Article
Mathematics, Applied
Tong Tang, Sarka Necasova
Summary: This paper rigorously derives the inviscid compressible Primitive Equations from the Euler system in a periodic channel by utilizing the relative entropy inequality. The Primitive Equations play an important role in geophysical research and mathematical analysis.
APPLIED MATHEMATICS LETTERS
(2023)
Article
Mathematics
Xiaoguang You, Aibin Zang
Summary: In this paper, the authors study the second-grade fluid equations in a 2D exterior domain with non-slip boundary conditions. The second-grade fluid model is a well-known non-Newtonian fluid model, with alpha representing the length scale and nu > 0 representing the viscosity. The authors prove that as nu and alpha tend to zero, the solution of the second-grade fluid equations converges to the solution of the Euler equations with suitable initial data, provided that nu = o(alpha(4)/(3)). The convergent rate is also obtained.
ACTA MATHEMATICA SCIENTIA
(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
Daomin Cao, Jie Wan, Guodong Wang, Weicheng Zhan
Summary: The study presents a family of rotating vortex patches with fixed angular velocity in a disk, where the limit of these patches as the vorticity strength approaches infinity is a rotating point vortex. The construction is based on solving a variational problem for the vorticity using an adaption of Arnold's variational principle. Nonlinear orbital stability of the set of maximizers in the variational problem is then proven under L-p perturbation for p in the range of [3/2, +infinity).
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics
Frank Merle, Pierre Raphael, Igor Rodnianski, Jeremie Szeftel
Summary: This paper and its sequel construct a set of finite energy smooth initial data for which the corresponding solutions to the compressible three-dimensional Navier-Stokes and Euler equations implode at a later time and point, and describe the formation of singularity. The existence of smooth self-similar profiles for the barotropic Euler equations in dimension d >= 2 with decaying density at spatial infinity is studied. The phase portrait of the nonlinear ODE for spherically symmetric self-similar solutions allows the construction of global profiles, but they are generically non-smooth. The existence of non-generic C-infinity self-similar solutions with suitable decay at infinity is proved, leading to the construction of finite energy blow up solutions of the compressible Euler and Navier-Stokes equations in dimensions d = 2, 3.
ANNALS OF MATHEMATICS
(2022)
Article
Mathematics, Applied
Guodong Wang, Bijun Zuo
Summary: This note discusses the radial symmetry property of rotating vortex patches for the 2D incompressible Euler equations in the unit disk. It shows that under certain conditions, a rotating vortex patch must be a disk.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2021)
Article
Mathematics, Applied
Cheng Yang
Summary: The paper investigates planar point vortex motion of the Euler equations, proves the non-collision property of the 2-vortex system in the half-plane, and demonstrates that the N-vortex system in the half-plane is non-integrable for N > 2. It also discusses the similarities between different vortex motions in various geometries and raises some open questions.
JOURNAL OF NONLINEAR SCIENCE
(2021)
Article
Mathematics
Jaemin Park
Summary: This paper provides quantitative estimates for uniformly-rotating vortex patches. It proves that if a non-radial simply-connected patch D has a small angular velocity 0 < Omega << 1, the distance from the center of rotation to the outmost point of the patch must be at least of order Omega-1/2. For m-fold symmetric simply-connected rotating patches, the paper shows that their angular velocity must be close to 12 for m >> 1, with a difference at most O(1/m), and also obtains estimates on the L infinity norm of the polar graph which parametrizes the boundary.
ADVANCES IN MATHEMATICS
(2022)
Article
Mathematics, Applied
Richard M. Hoefer
Summary: This paper studies the solution to the Navier-Stokes equations in perforated media with small particles and no-slip boundary conditions. The behavior of the solution is examined for small values of ε, which depends on the diameter of the particles and the viscosity of the fluid. The results demonstrate that when the local Reynolds number at the particles is negligible, the particles exert an approximately linear friction force on the fluid. The effective macroscopic equations obtained depend on the magnitude of the collective friction.
Article
Mathematics, Applied
Christian Zillinger
Summary: The article discusses linear inviscid damping in Gevrey regularity for compactly supported Gevrey regular shear flows in a finite channel, providing an alternative proof of stability using Fourier-based Lyapunov functional. The stability in L-2 by Fourier methods immediately upgrades to stability in Gevrey regularity for certain flows, even without assuming compact support.
