Article
Mechanics
Alexey Zhokh, Peter Strizhak
Summary: The study found that in composite porous particles, different classes of diffusing agents exhibit different transport mechanisms in the short and long term, with long-term transport following either Fickian or non-Fickian kinetics depending on the diffusing agent's adsorption energy. The occurrence of non-Fickian transport may be attributed to strong irreversible adsorption sticking of diffusing molecules on the surface of the porous particle.
Article
Mechanics
Yanli Chen, Wenwen Jiang, Xueqing Zhang, Yuanyuan Geng, Guiqiang Bai
Summary: By linking the fractional operators to the microstructure of pore porous media, this study develops a spatial fractional permeability model and a fractional thermal conductivity model for non-Newtonian fluids in porous media. The accuracy of these models is higher than that of the conventional capillary model and they reveal the relationship between nonlocal memory and microstructural properties of complex fluids.
Article
Mathematics, Applied
Belen Lopez, Hanna Okrasinska-Plociniczak, Lukasz Plociniczak, Juan Rocha
Summary: In this study, the self-similar solutions of the time-fractional porous medium equation were investigated. It was proved that there exists a unique solution in the one-dimensional setting and a backward shooting method was used to propose an efficient numerical scheme. The convergence of the problem and several error estimates were also demonstrated. The theoretical findings were verified through numerical simulations.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2024)
Article
Mathematics
Nikolaos Roidos, Yuanzhen Shao
Summary: This article is the second of a series of two papers that investigate the fractional porous medium equation on a Riemannian manifold with isolated conical singularities. The first part focuses on deriving useful properties for the Mellin-Sobolev spaces and the second part studies the Markovian extensions of the conical Laplacian operator. Based on the obtained results, the article establishes the existence and uniqueness of a global strong solution for L-infinity initial data and all m > 0, and further investigates properties such as the comparison principle and conservation of mass. The approach presented in this article is applicable to a variety of similar problems on manifolds with more general singularities.
MATHEMATISCHE NACHRICHTEN
(2023)
Article
Mathematics, Applied
Hanna Okrasinska-Plociniczak, Lukasz Plociniczak
Summary: This study investigates a time-fractional porous medium equation that is important in applications such as hydrology and material sciences. The study reveals that solutions of the free boundary Dirichlet, Neumann, and Robin problems on the half-line satisfy a Volterra integral equation with a non-Lipschitz nonlinearity. Based on this result, the study proves the existence, uniqueness, and constructs a family of numerical methods that outperform the usual finite difference approach. Furthermore, the study demonstrates the convergence of these methods and supports the theory with several numerical examples.
APPLIED MATHEMATICS AND COMPUTATION
(2022)
Article
Mathematics, Applied
Young-Pil Choi, In-Jee Jeong
Summary: In this study, asymptotic analysis is performed for the Euler-Riesz system in high-force regime, resulting in a quantified relaxation limit to the fractional porous medium equation. A unified approach is provided for asymptotic analysis in the presence of repulsive Riesz interactions, while a lower bound estimation on modulated internal energy is considered for attractive Riesz interactions in a periodic domain.
JOURNAL OF NONLINEAR SCIENCE
(2021)
Article
Engineering, Multidisciplinary
O. Nikan, Z. Avazzadeh, J. A. Tenreiro Machado
Summary: The paper introduces an accurate and robust meshless technique for approximating the solution of the time fractional Cattaneo model describing heat conduction in a porous medium. The method involves using fractional derivatives and a local meshless approach, resulting in sparse matrices that do not suffer from ill-conditioning and computational burden. Computational examples illustrate the accuracy and effectiveness of the proposed method.
APPLIED MATHEMATICAL MODELLING
(2021)
Article
Mathematics, Applied
Li Chen, Alexandra Holzinger, Ansgar Juengel, Nicola Zamponi
Summary: This paper performs a mean-field-type limit from stochastic moderately interacting many-particle systems with singular Riesz potential, resulting in nonlocal porous-medium equations in the whole space. The nonlocality is given by the inverse of a fractional Laplacian, and the limit equation can be interpreted as a transport equation with a fractional pressure. The paper also provides an existence analysis of the fractional porous-medium equation and concludes the propagation of chaos property.
COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS
(2022)
Article
Engineering, Marine
R. Meher, J. Kesarwani, Z. Avazzadeh, O. Nikan
Summary: This paper proposes a temporal-fractional porous medium model for describing co-current and counter-current imbibition. The correlation between imbibition in fractures and porous matrix is examined to determine the saturation and recovery rate. The effects of different fractional orders, wettability, and inclination on saturation and recovery rate are also studied.
