304) +[PDF](523.72KB)
Abstract:

We study phase concentration for the Kuramoto-Sakaguchi(K-S) equation with frustration via detailed estimates on the dynamics of order parameters. The Kuramoto order parameters measure the overall degree of phase concentrations. When the coupling strength is sufficiently large and the size of frustration parameter is sufficiently small, we show that the amplitude order parameter has a positive lower bound uniformly in time, and we also show that the total mass concentrates on the translated phase order parameter by a frustration parameter asymptotically, whereas the mass in the region around the antipodal point decays to zero exponentially fast.

Collective behaviors of the Lohe Hermitian sphere model with inertia
Seung-Yeal Ha, Myeongju Kang and Hansol Park
2021, 20(7&8): 2613-2641 doi: 10.3934/cpaa.2021046 +[Abstract](484) +[HTML](221) +[PDF](482.58KB)
Abstract:

We present a second-order extension of the first-order Lohe Hermitian sphere (LHS) model and study its emergent asymptotic dynamics. Our proposed model incorporates an inertial effect as a second-order extension. The inertia term can generate an oscillatory behavior of particle trajectory in a small time interval(initial layer) which causes a technical difficulty for the application of monotonicity-based arguments. For emergent estimates, we employ two-point correlation function which is defined as an inner product between positions of particles. For a homogeneous ensemble with the same frequency matrix, we provide two sufficient frameworks in terms of system parameters and initial data to show that two-point correlation functions tend to the unity which is exactly the same as the complete aggregation. In contrast, for a heterogeneous ensemble with distinct frequency matrices, we provide a sufficient framework in terms of system parameters and initial data, which makes two-point correlation functions be close to unity by increasing the principal coupling strength.

On a supersonic-sonic patch arising from the frankl problem in transonic flows
Yanbo Hu and Jiequan Li
2021, 20(7&8): 2643-2663 doi: 10.3934/cpaa.2021015 +[Abstract](531) +[HTML](283) +[PDF](490.52KB)
Abstract:

We construct a supersonic-sonic smooth patch solution for the two dimensional steady Euler equations in gas dynamics. This patch is extracted from the Frankl problem in the study of transonic flow with local supersonic bubble over an airfoil. Based on the methodology of characteristic decompositions, we establish the global existence and regularity of solutions in a partial hodograph coordinate system in terms of angle variables. The original problem is solved by transforming the solution in the partial hodograph plane back to that in the physical plane. Moreover, the uniform regularity of the solution and the regularity of an associated sonic curve are also verified.

Radon measure solutions for steady compressible hypersonic-limit Euler flows passing cylindrically symmetric conical bodies
Yunjuan Jin, Aifang Qu and Hairong Yuan
2021, 20(7&8): 2665-2685 doi: 10.3934/cpaa.2021048 +[Abstract](643) +[HTML](313) +[PDF](578.0KB)
Abstract:

We study steady uniform hypersonic-limit Euler flows passing a finite cylindrically symmetric conical body in the Euclidean space \begin{document}$ \mathbb{R}^3 $\end{document}, and its interaction with downstream static gas lying behind the tail of the body. Motivated by Newton's theory of infinite-thin shock layers, we propose and construct Radon measure solutions with density containing Dirac measures supported on surfaces and prove the Newton-Busemann pressure law of hypersonic aerodynamics. It happens that if the pressure of the downstream static gas is quite large, the Radon measure solution terminates at a finite distance from the tail of the body. The main difficulty of the analysis is a correct definition of Radon measure solutions. The results are helpful to understand mathematically some physical phenomena and formulas about hypersonic inviscid flows.

Blow-up results for effectively damped wave models with nonlinear memory
Tayeb Hadj Kaddour and Michael Reissig
2021, 20(7&8): 2687-2707 doi: 10.3934/cpaa.2020239 +[Abstract](1129) +[HTML](480) +[PDF](426.16KB)
Abstract:

In this paper, we study the Cauchy problem for a special family of effectively damped wave models with nonlinear memory on the right-hand side. Our goal is to prove blow-up results for local (in time) Sobolev solutions. Due to the effective dissipation the model is parabolic like from the point of view of energy decay estimates of the corresponding linear Cauchy problem with vanishing right-hand side. For this reason we apply the test function method for proving our results.

Subellipticity of some complex vector fields related to the Witten Laplacian
Wei Xi Li and Chao Jiang Xu
2021, 20(7&8): 2709-2724 doi: 10.3934/cpaa.2021047 +[Abstract](469) +[HTML](198) +[PDF](401.57KB)
Abstract:

We consider some system of complex vector fields related to the semi-classical Witten Laplacian, and establish the local subellipticity of this system basing on condition \begin{document}$ (\Psi) $\end{document}.

Uniform regularity and vanishing viscosity limit for the incompressible non-resistive MHD system with TMF
Cheng-Jie Liu, Feng Xie and Tong Yang
2021, 20(7&8): 2725-2750 doi: 10.3934/cpaa.2021073 +[Abstract](477) +[HTML](171) +[PDF](482.27KB)
Abstract:

This paper is concerned with the vanishing viscosity limit for the incompressible MHD system without magnetic diffusion effect in the half space under the influence of a transverse magnetic field on the boundary. We prove that the solution to the incompressible MHD system is uniformly bounded in both conormal Sobolev norm and \begin{document}$ L^\infty $\end{document} norm in a fixed time interval independent of the viscosity coefficient. As a direct consequence, the inviscid limit from the viscous MHD system to the ideal MHD system is established in \begin{document}$ L^\infty $\end{document}-norm. In addition, the analysis shows that the boundary layer effect is weak because of the transverse magnetic field.

