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卷 85, 编号 1 (2021)
Articles
Kolmogorov widths of intersections of weighted Sobolev classes on an interval with conditions on the zeroth and first derivatives
摘要
In the paper, we obtain order estimates for the Kolmogorov widths of the intersectionof weighted Sobolev classes with conditions on the first and zerothderivatives; the weights have power-law form.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2021;85(1):3-26
3-26
Representation of solutions of the Cauchy problem for a one dimensional Schrödinger equationwith a smooth bounded potential by quasi-Feynman formulae
摘要
We consider the Cauchy problem for a Schrödinger equation whose Hamiltonian is the difference of the operatorof multiplication by the potential and the operator of taking the second derivative. Here the potential is a realdifferentiable function of a real variable such that this function and its derivative are bounded. This equationhas been studied since the advent of quantum mechanics and is still a good model case for variousmethods of solving partial differential equations. We find solutions of the Cauchy problem in the form of quasi-Feynman formulae by using Remizov's theorem. Quasi-Feynman formulae are relatives of Feynmanformulae containing multiple integrals of infinite multiplicity. Their proof is easier than that of Feynman formulae butthey give longer expressions for the solutions. We provide detailed proofs of all theorems and deliberately restrict thespectrum of our results to the domain of classical mathematical analysis and elements of real analysis trying to avoidgeneral methods of functional analysis. As a result, the paper is long but accessible to readers whoare not experts in the field of functional analysis.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2021;85(1):27-65
27-65
Investigation of the weak solubility of the fractional Voigt alpha-model
摘要
This paper is devoted to investigating the weak solubility of the alpha-model for a fractional viscoelastic Voigt medium.The model involves the Voigt rheological relation with a left Riemann–Liouville fractional derivative, whichaccounts for the medium's memory. The memory is considered along the trajectories of fluid particlesdetermined by the velocity field. Since the velocity field is not smooth enough to uniquely determine the trajectoriesfor every initial value, we introduce weak solutions of this problem using regular Lagrangian flows. On the basis ofthe approximation-topological approach to the study of hydrodynamical problems, we prove the existence of weaksolutions of the alpha-model and establish the convergence of solutions of the alpha-model to solutions ofthe original model as the parameter $\alpha$ tends to zero.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2021;85(1):66-97
66-97
Interior estimates for solutions of linear elliptic inequalities
摘要
We study the wedge of solutions of the inequality $A(u) \ge 0$, where $A$ is a linear elliptic operator of order $2m$ acting on functions \linebreak of $n$ variables. We establish interior estimates of the form $\|u; W_p^{2m-1}(\omega)\| \le C(\omega,\Omega) \|u;L(\Omega)\|$ for the elements of this wedge, where $\omega$ is a compact subdomain of $\Omega$, $W_p^{2 m-1}(\omega)$ is the Sobolev space, $p (n-1) < n$, $L(\Omega)$ is the Lebesgue space of integrable functions, and the constant $C(\omega,\Omega)$ is independent of $u$.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2021;85(1):98-117
98-117
On the critical exponent “instantaneous blow-up” versus “local solubility” in the Cauchy problem for a model equation of Sobolev type
摘要
We consider the Cauchy problem for a model partial differential equation of order three with a non-linearity of the form$|\nabla u|^q$. We prove that when $q\in(1,3/2]$ the Cauchy problem in $\mathbb{R}^3$ has no local-in-time weak solution for a large class of initial functions, while when $q>3/2$ there is a local weak solution.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2021;85(1):118-153
118-153
On the standard conjecture for projective compactifications of Neron models of $3$-dimensionalAbelian varieties
摘要
We prove that the Grothendieck standard conjecture of Lefschetz type holdsfor a smooth complex projective $4$-dimensional variety $X$fibred by Abelian varieties (possibly, with degeneracies)over a smooth projective curve if the endomorphism ring $\operatorname{End}_{\overline{\kappa(\eta)}} (X_\eta\otimes_{\kappa(\eta)}\overline{\kappa(\eta)})$ of the genericgeometric fibre is not an orderof an imaginary quadratic field. This conditionholds automatically in the cases when the reduction of the generic scheme fibre $X_\eta$ at someplace of the curve is semistable in the sense of Grothendieck and hasodd toric rank or the generic geometric fibre is not a simple Abelian variety.
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya. 2021;85(1):154-186
154-186