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
(2021)
Article
Mathematics, Applied
Eliseo Luongo
Summary: We study the convergence of the solution of the stochastic second-grade fluid equations with transport noise and no-slip boundary conditions to the solution of the corresponding Euler equations in a smooth bounded and simply connected two dimensional domain. We prove that, under proper regularity of the initial conditions of the Euler equations and appropriate behavior of the parameters ? and a, the inviscid limit holds without requiring specific dissipation of the energy of the solutions in the boundary layer.
STOCHASTICS AND PARTIAL DIFFERENTIAL EQUATIONS-ANALYSIS AND COMPUTATIONS
(2023)
Article
Mathematics, Applied
Gerasim Krivovichev
Summary: This paper compares one-dimensional blood flow models for solving model problems. The non-Newtonian properties of blood and the viscid case are considered. Analytical and numerical methods are proposed for solving problems. The effects of viscosity and velocity profile on blood flow are analyzed through model comparison.
APPLIED MATHEMATICS AND COMPUTATION
(2022)
Article
Physics, Fluids & Plasmas
Cornelius Rampf, Uriel Frisch, Oliver Hahn
Summary: This study explores the complex singularities in the temporal domain of the inviscid Burgers equation with sine-wave initial conditions, revealing a unique eye-shaped arrangement of singularities centered around the origin. These complex-time singularities become physically relevant before the preshock, showcasing the importance of early-time phenomena in fluid dynamics. Various methods are employed to reduce the amplitude of these early-time singularities, including tyger purging and iterative UV completion, demonstrating potential applications in higher dimensions and other hydrodynamic equations.
PHYSICAL REVIEW FLUIDS
(2022)
Article
Mathematics
David M. Ambrose, Elaine Cozzi, Daniel Erickson, James P. Kelliher
Summary: This article establishes the short-time existence of solutions to the surface quasi-geostrophic (SQG) equation in the Holder spaces Cr (][82) for r > 1. To avoid an integrability assumption, a generalization of the SQG constitutive law is introduced. Using these solutions, the existence of solutions of SQG in another class of non-decaying function spaces, the uniformly local Sobolev spaces Hsul(][82) for s >= 3, is proven. The short-time existence of the three-dimensional Euler equations in uniformly local Sobolev spaces is also obtained using similar methods as the SQG equation. (c) 2023 Elsevier Inc. All rights reserved.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2023)
Article
Engineering, Multidisciplinary
Nadeem Abbas, Maryam Tumreen, Wasfi Shatanawi, Muhammad Qasim, Taqi A. M. Shatnawi
Summary: The three-dimensional flow of non-Newtonian fluid over a slendering stretching sheet considering radiative and chemical reaction effects is studied. The Buongiorno model is used to analyze the fluid flow region, while a numerical scheme in Matlab is employed to solve the governing model. The results show that higher values of heat generation improves the temperature curves, and radiation and heat source enhance the temperature of the fluid.
ALEXANDRIA ENGINEERING JOURNAL
(2023)
Article
Mathematics, Applied
Aseel Farhat, Evelyn Lunasin, Edriss S. Titi
JOURNAL OF NONLINEAR SCIENCE
(2017)
Article
Mathematics, Applied
Ciprian Foias, Michael S. Jolly, Dan Lithio, Edriss S. Titi
JOURNAL OF NONLINEAR SCIENCE
(2017)
Article
Mechanics
Adam Larios, Mark R. Petersen, Edriss S. Titi, Beth Wingate
THEORETICAL AND COMPUTATIONAL FLUID DYNAMICS
(2018)
Article
Mathematics, Applied
Claude Bardos, Piotr Gwiazda, Agnieszka Swierczewska-Gwiazda, Edriss S. Titi, Emil Wiedemann
JOURNAL OF NONLINEAR SCIENCE
(2019)
Article
Mathematics, Applied
Xin Liu, Edriss S. Titi
Summary: This study demonstrates the local-in-time well-posedness of strong solutions to the three-dimensional compressible primitive equations of atmospheric dynamics. It shows that strong solutions exist, are unique, and depend continuously on the initial data for a short period in two cases: with gravity but without vacuum, and with vacuum but without gravity.
