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Convergence of the one-dimensional Cahn-Hilliard equation
SIAM Journal on Mathematical Analysis. 2012. Vol. 44. No. 5. P. 3458–3480.
We consider the Cahn-Hilliard equation in one space dimension with scaling parameter epsilon, i.e., u(t) = (W'(u) - epsilon(2)u(xx))(xx), where W is a nonconvex potential. In the limit epsilon down arrow 0, under the assumption that the initial data are energetically well prepared, we show the convergence to a Stefan problem. The proof is based on variational methods and exploits the gradient flow structure of the Cahn-Hilliard equation.
Levashev V., / Series arXiv "math". 2026. No. 2609.06010.
We prove that continuous A-bilinear pairings on the ring of Laurent series that are invariant under continuous automorphisms coincide, up to a constant, with the pairing given by the residue of a differential form over any commutative associative ring with identity. ...
Added: September 24, 2026
Pyatov P. N., Pivovarov P. A., / Series math "arxiv.org". 2026. No. 2609.06274.
We investigate a special ansats that allows for an iterative solution of the constant Yang-Baxter equation. Testing this ansatz, we construct four sequences of the constant R-matrices. In each sequence the R-matrices act on the tensor squares of vector spaces of linearly growing dimensions. Each R-matrix also depends on a single complex parameter.
By analyzing the ...
Added: September 24, 2026
Medvedev V., / Series arXiv "math". 2026.
We study complete static manifolds with boundary admitting a nowhere-vanishing static potential. Our main result shows that, under a natural lower bound relating the scalar curvature and the boundary mean curvature, a simple static manifold with boundary must in fact have positive scalar curvature, negative boundary mean curvature, and be compact; we also obtain explicit ...
Added: September 19, 2026
Poddiakov A., / Series Social Science Research Network "Social Science Research Network". 2026. No. 7437658.
Clarity of knowledge and reasoning is necessary in many cases. Yet vagueness in scientific thinking related to surprise, curiosity, "ability to engage with not-knowing" (de Freitas) and abductive reasoning is also a crucially important source of scientific creativity which supplements combinatorial logic when dealing with the already known. Starting from studies by C. S. Peirce ...
Added: September 15, 2026
Glutsyuk A., / Series arXiv "math". 2026.
B.Josephson (Nobel Prize, 1973) predicted a tunnelling effect for a system of two superconductors separated by a narrow dielectric (such a system is called Josephson junction): existence of a supercurrent through it and equations governing it. The overdamped Josephson junction is modeled by the family of differential equations on the 2-torus, dθdτ=1ω(cosθ+B+Acosτ), which is known as ...
Added: September 8, 2026
Pochinka O., Shmukler V., / Series math.RT "arXiv:1808.06395 [math.RT]". 2026.
Anosov flows have a long and rich history, firstly motivated by the
study of geodesic flows in negative curvature surface by Anosov and Sinai.
Not every closed manifold admits an Anosov flow for well-known reasons:
the fundamental group of a 3-manifold M admitting an Anosov flow must
have exponential growth, and M must be universally covered by R3. Nevertheless,
there ...
Added: August 31, 2026
Loubenets E. R., / Series arxiv.org "quant-ph". 2026. No. 2607.18050.
In many quantum applications it is important to know whether or not a Bell nonlocal two-qudit state exhibits its nonlocality under correlation scenarios with some given numbers S1,S2≥1 of generalized quantum measurements at two sites. In the present article, we find analytically a new general locality condition sufficient for a nonseparable Werner state with a ...
Added: July 21, 2026
Bolbachan V., / Series math "arxiv.org". 2024.
Chow polylogarithms are some special functions arising in explicit description of the Beilinson regulator map. The most interesting functional equation for this function reflects its vanishing on the boundary in the Bloch's cycle complex. We show that this functional equation formally follows from more simple ones, namely skew-symmetry, functoriality and multiplicativity.
To prove this, we study ...
Added: July 16, 2026
Bolbachan V., / Series math "arxiv.org". 2024.
Let K be a field of characteristic zero. We prove that its motivic cohomology in degree m−1 and weight m is rationally isomorphic to the cohomology of the polylogarithmic complex. This gives a partial extension of A. Suslin theorem describing the indecomposable K3 of a field. ...
Added: July 16, 2026
Panov V., Ryabchenko A., / Series arXiv "stat.ME". 2026. No. 2607.05048.
This paper investigates the problem of statistical inference for a mixture distribution consisting of a discrete and a continuous component, with a particular focus on the class of rational-infinitely divisible distributions. We consider non-parametric estimation of both components of the mixture as well as the quasi-L{é}vy measure, assuming that the mixture belongs to the class ...
Added: July 9, 2026
Konakov V., Kucher D., Mammen E., / Series arXiv "math". 2026. No. 2606.11142v1.