On the invariant region for compressible Euler equations with a general equation of state
Hailiang Liu and Ferdinand Thein
2021, 20(7&8): 2751-2763 doi: 10.3934/cpaa.2021084 +[Abstract](520) +[HTML](173) +[PDF](355.24KB)
Abstract:

The state space for solutions of the compressible Euler equations with a general equation of state is examined. An arbitrary equation of state is allowed, subject only to the physical requirements of thermodynamics. An invariant region of the resulting Euler system is identified and the convexity property of this region is justified by using only very minimal thermodynamical assumptions. Finally, we show how an invariant-region-preserving (IRP) limiter can be constructed for use in high order finite-volume type schemes to solve the compressible Euler equations with a general constitutive relation.

Remark on 3-D Navier-Stokes system with strong dissipation in one direction
Yanlin Liu and Ping Zhang
2021, 20(7&8): 2765-2787 doi: 10.3934/cpaa.2020244 +[Abstract](992) +[HTML](440) +[PDF](488.24KB)
Abstract:

In this paper, we consider 3D anisotropic incompressible Navier-Stokes equations with strong dissipation in the vertical direction. We shall prove that this system has a unique global strong solution and the norm of the vertical component of the velocity field can be controlled by the norm of the corresponding component to the initial data. Similar result can also be obtained for the horizontal components of the vorticity. In particular, we simplify our proofs to the well-posedness result in our previous paper [11, 13].

Frequency domain approach to decay rates for a coupled hyperbolic-parabolic system
Bopeng Rao and Xu Zhang
2021, 20(7&8): 2789-2809 doi: 10.3934/cpaa.2021119 +[Abstract](302) +[HTML](98) +[PDF](458.02KB)
Abstract:

We consider the asymptotic behavior of a linear model arising in fluid-structure interactions. The system is formed by a heat equation and a wave equation in two distinct domains, which are coupled by atransmission condition along the interface of the domains. By means of the frequency domain approach, we establish some decay rates for the whole system. Our results also showthat the decay of the fluid-structure interaction depends not only on the transmission of the damping from the heat equation to the wave equation, but also on the location of the damping for the wave equation.

Large-time behaviors of the solution to 3D compressible Navier-Stokes equations in half space with Navier boundary conditions
Teng Wang and Yi Wang
2021, 20(7&8): 2811-2838 doi: 10.3934/cpaa.2021080 +[Abstract](444) +[HTML](178) +[PDF](490.38KB)
Abstract:

We are concerned with the large-time asymptotic behaviors towards the planar rarefaction wave to the three-dimensional (3D) compressible and isentropic Navier-Stokes equations in half space with Navier boundary conditions. It is proved that the planar rarefaction wave is time-asymptotically stable for the 3D initial-boundary value problem of the compressible Navier-Stokes equations in \begin{document}$ \mathbb{R}^+\times \mathbb{T}^2 $\end{document} with arbitrarily large wave strength. Compared with the previous work [17, 16] for the whole space problem, Navier boundary conditions, which state that the impermeable wall condition holds for the normal velocity and the fluid tangential velocity is proportional to the tangential component of the viscous stress tensor on the boundary, are crucially used for the stability analysis of the 3D initial-boundary value problem.

Remarks on global weak solutions to a two-fluid type model
Huanyao Wen and Changjiang Zhu
2021, 20(7-8): 2839-2856 doi: 10.3934/cpaa.2021072 +[Abstract](419) +[HTML](172) +[PDF](499.82KB)
Abstract:

The present paper aims to give a review of a two-fluid type model mostly on large-data solutions. Some derivations of the model arising in different physical background will be introduced. In addition, we will sketch the proof of global existence of weak solutions to the Dirichlet problem for the model in one dimension with more general pressure law which can be non-monotone, in the context of allowing unconstrained transition to single-phase flow.

The two-component Novikov-type systems with peaked solutions and $ H^1 $-conservation law
Min Zhao and Changzheng Qu
2021, 20(7&8): 2857-2883 doi: 10.3934/cpaa.2020245 +[Abstract](1038) +[HTML](474) +[PDF](468.52KB)
Abstract:

In this paper, we provide a classification to the general two-component Novikov-type systems with cubic nonlinearities which admit multi-peaked solutions and \begin{document}$ H^1 $\end{document}-conservation law. Local well-posedness and wave breaking of solutions to the Cauchy problem of a resulting system from the classification are studied. First, we carry out the classification of the general two-component Novikov-type system based on the existence of two peaked solutions and \begin{document}$ H^1 $\end{document}-conservation law. The resulting systems contain the two-component integrable Novikov-type systems. Next, we discuss the local well-posedness of Cauchy problem to the resulting systems in Sobolev spaces \begin{document}$ H^s({\mathbb R}) $\end{document} with \begin{document}$ s>3/2 $\end{document}, the approach is based on the new invariant properties, certain estimates for transport equations of the system. In addition, blow up and wave-breaking to the Cauchy problem of a system are studied.

2020 Impact Factor: 1.916
5 Year Impact Factor: 1.510
2020 CiteScore: 1.9

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