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
(2021)
Article
Mathematics
Slim Ibrahim, Quyuan Lin, Edriss S. Titi
Summary: This study investigates the dynamics of inviscid primitive equations with rotation, proving their ill-posedness in Sobolev spaces and suggesting that a suitable space for well-posedness is Gevrey class of order s = 1.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics
Jinkai Li, Edriss S. Titi, Guozhi Yuan
Summary: This paper rigorously justifies the hydrostatic approximation and derivation of primitive equations as the small aspect ratio limit of the incompressible three-dimensional Navier-Stokes equations in anisotropic horizontal viscosity regime. The study shows that for well-prepared initial data, the solutions converge strongly to corresponding solutions of anisotropic primitive equations with only horizontal viscosities as epsilon tends to zero, with a convergence rate of O (epsilon(beta/2)). The convergence rate is determined by the relationship between the parameters alpha and beta.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2022)
Article
Meteorology & Atmospheric Sciences
Mohamad Abed El Rahman Hammoud, Edriss S. Titi, Ibrahim Hoteit, Omar Knio
Summary: Generating high-resolution flow fields is crucial for engineering and climate science applications. We propose a physics-informed deep neural network (PI-DNN) to predict fine-scale flow fields using coarse-scale data. Numerical results show that the predictions of the PI-DNN are comparable to those obtained by dynamical downscaling.
JOURNAL OF ADVANCES IN MODELING EARTH SYSTEMS
(2022)
Article
Mathematics
Xin Liu, Edriss S. Titi
Summary: In this work, we investigate the zero Mach number limit of compressible primitive equations in the domains R2 x 2T or T2 x 2T. The limit equations are identified to be the primitive equations with the incompressible condition. The convergence behaviors are studied in both R2 x 2T and T2 x 2T, respectively. This paper considers the presence of high oscillating acoustic waves and extends previous research in [29].
JOURNAL OF DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Sabine Hittmeir, Rupert Klein, Jinkai Li, Edriss S. Titi
Summary: In this work, we investigate the global solvability of moisture dynamics with phase changes for warm clouds, taking into account the different gas constants and heat capacities of dry air, water vapor, and liquid water. This refined thermodynamic setting has been proven essential in previous studies and requires careful derivations of a priori estimates for proving global existence and uniqueness of solutions.
JOURNAL OF NONLINEAR SCIENCE
(2023)
Article
Mathematics, Applied
Yu Cao, Michael S. Jolly, Edriss S. Titi, Jared P. Whitehead
Summary: The Rayleigh-Benard system with stress-free boundary conditions is shown to have a global attractor in each affine space where velocity has fixed spatial average. The physical problem is proven to be equivalent to one with periodic boundary conditions and certain symmetries. A Gronwall estimate on enstrophy leads to bounds on the L-2 norm of the temperature gradient on the global attractor. By finding a bounding region for the attractor in the enstrophy-palinstrophy plane, all final bounds are algebraic in the viscosity and thermal diffusivity, which is a significant improvement over previously established estimates.
Article
Mathematics, Applied
Chongsheng Cao, Yanqiu Guo, Edriss S. Titi
Summary: In this model, the dynamics of the velocity field occur at a much faster time scale than the temperature fluctuation, and the velocity field formally adjusts instantaneously to the thermal fluctuation at the limit. The global well-posedness of weak solutions and strong solutions to this model has been proven.
JOURNAL OF EVOLUTION EQUATIONS
(2021)
Article
Mathematics, Applied
Claude Bardos, Edriss S. Titi
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
(2018)
Article
Computer Science, Interdisciplinary Applications
M. U. Altaf, E. S. Titi, T. Gebrael, O. M. Knio, L. Zhao, M. F. McCabe, I. Hoteit
COMPUTATIONAL GEOSCIENCES
(2017)
Article
Mathematics, Applied
Michael S. Jolly, Vincent R. Martinez, Edriss S. Titi
ADVANCED NONLINEAR STUDIES
(2017)
