The main purpose of this work is to show that the gravity term of the segregation-mixing equation of fine mono-disperse particles in a fluid can be derived from first-principles （i.e., elementary physics）. Our deriv...The main purpose of this work is to show that the gravity term of the segregation-mixing equation of fine mono-disperse particles in a fluid can be derived from first-principles （i.e., elementary physics）. Our derivation of the gravity-driven flux of particles leads to the simplest case of the Richardson and Zaki correlation. Stokes velocity also naturally appears from the physical parameters of the particles and fluid by means of derivation only. This derivation from first-principle physics has never been presented before. It is applicable in small concentrations of fine particles.展开更多
Let q∈(0,∞]andϕbe a Musielak-Orlicz function with uniformly lower type p_(ϕ)^(−)∈(0,∞)and uniformly upper type p_(ϕ)^(−)∈(0,∞).In this article,the authors establish various realvariable characterizations of the ...Let q∈(0,∞]andϕbe a Musielak-Orlicz function with uniformly lower type p_(ϕ)^(−)∈(0,∞)and uniformly upper type p_(ϕ)^(−)∈(0,∞).In this article,the authors establish various realvariable characterizations of the Musielak-Orlicz-Lorentz Hardy space H^(ϕ,q)(R^(n)),respectively,in terms of various maximal functions,finite atoms,and various Little wood-Paley functions.As applications,the authors obtain the dual space of Hϕ,q(Rn)and the summability of Fourier transforms from Hϕ,q(Rn)to the Musielak-Orlicz-Lorentz space L^(ϕ,q)(R^(n))when q∈(0,∞)or from the Musielak-Orlicz Hardy space Hϕ(Rn)to Lϕ,∞(Rn)in the critical case.These results are new when q∈(0,∞)and also essentially improve the existing corresponding results(if any)in the case q=∞via removing the original assumption thatϕis concave.To overcome the essential obstacles caused by both thatϕmay not be concave and that the boundedness of the powered Hardy-Littlewood maximal operator on associated spaces of Musielak-Orlicz spaces is still unknown,the authors make full use of the obtained atomic characterization of H^(ϕ,q)(R^(n)),the corresponding results related to weighted Lebesgue spaces,and the subtle relation between Musielak-Orlicz spaces and weighted Lebesgue spaces.展开更多
An existing phase-fieldmodel of two immiscible fluids with a single soluble surfactant present is discussed in detail.We analyze the well-posedness of the model and provide strong evidence that it is mathematically il...An existing phase-fieldmodel of two immiscible fluids with a single soluble surfactant present is discussed in detail.We analyze the well-posedness of the model and provide strong evidence that it is mathematically ill-posed for a large set of physically relevant parameters.As a consequence,critical modifications to the model are suggested that substantially increase the domain of validity.Carefully designed numerical simulations offer informative demonstrations as to the sharpness of our theoretical results and the qualities of the physical model.A fully coupled hydrodynamic test-case demonstrates the potential to capture also non-trivial effects on the overall flow.展开更多
A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier transforms. Under some conditions on θ, it is proved that the maximal operator of the θ-means defined in a cone ...A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier transforms. Under some conditions on θ, it is proved that the maximal operator of the θ-means defined in a cone is bounded from the amalgam Hardy space W（hp, e∞） to W（Lp,e∞）. This implies the almost everywhere convergence of the θ-means in a cone for all f ∈ W（L1, e∞） velong to L1.展开更多
The numerical approximation of high frequency wave propagation is important in many applications.Examples include the simulation of seismic,acoustic,optical waves and microwaves.When the frequency of the waves is high...The numerical approximation of high frequency wave propagation is important in many applications.Examples include the simulation of seismic,acoustic,optical waves and microwaves.When the frequency of the waves is high,this is a difficult multiscale problem.The wavelength is short compared to the overall size of the computational domain and direct simulation using the standard wave equations is very expensive.Fortunately,there are computationally much less costly models,that are good approximations of many wave equations precisely for very high frequencies.Even for linear wave equations these models are often nonlinear.The goal of this paper is to review such mathematical models for high frequency waves,and to survey numerical methods used in simulations.We focus on the geometrical optics approximation which describes the infinite frequency limit of wave equations.We will also discuss finite frequency corrections and some other models.展开更多
An adaptive numerical scheme is developed for the propagation of an interface in a velocity field based on the fast interface tracking method proposed in [2]. A multiresolution stategy to represent the interface inste...An adaptive numerical scheme is developed for the propagation of an interface in a velocity field based on the fast interface tracking method proposed in [2]. A multiresolution stategy to represent the interface instead of point values, allows local grid refinement while controlling the approximation error on the interface. For time integration, we use an explicit Runge-Kutta scheme of second-order with a multiseale time step, which takes longer time steps for finer spatial scales. The implementation of the algorithm uses a dynamic tree data structure to represent data in the computer memory. We briefly review first the main algorithm, describe the essential data structures, highlight the adaptive scheme, and illustrate the computational efficiency by some numerical examples.展开更多
文摘The main purpose of this work is to show that the gravity term of the segregation-mixing equation of fine mono-disperse particles in a fluid can be derived from first-principles （i.e., elementary physics）. Our derivation of the gravity-driven flux of particles leads to the simplest case of the Richardson and Zaki correlation. Stokes velocity also naturally appears from the physical parameters of the particles and fluid by means of derivation only. This derivation from first-principle physics has never been presented before. It is applicable in small concentrations of fine particles.