In this paper, we construct strong approximations for discrete-time Markov chains weakly converging to continuous diffusion processes, as well as for their perturbed counterparts. Under the assumption of bounded coefficients, we construct closely coupled versions of these processes on a shared probability space. In particular, for both non-degenerate and degenerate cases, we maximize the probability ...
Added: June 11, 2026
Dorovskiy A., / Series arXiv "math". 2026.
In this paper the structural stability of generic families of vector fields of the PC-HC class on the two-dimensional sphere is proved. A classification of these families up to moderate equivalence in neighborhoods of their large bifurcation supports is presented, based on such invariants as the configuration and the characteristic set. The realization lemma is proved. ...
Added: May 14, 2026
Taletskii D., / Series arXiv "math". 2026.
A vertex subset of a graph is called a \textit{distance-$k$ independent set} if the distance between any two of its distinct vertices is at least $k + 1$. For all $n,k \geq 1$, we determine the minimum possible number of inclusion-wise maximal distance-$k$ independent sets among all $n$-vertex trees. It equals~$n$ if $n \leq k ...
Added: May 1, 2026
Ovcharenko M., / Series arXiv "math". 2026.
We introduce an explicit class of tempered Laurent polynomials in the sense of Villegas and Doran--Kerr in n⩽4 variables including all Landau--Ginzburg models for smooth Fano threefolds with very ample anticanonical class. We check that it contains Landau--Ginzburg models for various Fano fourfolds which are complete intersections in smooth toric varieties and Grassmannians of planes, ...
Added: April 30, 2026
Zlotnik Alexander, / Series arXiv "math". 2026. No. 2602.03481v1.
We deal with the global in time weak solutions to the 1D compressible Navier-Stokes system of equations for large discontinuous initial data and nonhomogeneous boundary conditions of three standard types. We prove the Lipschitz-type continuous dependence of the solution $(\eta,u,\theta)$, in a norm slightly stronger than $L^{2,\infty}(Q)\times L^2(Q)\times L^2(Q)$, on the initial data $(\eta^0,u^0,e^0)$ in a ...
Added: April 18, 2026
Zlotnik A., Lomonosov T., Математическое моделирование 2021 Т. 33 № 5 С. 16–34
We study explicit two-level finite-difference schemes on staggered meshes for two known regularizations of 1D barotropic gas dynamics equations including schemes with discretizations in x that possess the dissipativity property with respect to the total energy. We derive criterions of L^2-dissipativity in the Cauchy problem for their linearizations at a constant solution with zero background ...
Added: April 1, 2021
Balashov V., Zlotnik A., Journal of Scientific Computing 2021 Vol. 86 Article 33
We construct a new spatial finite-difference discretization for a regularized 3D Navier-Stokes-Cahn-Hilliard system of equations. The system {can be attributed to phase field type models} and describes flows of a viscous
compressible isothermal two-component two-phase fluid with surface effects; the potential body force is also taken into account. In the discretization, the main sought functions are defined on one ...
Added: December 20, 2020
Balashov V., Zlotnik A., Journal of Computational Dynamics 2020 Vol. 7 No. 2 P. 291–312
We consider the initial-boundary value problem for the 3D regularized compressible isothermal Navier-Stokes-Cahn-Hilliard equations describing flows of a two-component two-phase mixture taking into account capillary effects. We construct a new numerical semi-discrete finite-difference method using staggered meshes for the main unknown functions. The method allows one to improve qualitatively the computational flow dynamics by eliminating the so-called parasitic currents and ...
Added: April 28, 2020
Balashov V., Zlotnik A., Mathematical Modelling and Analysis 2020 Vol. 25 No. 1 P. 110–129
We consider the regularized 3D Navier-Stokes-Cahn-Hilliard equations describing isothermal flows of viscous compressible two-component fluids with interphase effects. We construct for them a new energy dissipative finite-difference discretization in space, i.e., with the non-increasing total energy in time. This property is preserved in the absence of a regularization. In addition, the discretization is well-balanced for equilibrium flows and the ...
Added: October 27, 2019
Balashov V., Savenkov E., Zlotnik A., Russian Journal on Numerical Analysis and Mathematical Modelling 2019 Vol. 34 No. 1 P. 1–13
A numerical algorithm is proposed for modeling two-component viscous compressible isothermal flows with surface effects in 3D domains of complex shape with a voxel representation of geometry. The regularized system of Navier-Stokes-Kahn-Hillard equations is used as a basic mathematical model. Modeling of the drop spreading on a flat substrate and the displacement of one liquid by ...
Added: September 28, 2018
Balashov V., Zlotnik Alexander, Savenkov E., Russian Journal on Numerical Analysis and Mathematical Modelling 2017 Vol. 32 No. 6 P. 347–358
A study of a regularized quasi-hydrodynamic system of equations for the two-component isothermal mixture with the diffuse interface is accomplished. Under general assumptions on the Helmholtz free energy of the mixture, the energy balance equation with non-positive energy production and its corollary, the law of non-increasing total energy, are derived. The necessary and sufficient conditions ...
Added: February 26, 2017