JOURNAL OF OCEAN ENGINEERING AND SCIENCE
(2023)
Article
Engineering, Marine
Anand Kumar Yadav
Summary: The research article analyzes the wave propagation in an initially stressed micropolar fractional order derivative thermoelastic diffusion medium with voids. Different types of plane waves are identified, each exhibiting unique characteristics influenced by the presence of thermal, diffusion, and void parameters in the medium. The speeds of the plane waves were determined for a specific material and plotted against various parameters for further analysis.
JOURNAL OF OCEAN ENGINEERING AND SCIENCE
(2021)
Article
Multidisciplinary Sciences
Bangti Jin, Yavar Kian
Summary: This study investigates an inverse problem of recovering multiple orders in a time-fractional diffusion model from data observed at a single point on the boundary. The unique recovery of the orders and weights is proven without needing full knowledge of domain or medium properties. The proof is based on Laplace transform and asymptotic expansion, and a numerical procedure for recovering these parameters is proposed based on nonlinear least-squares fitting.
PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES
(2021)
Article
Mathematics, Applied
Inwon Kim, Yuming Paul Zhang
Summary: In this study, regularity properties of the free boundary for solutions of the porous medium equation with drift were investigated. The C-1, C-alpha regularity of the free boundary was shown in the presence of directional monotonicity in space variable in a local neighborhood. The main challenge was establishing a local non-degeneracy estimate, which is new even in the case of zero drift.
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
(2021)
Article
Thermodynamics
Milad Mozafarifard, Davood Toghraie, Hossein Sobhani
Summary: The study utilized the Caputo fractional subdiffusion model to analyze the fast-transient process of heat flow in a porous medium, demonstrating its effectiveness in simulating solid-fluid interactions and fast transient processes.
INTERNATIONAL COMMUNICATIONS IN HEAT AND MASS TRANSFER
(2021)
Article
Engineering, Electrical & Electronic
Xuejiao Zhao, Shipeng Wang, Qiong Wu, Yanju Ji
Summary: The theory of magnetic-source electromagnetic anomalous diffusion is developed, presenting a multiscale electromagnetic anomalous diffusion theory with improved generalized electrical conductivity. This theory is applied to propose a 3-D modeling method for complex electromagnetic propagation in rough geologic medium, demonstrating improved efficiency and accuracy in detection.
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION
(2021)
Article
Mathematics, Applied
Pedro Cardoso, Renato de Paula, Patricia Goncalves
Summary: In this article, we obtain the fractional porous medium equation for any power of the fractional Laplacian by studying the microscopic dynamics of random particles with long range interactions. However, the jump rate is highly influenced by the occupancy near the sites where the interactions occur.
Article
Mathematics, Applied
Felix del Teso, Jorgen Endal, Juan Luis Vazquez
Summary: The study investigates the existence and properties of solutions and free boundaries of the one-phase Stefan problem with fractional diffusion posed in Double-struck capital R-N. We prove the existence of a continuous and bounded selfsimilar solution with a free boundary at the change-of-phase level. The study also provides well-posedness and basic properties of very weak solutions for general bounded data in several dimensions, and explores limits and connections with other diffusion problems.
MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES
(2021)
Article
Mathematics, Applied
Lorenzo Brasco, David Gomez-Castro, Juan Luis Vazquez
Summary: The study aims to characterize the homogeneous fractional Sobolev-Slobodecki.i spaces D-s,D-p(R-n) and their embeddings, showing isomorphisms to suitable function spaces or space of equivalence classes of functions. The Morrey-Campanato inequality is presented as a main tool in the analysis.
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
R. Ferreira, A. de Pablo
Summary: The study focuses on the speed of convergence to infinity of nonglobal solutions to the fractional heat equation ut |(-Delta) (alpha/2)u = u(p). It is proven that under certain conditions, the behavior of the solution near the blow-up time can be characterized. Elementary tools such as rescaling or comparison arguments are used in the proofs.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
(2021)
Article
Mathematics
Hardy Chan, David Gomez-Castro, Juan Luis Vazquez
Summary: The paper presents a linear theory of nonlocal eigenvalue problems involving integro-differential operators posed in bounded domains with homogeneous Dirichlet exterior condition, discussing both cases with and without singular boundary data. Various aspects are explored, including the Fredholm alternative, large solutions, possible blow-up limits, and the behavior as the first eigenvalue is approached. The transfer of orthogonality, existence of large eigenfunctions, and maximum principle for weighted solutions are also highlighted.
