Article
Mathematics
Jitao Liu, Yanqing Wang, Yulin Ye
Summary: In this paper, we investigate the influence of Lp control of vorticity on the energy conservation for the incompressible homogeneous and nonhomogeneous Euler equations, referring to the previous works. For the homogeneous flow in the periodic domain or whole space, we generalize the criterion ω = curl v ∈ L3 (0, T; L n+23n (2)) (n = 2, 3) and provide a self-contained proof, which extends the corresponding result in [8]. For the nonhomogeneous flow, we show that the energy is conserved as long as the vorticity lies in the same space as before and the backward difference √ρ belongs to L∞(0, T ; Ln(Tn)) (n = 2, 3), giving an affirmative answer to the problem proposed by Chen and Yu in [5].
JOURNAL OF DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Quansen Jiu, Jitao Liu, Dongjuan Niu
Summary: This paper proves the global existence of weak solutions to the incompressible axisymmetric Euler equations without swirl under specific conditions on the initial vorticity, without the requirement of finite initial energy. Finite initial velocity belonging to certain function spaces is considered. The key part of the proof involves establishing new estimates for velocity fields.
JOURNAL OF NONLINEAR SCIENCE
(2021)
Article
Mathematics, Applied
Francisco Mengual, Laszlo Szekelyhidi
Summary: We construct infinitely many admissible weak solutions to the 2D incompressible Euler equations for vortex sheet initial data. Our solutions, obtained through convex integration, are smooth outside a turbulence zone which grows linearly in time around the vortex sheet. Furthermore, we show that the growth of the turbulence zone is controlled by the local energy inequality and measures the maximal initial dissipation rate in terms of the vortex sheet strength.
COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS
(2023)
Article
Multidisciplinary Sciences
Yann Brenier, Ivan Moyano
Summary: This article explores a more relevant set of relaxed Euler equations by studying the multi-stream pressure-less gravitational Euler-Poisson system, which allows for the recovery of a large class of smooth solutions for short enough times.
PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES
(2022)
Article
Mathematics, Applied
Yanghai Yu, Jinlu Li, Zhaoyang Yin
Summary: In this paper, a new smallness hypothesis of initial data for the three-dimensional incompressible Navier-Stokes equations is derived and used to prove the existence of a unique global solution. Additionally, two examples of initial data satisfying the smallness condition are constructed, demonstrating that the norm of the initial data can be arbitrarily large.
APPLIED MATHEMATICS LETTERS
(2022)
Article
Engineering, Mechanical
Junchao Chen, Manwai Yuen
Summary: This paper constructs two types of exact global solutions using elementary functions to describe the three-dimensional incompressible MHD equations without viscosity. These solutions correspond to a generalization of the well-known Arnold-Beltrami-Childress (ABC) flow and exhibit interesting local behaviors with infinite energy. Under special parameter values, these solutions can be reduced to those of the incompressible Euler equations.
NONLINEAR DYNAMICS
(2021)
Article
Mathematics, Applied
Zhipeng Zhang
Summary: In this paper, the principle of energy conservation for weak solutions of the incompressible inhomogeneous Euler-Korteweg equations is investigated. Two sufficient conditions on the regularity of weak solutions in Besov space are provided to ensure energy conservation.
ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK
(2022)
Article
Mathematics
Tarek M. Elgindi
Summary: The 3D incompressible Euler equation is locally well-posed for velocity fields with Ho center dot lder continuous gradient and suitable decay at infinity. However, it is shown that these local solutions can develop singularities in finite time, even for some of the simplest three-dimensional flows.
ANNALS OF MATHEMATICS
(2021)
Article
Mathematics, Applied
Haibo Cui, Lei Yao
Summary: This paper considers the initial-boundary value problem of the coupled inhomogeneous incompressible Navier-Stokes equations and Vlasov-Boltzmann equation for the moderately thick spray in three-dimensional space. The global existence of weak solutions is established using an approximation scheme, a fixed point argument, and the weak convergence method.
APPLIED MATHEMATICS LETTERS
(2021)
Article
Mathematics
Alberto Enciso, Daniel Peralta-Salas, Francisco Torres de Lizaur
Summary: Building on previous work, we construct continuous and infinitely differentiable quasi-periodic solutions to the incompressible Euler equations in three and even dimensions with periodic boundary conditions. These solutions are high-dimensional and have the interesting property of being dense on tori of arbitrary dimension. Additionally, in the two-dimensional case, we prove that quasi-periodic solutions are dense in the phase space of the Euler equations. This provides important insights into the existence of global solutions for high-dimensional initial data.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics
Luigi De Rosa, Marco Inversi, Giorgio Stefani
Summary: In the class of admissible weak solutions, we establish a weak-strong uniqueness result for the incompressible Euler equations under the assumption that the symmetric part of the gradient belongs to Lac([0, +infinity); Lexp(Rd ; Rdxd)), where Lexp denotes the Orlicz space of exponentially integrable functions. Additionally, assuming the same conditions on the limit solution to the Euler system, we prove the convergence of vanishing-viscosity Leray-Hopf weak solutions of the Navier-Stokes equations.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
In-Jee Jeong, Tsuyoshi Yoneda
Summary: The study on vortex thinning mechanism in two-dimensional turbulence reveals that it may result in inverse energy cascade and enhanced dissipation rate. The research also demonstrates that the enhanced dissipation induced by vortex thinning has a vanishing order of enstrophy dissipation slower than Re-1.
Article
Mathematics, Applied
Daniel Ginsberg
Summary: In this study, a continuation criterion for incompressible liquids with free surface boundary in a bounded region was proven by combining energy estimates and velocity gradient estimates in terms of vorticity. The solution can be continued as long as certain physical quantities on the free boundary remain bounded.
SIAM JOURNAL ON MATHEMATICAL ANALYSIS
(2021)
Article
Mathematics, Applied
Senming Chen
Summary: This article studies the energy conservation property of the weak solution of the 3D Euler-Navier-Stokes system. Two sufficient conditions are provided to ensure the conservation of total energy, including the initial energy. A variant method is adopted to deal with the density of Euler flows and velocity of Navier-Stokes fluid due to the coupling structure between the inhomogeneous Euler equation and the incompressible Navier-Stokes system.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics
Wentao Cao
Summary: This paper focuses on the global existence of weak solutions in L-p with finite energy to a type of one-dimensional compressible Euler-Vlasov equations, which models the interaction between the isentropic gas and dispersed particles. Approximate solutions are constructed by adding artificial viscosity. Then the uniform L-p estimates of the approximate solutions with respect to the artificial viscosity are established through some subtle analysis on level sets of density and relative velocity. The convergence of approximate solutions to the desired weak solutions is guaranteed by the L-p compensated compactness framework.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2022)
