Article
Mathematics, Applied
M. Barinova, V Grines, O. Pochinka, B. Yu
Summary: This paper continues research on constructing energy functions for discrete dynamical systems, with a focus on A-diffeomorphisms of three-dimensional closed orientable manifolds. The authors demonstrate the existence of an energy function where the non-wandering set includes a chaotic one-dimensional canonically embedded surface attractor and repeller.
Article
Mathematics
S. D. Glyzin, A. Yu. Kolesov
Summary: In this study, we solve a classical problem in the theory of dynamical systems regarding the minimality of the shift on an infinite-dimensional torus. More specifically, we provide sufficient conditions that guarantee the absence of the minimality property.
DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics
Tobias Barker, Wendong Wang
Summary: In this paper, the authors systematically investigate the supercritical conditions on the pressure pi associated with a Navier-Stokes solution v in three dimensions. They show that if the pressure pi satisfies certain endpoint scale invariant conditions, then the Hausdorff dimension of the singular set at a first potential blow-up time can be arbitrarily small. The authors establish a higher integrability result for the Navier-Stokes equations and a convenient epsilon-regularity criterion involving space-time integrals.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Alexis Leculier, Jean-Michel Roquejoffre
Summary: This paper investigates the existence and uniqueness of a non-trivial bounded steady state of a Fisher-KPP equation involving a fractional Laplacian in a fragmented domain with exterior Dirichlet conditions. The rigidity of the steady states due to nonlocal dispersion is of particular interest. The results also provide criteria for the subsistence of a species subject to nonlocal diffusion in a fragmented area. Furthermore, the paper presents further effects, such as the continuity of the principal eigenvalue with respect to the distance between two compact patches in the one-dimensional case.
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Elio Marconi
Summary: We study weak solutions to the scalar conservation law with finite entropy production. Under suitable nonlinearity assumption, we prove that the set of non Lebesgue points of the solution has Hausdorff dimension at most d. We introduce the notion of Lagrangian representation for this class of solutions, which provides a new interpretation of the entropy dissipation measure.
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
(2022)
Article
Mathematics
Sayani Bera, Ratna Pal, Kaushal Verma
Summary: This paper studies the sub-level sets of Henon maps and their associated holomorphic automorphism groups. It is found that although the sub-level sets have flat boundaries and can be exhausted by biholomorphic images of the unit ball, their automorphism groups are not too large.
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
(2023)
Article
Mathematics
R. McOwen, P. Topalov
Summary: This study focuses on perfect fluid flows in weighted Sobolev spaces, showing that under sufficient spatial decay of the initial velocity, the fluid velocity can develop non-vanishing asymptotic terms expansion at infinity analytically dependent on time and initial data.
MATHEMATISCHE ANNALEN
(2022)
Article
Mathematics
Yaohui Xue, Rencai Lu
Summary: In this paper, we classify the simple bounded weight modules of the Lie algebra of vector fields on C-n. The classification result shows that any simple bounded weight module is isomorphic to the simple quotient of a tensor module F(P, M), where P is a simple weight module over the Weyl algebra and M is a finite-dimensional simple gl(n) (C) module.
ISRAEL JOURNAL OF MATHEMATICS
(2023)
Article
Mathematics, Applied
Zujin Zhang, Yali Zhang
Summary: It is proven that the solution to the Navier-Stokes system is smooth under specific conditions regarding the Sobolev spaces L-p and L-q, which improves upon previous research findings.
ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK
(2021)
Article
Mathematics
Thapakorn Pantarak, Yanisa Chaiya
Summary: In this article, we investigate the properties of the partial transformation semigroup PT(X,Y) and provide the necessary and sufficient conditions for its elements to be regular, left regular, and right regular. We also describe the relationships between these elements and determine their number in the case of a finite set X. Additionally, we demonstrate that PT(X,Y) is always abundant.
Article
Computer Science, Artificial Intelligence
Ganesh Sundaramoorthi, Anthony Yezzi, Minas Benyamin
Summary: This research considers the optimization of cost functionals on the infinite dimensional manifold of diffeomorphisms. It presents a new class of optimization methods that generalize Nesterov accelerated optimization to the manifold of diffeomorphisms. By establishing connections with simple mechanical principles from fluid mechanics, it derives surprisingly simple continuum evolution equations for accelerated gradient descent. The approach has natural connections to the optimal mass transport problem.
SIAM JOURNAL ON IMAGING SCIENCES
(2022)
Article
Mathematics
Dongsheng Li, Xuemei Li, Kai Zhang
Summary: A new method is introduced in this paper to investigate boundary W-2,W-p estimates for elliptic equations, which derives boundary W(2,p) estimates from interior W-2,W-p estimates using Whitney decomposition. This method is used to obtain W-2,W-p estimates on C-1,C-alpha domains for nondivergence form linear elliptic equations, and also considers fully nonlinear elliptic equations.
MATHEMATISCHE ANNALEN
(2023)
Article
Mathematics
Laurent Meersseman, Marcel Nicolau, Javier Ribon
Summary: The structure of diffeomorphisms preserving a foliation is studied, and an example of a C-infinity foliation is given whose diffeomorphism group does not have a natural Lie group structure. On the positive side, it is proven that the automorphism group of a transversely holomorphic foliation or a Riemannian foliation is a strong ILH Lie group in the sense of Omori. The relationship between these considerations and deformation problems in foliation theory is also investigated.
MATHEMATISCHE ZEITSCHRIFT
(2022)
Article
Mathematics, Applied
Jiri Neustupa, Minsuk Yang
Summary: This paper investigates the pressure and regularity of weak solutions to the MHD equations, showing that pressure can always be assigned to a weak solution under certain conditions. It also provides integrability conditions and regularity criteria for the pressure function, as well as remarks on similar results for different types of boundary conditions.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2021)
Article
Mathematics
Juan Alonso, Joaquin Brum, Alejandro Passeggi
Summary: In this study, we establish the relationship between the genus and possible geometries for homological rotation sets of maps on closed oriented surfaces with a genus g >= 2. We demonstrate that this invariant for Smale diffeomorphisms can be described as the union of at most 2(5g-3) convex sets, all containing zero. By utilizing the theory of hyperbolic dynamics, we extend this bound to a C-0-open and dense set of homeomorphisms, indicating its general validity. Additionally, we provide examples that illustrate the sharpness of this asymptotic order.
JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES
(2023)