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Mathematics
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Asymptotics for Differential Equations
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Asymptotics for Differential Equations

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "E4: Mathematical Physics".

Deadline for manuscript submissions: closed (30 June 2022) | Viewed by 11671

Special Issue Editor

Prof. Dr. Denis Borisov

E-Mail Website
Guest Editor
Department of Differential Equations Institute of Mathematics, Ufa Scientific Center, Russian Academy of Sciences, Ufa, Russia
Interests: mathematical physics; spectral theory; theory of waveguides; asymptotic analysis; PT-symmetric equatiosn; homogenization theory; theory of singular perturbations
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Differential equations play a more than very important role not only in mathematics but also in many natural sciences. Very often it is important to know how the solutions of differential equations behave with respect to some parameter and many interesting phenomena can be hidden in such behavior. As an example we can mention various perturbations of differential equations generating plenty of interesting and intriguing scenario for the behavior of the solution, studying of asymptotics of spectral characteristics for various differential operators, stability and bifurcations in dynamical systems, homogenization of boundary value problems and many others.

This issue is aimed on collecting papers with very recent results on asymptotic properties of solutions for various differential equations in modern hot topics in mathematics and mathematical physics as well as in other natural sciences.

Prof. Dr. Denis Borisov
Guest Editor

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Keywords

  • Differential equation
  • Differential operator
  • Asymptotics
  • Bifurcation
  • Dynamical system

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Published Papers (5 papers)

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Research

15 pages, 316 KiB  
Article
Heat Kernels Estimates for Hermitian Line Bundles on Manifolds of Bounded Geometry
by Yuri A. Kordyukov
Mathematics 2021, 9(23), 3060; https://doi.org/10.3390/math9233060 - 28 Nov 2021
Cited by 1 | Viewed by 1837
Abstract
We consider a family of semiclassically scaled second-order elliptic differential operators on high tensor powers of a Hermitian line bundle (possibly, twisted by an auxiliary Hermitian vector bundle of arbitrary rank) on a Riemannian manifold of bounded geometry. We establish an off-diagonal Gaussian [...] Read more.
We consider a family of semiclassically scaled second-order elliptic differential operators on high tensor powers of a Hermitian line bundle (possibly, twisted by an auxiliary Hermitian vector bundle of arbitrary rank) on a Riemannian manifold of bounded geometry. We establish an off-diagonal Gaussian upper bound for the associated heat kernel. The proof is based on some tools from the theory of operator semigroups in a Hilbert space, results on Sobolev spaces adapted to the current setting, and weighted estimates with appropriate exponential weights. Full article
(This article belongs to the Special Issue Asymptotics for Differential Equations)
11 pages, 1677 KiB  
Article
The Meyers Estimates for Domains Perforated along the Boundary
by Gregory A. Chechkin
Mathematics 2021, 9(23), 3015; https://doi.org/10.3390/math9233015 - 24 Nov 2021
Cited by 11 | Viewed by 1679
Abstract
In this paper, we consider an elliptic problem in a domain perforated along the boundary. By setting a homogeneous Dirichlet condition on the boundary of the cavities and a homogeneous Neumann condition on the outer boundary of the domain, we prove higher integrability [...] Read more.
In this paper, we consider an elliptic problem in a domain perforated along the boundary. By setting a homogeneous Dirichlet condition on the boundary of the cavities and a homogeneous Neumann condition on the outer boundary of the domain, we prove higher integrability of the gradient of the solution to the problem. Full article
(This article belongs to the Special Issue Asymptotics for Differential Equations)
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21 pages, 1906 KiB  
Article
Inviscid Modes within the Boundary-Layer Flow of a Rotating Disk with Wall Suction and in an External Free-Stream
by Bashar Al Saeedi and Zahir Hussain
Mathematics 2021, 9(22), 2967; https://doi.org/10.3390/math9222967 - 21 Nov 2021
Cited by 1 | Viewed by 3716
Abstract
The purpose of this paper is to investigate the linear stability analysis for the laminar-turbulent transition region of the high-Reynolds-number instabilities for the boundary layer flow on a rotating disk. This investigation considers axial flow along the surface-normal direction, by studying analytical expressions [...] Read more.
The purpose of this paper is to investigate the linear stability analysis for the laminar-turbulent transition region of the high-Reynolds-number instabilities for the boundary layer flow on a rotating disk. This investigation considers axial flow along the surface-normal direction, by studying analytical expressions for the steady solution, laminar, incompressible and inviscid fluid of the boundary layer flow due to a rotating disk in the presence of a uniform injection and suction. Essentially, the physical problem represents flow entrainment into the boundary layer from the axial flow, which is transferred by the spinning disk surface into flow in the azimuthal and radial directions. In addition, through the formation of spiral vortices, the boundary layer instability is visualised which develops along the surface in spiral nature. To this end, this study illustrates that combining axial flow and suction together may act to stabilize the boundary layer flow for inviscid modes. Full article
(This article belongs to the Special Issue Asymptotics for Differential Equations)
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21 pages, 514 KiB  
Article
Periodic Solutions in Slowly Varying Discontinuous Differential Equations: The Generic Case
by Flaviano Battelli and Michal Fečkan
Mathematics 2021, 9(19), 2449; https://doi.org/10.3390/math9192449 - 2 Oct 2021
Cited by 3 | Viewed by 1553
Abstract
We study persistence of periodic solutions of perturbed slowly varying discontinuous differential equations assuming that the unperturbed (frozen) equation has a non singular periodic solution. The results of this paper are motivated by a result of Holmes and Wiggins where the authors considered [...] Read more.
We study persistence of periodic solutions of perturbed slowly varying discontinuous differential equations assuming that the unperturbed (frozen) equation has a non singular periodic solution. The results of this paper are motivated by a result of Holmes and Wiggins where the authors considered a two dimensional Hamiltonian family of smooth systems depending on a scalar variable which is the solution of a singularly perturbed equation. Full article
(This article belongs to the Special Issue Asymptotics for Differential Equations)
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24 pages, 990 KiB  
Article
Spectra of Elliptic Operators on Quantum Graphs with Small Edges
by Denis I. Borisov
Mathematics 2021, 9(16), 1874; https://doi.org/10.3390/math9161874 - 6 Aug 2021
Cited by 6 | Viewed by 1855
Abstract
We consider a general second order self-adjoint elliptic operator on an arbitrary metric graph, to which a small graph is glued. This small graph is obtained via rescaling a given fixed graph γ by a small positive parameter ε. The coefficients in [...] Read more.
We consider a general second order self-adjoint elliptic operator on an arbitrary metric graph, to which a small graph is glued. This small graph is obtained via rescaling a given fixed graph γ by a small positive parameter ε. The coefficients in the differential expression are varying, and they, as well as the matrices in the boundary conditions, can also depend on ε and we assume that this dependence is analytic. We introduce a special operator on a certain extension of the graph γ and assume that this operator has no embedded eigenvalues at the threshold of its essential spectrum. It is known that under such assumption the perturbed operator converges to a certain limiting operator. Our main results establish the convergence of the spectrum of the perturbed operator to that of the limiting operator. The convergence of the spectral projectors is proved as well. We show that the eigenvalues of the perturbed operator converging to limiting discrete eigenvalues are analytic in ε and the same is true for the associated perturbed eigenfunctions. We provide an effective recurrent algorithm for determining all coefficients in the Taylor series for the perturbed eigenvalues and eigenfunctions. Full article
(This article belongs to the Special Issue Asymptotics for Differential Equations)
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