JOURNAL OF FUNCTIONAL ANALYSIS
(2021)
Article
Mathematics, Applied
Jose A. Carrillo, David Gomez-Castro, Juan Luis Vazquez
Summary: This paper studies radial solutions of an aggregation-diffusion model that combines nonlinear fast diffusion with a convection term driven by the gradient of a potential. Depending on the exponent of fast diffusion and the potential, the steady state is given by the sum of an explicit integrable function, plus a Dirac delta at the origin containing the rest of the mass of the initial datum, showing a splitting phenomenon.
JOURNAL DE MATHEMATIQUES PURES ET APPLIQUEES
(2022)
Article
Mathematics
Carmen Cortazar, Fernando Quiros, Noemi Wolanski
Summary: The study focuses on the large-time behavior in all L-p norms of solutions to a heat equation with a Caputo alpha-time derivative, specifically for subdiffusion equations and integrable initial data. The decay rate in all L-p norms is found to greatly depend on the space-time scale, explaining the critical dimension phenomenon. Additionally, the final profiles strongly depend on scale, with the final profile in compact sets being a multiple of the Newtonian potential of the initial datum. These results differ significantly from classical diffusion equations and indicate that these equations are good models for systems with sticking and trapping phenomena or fluids with memory.
JOURNAL OF FUNCTIONAL ANALYSIS
(2021)
Article
Mathematics, Applied
Juan Luis Vazquez
Summary: The study establishes the existence, uniqueness, and quantitative estimates for solutions to the fractional nonlinear diffusion equation, focusing on a certain range of exponents. By obtaining weighted global integral estimates, the existence of solutions for a class of large data is proved, and the dichotomy positivity versus extinction for nonnegative solutions at any given time is established. The analysis includes the conditions for extinction in finite time.
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
(2022)
Article
Mathematics, Applied
Arturo de Pablo, Fernando Quiros, Antonella Ritorto
Summary: The study proves the existence of an extremal function in the Hardy-Littlewood-Sobolev inequality for the energy associated with a stable operator, and obtains a concentration-compactness principle for stable processes in R-N.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2022)
Article
Mathematics
Hardy Chan, David Gomez-Castro, Juan Luis Vazquez
Summary: In this paper, we investigate the existence and uniqueness of solutions for the parabolic problem with singular data, as well as the impact of initial data and forcing term. When the boundary data is zero, the results coincide with the standard fractional heat semigroup. Furthermore, we explore the spectral theory of the fractional heat semigroup and obtain bounds on the fractional heat kernel.
REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS
(2022)
Article
Mathematics, Applied
Filomena Feo, Juan Luis Vazquez, Bruno Volzone
Summary: We prove the existence of self-similar fundamental solutions (SSF) of the anisotropic porous medium equation in the suitable fast diffusion range. Each of such SSF solutions is uniquely determined by its mass. We also obtain the asymptotic behaviour of all finite mass solutions in terms of the family of self-similar fundamental solutions.
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
(2023)
Article
Mathematics, Applied
J. A. Carrillo, D. Gomez-Castro, J. L. Vazquez
Summary: This paper discusses the density solutions for gradient flow equations of the form u(t) = del center dot (gamma(u) backward difference N(u)), and compares the results for linear mobility and concave mobility cases. It is found that solutions with compactly supported initial data remain compactly supported and generate moving free boundaries. For the linear mobility case, a special solution in the form of a disk vortex is present, while for the concave mobility case, viscosity solutions exist in the whole space and display a fat tail at infinity.
ADVANCES IN NONLINEAR ANALYSIS
(2022)
Article
Mathematics, Interdisciplinary Applications
Carmen Cortazar, Fernando Quiros, Noemi Wolanski
Summary: This study focuses on the decay/growth rates in all L-p norms of solutions to an inhomogeneous nonlocal heat equation in a large dimension space, N > 4 beta. The rates are strongly influenced by the space-time scale and the time behavior of the spatial L-1 norm of the forcing term.
MATHEMATICS IN ENGINEERING
(2022)
Article
Mathematics, Interdisciplinary Applications
Raul Ferreira, Arturo de Pablo
Summary: We study the behaviour of solutions to a quasilinear heat equation with a reaction restricted to a half-line. We characterize the global existence exponent and Fujita exponent, and then examine the grow-up rate and blow-up rate under different conditions. We show that the grow-up rate differs from the case of global reaction if certain conditions are met.
MATHEMATICS IN ENGINEERING
(2022)
Article
Mathematics
Carmen Cortazar, Fernando Quiros, Noemi Wolanski
Summary: The study focuses on the large-time behavior of solutions to a nonlocal heat equation in R-N involving a Caputo alpha-time derivative and a power of the Laplacian (-Delta)(s), s is an element of (0, 1), extending recent results for the case s = 1. The main novelty lies in the behavior in fast scales, where results not available for the case s = 1 or the standard heat equation s = 1, alpha = 1 are obtained due to the fat tails of the fundamental solution of the equation.
