Article
Mathematics, Applied
Mikhail Roop
Summary: This paper considers a system of one-dimensional gas dynamics equations as a special case of Jacobi type systems, with a natural representation in terms of 2-forms on 0-jet space. It discovers a new class of multivalued solutions for arbitrary thermodynamic state models and discusses singularities of their projections in the space of independent variables in the case of an ideal gas, identifying caustics and discontinuity lines.
JOURNAL OF GEOMETRY AND PHYSICS
(2021)
Article
Mathematics, Applied
Ping Lv, Yanbo Hu
Summary: This paper investigates the singularity formation in smooth solutions of the one-dimensional rotating Euler equations of Chaplygin gases. The equations are a nonhomogeneous quasilinear hyperbolic system with linearly degenerate characteristic fields. The study shows that the density of the smooth solution tends to infinity in finite time for a specific type of initial data.
APPLIED MATHEMATICS LETTERS
(2023)
Article
Mathematics, Applied
Geng Lai, Qing Zhao
Summary: In this study, we investigate the conditions for the existence of global bounded smooth solutions to the 1D nonisentropic Euler system with large initial data. By deriving characteristic decompositions and utilizing them to establish conditions for global bounded classical solutions to the Cauchy problem, we also demonstrate a type of large initial data that leads to singularity formation in the system.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Acoustics
Saikumar R. Yeddula, Aimee S. Morgans
Summary: A semi-analytical solution is developed for the propagation of plane acoustic waves in a varying area duct with a temperature gradient. The second order differential equation for acoustic pressure derived from linearised Euler equations is solved using an iterative WKB approximation method. The obtained wave-like solution allows for superposition of downstream and upstream propagating plane wave amplitudes.
JOURNAL OF SOUND AND VIBRATION
(2021)
Article
Mathematics
Jeongho Kim
Summary: This paper introduces the kinetic description of the first-order Cucker-Smale (CS) flocking model on the real line, exploring the equivalent relation between measure-valued solutions and the behavior of the first-order kinetic CS equation. It also provides an efficient algorithm for obtaining asymptotic solutions without lengthy simulations, with numerical experiments supporting the analysis conducted.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
Yongqiang Fan, Lihui Guo, Xingya Fan, Shouke You
Summary: By considering the range of Mach numbers, we studied various solutions to the generalized piston problem in the Chaplygin Euler equations. Additionally, as the Mach number approaches infinity, the convergence of solutions and degeneration of equations were analyzed when the piston moves away from the tube.
APPLIED MATHEMATICS LETTERS
(2021)
Article
Mathematics, Applied
Paolo Antonelli, Pierangelo Marcati, Hao Zheng
Summary: In this paper, we analyze the multi-dimensional Quantum Hydrodynamics (QHD) system using an intrinsically hydrodynamic approach. We extend the analysis from the one-dimensional case to the multi-dimensional problem and consider two physically relevant classes of initial data. By assuming the continuity of the mass density and a quantization rule for the vorticity, we study the Cauchy problem and provide global finite energy weak solutions. We also obtain suitable dispersive estimates for rough solutions with finite energy and show the sequential stability of weak solutions with positive density.
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
(2023)
Article
Mechanics
Vladimir A. Dorodnitsyn, Evgeniy I. Kaptsov, Roman Kozlov, Sergey Meleshko, Potcharapol Mukdasanit
Summary: This paper considers the one-dimensional flows for magnetohydrodynamics in the mass Lagrangian coordinates. The symmetries, conservation laws, and variational form are examined for both finite and infinite electric conductivity cases. Lie group classification and direct computation are used to establish symmetry extensions and derive conservation laws. By utilizing the variational structure and Noether theorem, conservation laws are obtained in physical variables.
INTERNATIONAL JOURNAL OF NON-LINEAR MECHANICS
(2022)
Article
Mathematics, Applied
Pu Gao, Jianli Liu
Summary: This paper studies the flow description with mass addition in a one-dimensional duct and obtains the global stability of steady supersonic flows for one-dimensional isentropic compressible Euler equations with mass addition in a finite rectilinear duct with length L < Lm.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2023)
Article
Physics, Mathematical
Wenlong Sun, Yeping Li, Xiaoying Han
Summary: This study examines a viscous quantum hydrodynamic system involving particle density, current density, energy density, and electrostatic potential, coupled with a Poisson equation, in one-dimensional real space. The existence and uniqueness of a stationary solution in an appropriate Sobolev space are proven. Exponential stability of the stationary solution is established through an a priori estimate. The existence of a local-in-time solution is obtained by showing the existence of local-in-time solutions of a reformulated system via the iteration method. These results are of significant importance for the study of quantum hydrodynamics.
JOURNAL OF MATHEMATICAL PHYSICS
(2023)
Article
Mechanics
Vladimir A. Dorodnitsyn, Evgeniy I. Kaptsov, Roman Kozlov, Sergey Meleshko
Summary: This paper investigates symmetries and conservation laws in the mass Lagrangian coordinates of one-dimensional magnetohydrodynamics flows. It analyzes flows with cylindrical symmetry and assumes the medium to be inviscid and thermally non-conducting, modeled by a polytropic gas. The study identifies additional symmetries in cases of finite electric conductivity and presents conservation laws through direct computation. For cases with infinite electric conductivity, the study considers variational formulations and uses the Noether theorem to compute conservation laws.
INTERNATIONAL JOURNAL OF NON-LINEAR MECHANICS
(2023)
Article
Mathematics, Applied
David Cheban
Summary: This paper focuses on the dynamics of one-dimensional monotone nonautonomous (cocycle) dynamical systems. It provides a description of the structures of their invariant sets, omega limit sets, Bohr/Levitan almost periodic and almost automorphic motions, global attractors, pinched and minimal sets. An application of these general results is made to scalar differential and difference equations.
SCIENCE CHINA-MATHEMATICS
(2023)
Article
Physics, Multidisciplinary
Amirali Hannani, Francois Huveneers
Summary: In this paper, we derive Euler equations from a Hamiltonian microscopic dynamics for a one-dimensional disordered harmonic chain. By controlling the second moment of the fluctuations around the average, we strengthen the existing results.
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL
(2022)
Article
Mathematics
Tianfang Wu, Xiongfeng Yang
Summary: In this paper, the hydrodynamic limit of Boltzmann equations for gas mixture is studied using the Hilbert expansion method. The terms in the Hilbert expansion are derived formally according to different orders of the Knudsen number. The truncation of the expansion and the justification of the hydrodynamic limit are done by establishing uniform estimates of the remainder term. The approach is based on the L2 - L infinity framework, which is motivated by the study of the single Boltzmann equation in a previous work [22].
JOURNAL OF DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Roberta Bianchini, Charlotte Perrin
Summary: This article discusses the impact of singular pressure on the solutions of one-dimensional compressible Euler equations, and rigorously proves the singular limit towards the free-congested Euler equations for smooth solutions.