基金partially supported by the National Key Research and Development Program of China(Grant No.2020YFA0712900)the National Natural Science Foundation of China(Grant Nos.12371093,12071197,and 12122102)+1 种基金the Fundamental Research Funds for the Central Universities(Grant No.2233300008)partially supported by a McDevitt Endowment Fund at Georgetown University。
文摘Let q∈(0,∞]andϕbe a Musielak-Orlicz function with uniformly lower type p_(ϕ)^(−)∈(0,∞)and uniformly upper type p_(ϕ)^(−)∈(0,∞).In this article,the authors establish various realvariable characterizations of the Musielak-Orlicz-Lorentz Hardy space H^(ϕ,q)(R^(n)),respectively,in terms of various maximal functions,finite atoms,and various Little wood-Paley functions.As applications,the authors obtain the dual space of Hϕ,q(Rn)and the summability of Fourier transforms from Hϕ,q(Rn)to the Musielak-Orlicz-Lorentz space L^(ϕ,q)(R^(n))when q∈(0,∞)or from the Musielak-Orlicz Hardy space Hϕ(Rn)to Lϕ,∞(Rn)in the critical case.These results are new when q∈(0,∞)and also essentially improve the existing corresponding results(if any)in the case q=∞via removing the original assumption thatϕis concave.To overcome the essential obstacles caused by both thatϕmay not be concave and that the boundedness of the powered Hardy-Littlewood maximal operator on associated spaces of Musielak-Orlicz spaces is still unknown,the authors make full use of the obtained atomic characterization of H^(ϕ,q)(R^(n)),the corresponding results related to weighted Lebesgue spaces,and the subtle relation between Musielak-Orlicz spaces and weighted Lebesgue spaces.
文摘An existing phase-fieldmodel of two immiscible fluids with a single soluble surfactant present is discussed in detail.We analyze the well-posedness of the model and provide strong evidence that it is mathematically ill-posed for a large set of physically relevant parameters.As a consequence,critical modifications to the model are suggested that substantially increase the domain of validity.Carefully designed numerical simulations offer informative demonstrations as to the sharpness of our theoretical results and the qualities of the physical model.A fully coupled hydrodynamic test-case demonstrates the potential to capture also non-trivial effects on the overall flow.
基金Supported by the Hungarian Scientific Research Funds (OTKA) No. K67642
文摘A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier transforms. Under some conditions on θ, it is proved that the maximal operator of the θ-means defined in a cone is bounded from the amalgam Hardy space W（hp, e∞） to W（Lp,e∞）. This implies the almost everywhere convergence of the θ-means in a cone for all f ∈ W（L1, e∞） velong to L1.
文摘The numerical approximation of high frequency wave propagation is important in many applications.Examples include the simulation of seismic,acoustic,optical waves and microwaves.When the frequency of the waves is high,this is a difficult multiscale problem.The wavelength is short compared to the overall size of the computational domain and direct simulation using the standard wave equations is very expensive.Fortunately,there are computationally much less costly models,that are good approximations of many wave equations precisely for very high frequencies.Even for linear wave equations these models are often nonlinear.The goal of this paper is to review such mathematical models for high frequency waves,and to survey numerical methods used in simulations.We focus on the geometrical optics approximation which describes the infinite frequency limit of wave equations.We will also discuss finite frequency corrections and some other models.
文摘An adaptive numerical scheme is developed for the propagation of an interface in a velocity field based on the fast interface tracking method proposed in [2]. A multiresolution stategy to represent the interface instead of point values, allows local grid refinement while controlling the approximation error on the interface. For time integration, we use an explicit Runge-Kutta scheme of second-order with a multiseale time step, which takes longer time steps for finer spatial scales. The implementation of the algorithm uses a dynamic tree data structure to represent data in the computer memory. We briefly review first the main algorithm, describe the essential data structures, highlight the adaptive scheme, and illustrate the computational efficiency by some numerical examples.