VIETNAM JOURNAL OF MATHEMATICS
(2021)
Article
Mathematics
Matija Bucic, Richard Montgomery
Summary: This article improves upon previous research by showing that any n-vertex graph can be decomposed into O(n log* n) cycles and edges.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Laurentiu G. Maxim, Jose Israel Rodriguez, Botong Wang, Lei Wu
Summary: The paper investigates the relationship between linear optimization degree and geometric structure. By analyzing the geometric structure of the conormal variety of an affine variety, the Chern-Mather classes of the given variety can be completely determined. Additionally, the paper shows that these bidegrees coincide with the linear optimization degrees of generic affine sections.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
William Chan, Stephen Jackson, Nam Trang
Summary: Under the determinacy hypothesis, this paper completely characterizes the existence of nontrivial maximal almost disjoint families for specific cardinals kappa, considering the ideals of bounded subsets and subsets of cardinality less than kappa.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Zhenguo Liang, Zhiyan Zhao, Qi Zhou
Summary: This paper investigates the reducibility of the one-dimensional quantum harmonic oscillator perturbed by a time quasi-periodic quadratic form. It provides a description and upper bound for the growth of the Sobolev norms of the solution, and demonstrates the optimality of the upper bound.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Zhao Yu Ma, Yair Shenfeld
Summary: This study provides a new approach to understanding the extremal cases of Stanley's inequalities by establishing a connection between the combinatorics of partially ordered sets and the geometry of convex polytopes.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Laurent Laurent, Rosa M. Miro-Roig
Summary: This paper discusses the problem of constructing matrices of linear forms of constant rank by focusing on vector bundles on projective spaces. It introduces important examples of classical Steiner bundles and Drezet bundles, and uses the classification of globally generated vector bundles to describe completely the indecomposable matrices of constant rank up to six.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Nicoletta Cantarini, Fabrizio Caselli, Victor Kac
Summary: In this paper, we construct a duality functor in the category of continuous representations to study the Lie superalgebra E(4, 4). By constructing a specific type of Lie conformal superalgebra, we obtain that E(4, 4) is its annihilation algebra. Furthermore, we also obtain an explicit realization of E(4, 4) on a supermanifold in the process of studying.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Rotem Assouline, Bo'az Klartag
Summary: This article studies the horocyclic Minkowski sum of two subsets in the hyperbolic plane and its properties. It proves an inequality relating the area of the subsets when they are Borel-measurable, and provides a connection to other inequalities.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Alessio Porretta
Summary: This article discusses Fokker-Planck equations driven by Levy processes in the entire Euclidean space, under the influence of confining drifts, similar to the classical Ornstein-Ulhenbeck model. A new PDE method is introduced to obtain exponential or sub-exponential decay rates of zero average solutions as time goes to infinity, under certain diffusivity conditions on the Levy process, including the fractional Laplace operator as a model example. The approach relies on long-time oscillation estimates of the adjoint problem and applies to both local and nonlocal diffusions, as well as strongly or weakly confining drifts.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Weichao Qian, Yong Li, Xue Yang
Summary: In this paper, we investigate the persistence of resonant invariant tori in Hamiltonian systems with high-order degenerate perturbation, and prove a quasiperiodic Poincare theorem under high degeneracy, answering a long-standing conjecture on the persistence of resonant invariant tori in general situations.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Julius Ross, David Witt Nystroem
Summary: This article extends Prekopa's Theorem and the Brunn-Minkowski Theorem from convexity to F-subharmonicity, and applies it to the interpolation problem of convex functions and convex sets, introducing a new notion of harmonic interpolation.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Airi Takeuchi, Lei Zhao
Summary: In this article, we explore the connection between several integrable mechanical billiards in the plane through conformal transformations. We discuss the equivalence of free billiards and central force problems, as well as the correspondence between integrable Hooke-Kepler billiards. We also investigate the integrability of Kepler billiards and Stark billiards, and the relationship between billiard systems and Euler's two-center problems.
ADVANCES IN MATHEMATICS
(2024)
Article
Mathematics
Damiano Rossi
Summary: In this study, we prove new results in generalised Harish-Chandra theory by providing a description of the Brauer-Lusztig blocks using the p-adic cohomology of Deligne-Lusztig varieties. We then propose new conjectures for finite reductive groups by considering geometric analogues of the p-local structures. Our conjectures coincide with the counting conjectures for large primes, thanks to a connection established between p-structures and their geometric counterparts. Finally, we simplify our conjectures by reducing them to the verification of Clifford theoretic properties.
ADVANCES IN MATHEMATICS
(2024)
