We consider the initial value problem for the Navier–Stokes equations over ${mathbb R}^3 times [0,T]$ with time $T>0$ in the spatially periodic setting. We prove that it induces open injective mappings ${mathcal A}_scolon B^{s}_1 to B^{s-1}_2$ where $B^{s}_1$, $B^{s-1}_2$ are elements from scales of specially constructed function spaces of Bochner–Sobolev type parametrized with the smoothness index $s in mathbb N$. Finally, we prove that a map ${mathcal A}_s$ is surjective if and only if the inverse image ${mathcal A}_s ^{-1}(K)$ of any precompact set $K$ from the range of the map ${mathcal A}_s$ is bounded in the Bochner space $L^{mathfrak s} ([0,T], L^{{mathfrak r}} ({mathbb T}^3))$ with the Ladyzhenskaya–Prodi–Serrin numbers ${mathfrak s}$, ${mathfrak r}$.
{"title":"Inverse image of precompact sets and regular solutions to the Navier–Stokes equations","authors":"Shlapunov A.A., Tarkhanov N.N.","doi":"10.35634/vm220208","DOIUrl":"https://doi.org/10.35634/vm220208","url":null,"abstract":"We consider the initial value problem for the Navier–Stokes equations over ${mathbb R}^3 times [0,T]$ with time $T>0$ in the spatially periodic setting. We prove that it induces open injective mappings ${mathcal A}_scolon B^{s}_1 to B^{s-1}_2$ where $B^{s}_1$, $B^{s-1}_2$ are elements from scales of specially constructed function spaces of Bochner–Sobolev type parametrized with the smoothness index $s in mathbb N$. Finally, we prove that a map ${mathcal A}_s$ is surjective if and only if the inverse image ${mathcal A}_s ^{-1}(K)$ of any precompact set $K$ from the range of the map ${mathcal A}_s$ is bounded in the Bochner space $L^{mathfrak s} ([0,T], L^{{mathfrak r}} ({mathbb T}^3))$ with the Ladyzhenskaya–Prodi–Serrin numbers ${mathfrak s}$, ${mathfrak r}$.","PeriodicalId":43239,"journal":{"name":"Vestnik Udmurtskogo Universiteta-Matematika Mekhanika Kompyuternye Nauki","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74392503","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the present paper, two non-local initial-boundary value problems have been formulated for a partial differential equation of high even order with a Bessel operator in a rectangular domain. The correctness of one of the considered problems has been investigated. To do this, applying the method of separation of variables to the problem under consideration, the spectral problem was obtained for an ordinary differential equation of high even order. The self-adjointness of the last problem was proved, which implies the existence of the system of its eigenfunctions, as well as orthonormality and completeness of this system. Further, the Green's function of the spectral problem was constructed, with the help of which it was equivalently reduced to the Fredholm integral equation of the second kind with symmetrical kernel. Using this integral equation and Mercer's theorem, the uniform convergence of some bilinear series depending on found eigenfunctions has been studied. The order of the Fourier coefficients was established. The solution of the considered problem has been written as the sum of a Fourier series with respect to the system of eigenfunctions of the spectral problem. The uniform convergence of this series and also the series obtained from it by term-by-term differentiation was proved. Using the method of spectral analysis, the uniqueness of the solution of the problem was proved. An estimate for the solution of the problem was obtained, from which its continuous dependence on the given functions follows.
{"title":"On the solvability of nonlocal initial-boundary value problems for a partial differential equation of high even order","authors":"Urinov A.K., Azizov M.S.","doi":"10.35634/vm220206","DOIUrl":"https://doi.org/10.35634/vm220206","url":null,"abstract":"In the present paper, two non-local initial-boundary value problems have been formulated for a partial differential equation of high even order with a Bessel operator in a rectangular domain. The correctness of one of the considered problems has been investigated. To do this, applying the method of separation of variables to the problem under consideration, the spectral problem was obtained for an ordinary differential equation of high even order. The self-adjointness of the last problem was proved, which implies the existence of the system of its eigenfunctions, as well as orthonormality and completeness of this system. Further, the Green's function of the spectral problem was constructed, with the help of which it was equivalently reduced to the Fredholm integral equation of the second kind with symmetrical kernel. Using this integral equation and Mercer's theorem, the uniform convergence of some bilinear series depending on found eigenfunctions has been studied. The order of the Fourier coefficients was established. The solution of the considered problem has been written as the sum of a Fourier series with respect to the system of eigenfunctions of the spectral problem. The uniform convergence of this series and also the series obtained from it by term-by-term differentiation was proved. Using the method of spectral analysis, the uniqueness of the solution of the problem was proved. An estimate for the solution of the problem was obtained, from which its continuous dependence on the given functions follows.","PeriodicalId":43239,"journal":{"name":"Vestnik Udmurtskogo Universiteta-Matematika Mekhanika Kompyuternye Nauki","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80216452","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We investigate the non-resonant evolution of the axial tilt of hypothetical exo-Earth in the gravitational field of a star, planet's satellite (exo-Moon) and outer planet (exo-Jupiter). The exo-Earth is assumed to be rigid, axially symmetric ($A=B$) and almost spherical. We assume the orbits of the both exo-planets to be Keplerian ellipses with focus in the star, the orbit of exo-Moon to be an evolving Keplerian ellipse with slowly changing of ascending node longitude and periapsis argument. Assuming the frequencies of the unperturbed orbital elliptical motion to be of the order of unity, we obtain the canonical averaged equations describing the perturbed oscillations of the exo-Moon spin axis. These equations contain parameters changing slowly over time. Using the smallness of the planets' masses relative to the mass of the star, we have obtained simplified equations of oscillations of the exo-Earth spin axis by the small parameter method. Time integration of simplified equations gives the axial tilt of exo-Moon as a function of time. It is shown that the torques from the exo-Jupiter create a secular, long-period oscillation mode in axial tilt with a frequency equals to frequency of unperturbed spin axis precession of the exo-Earth. The impact of the exo-Moon on the evolution of the exo-Earth spin axis is that short-period harmonics appear in the oscillations of the axial tilt. The frequency of such oscillations is close to the precession frequency of the ascending node longitude of the exo-Moon orbit. We have calculated the evolution of exo-Earth axial tilt for two exo-planetary systems, i.e., for a system similar to the solar system, and for a planetary exo-system 7 Canis Majoris. The effect of destabilization (stabilization) of the exo-Earth tilt oscillations due to the torques exerted by exo-Moon and exo-Jupiter is described.
{"title":"On the exo-planet precession under torqes due to three celestial bodies with the evolution of the satellite's orbit","authors":"Krasil'nikov P.S.","doi":"10.35634/vm220210","DOIUrl":"https://doi.org/10.35634/vm220210","url":null,"abstract":"We investigate the non-resonant evolution of the axial tilt of hypothetical exo-Earth in the gravitational field of a star, planet's satellite (exo-Moon) and outer planet (exo-Jupiter). The exo-Earth is assumed to be rigid, axially symmetric ($A=B$) and almost spherical. We assume the orbits of the both exo-planets to be Keplerian ellipses with focus in the star, the orbit of exo-Moon to be an evolving Keplerian ellipse with slowly changing of ascending node longitude and periapsis argument.\u0000Assuming the frequencies of the unperturbed orbital elliptical motion to be of the order of unity, we obtain the canonical averaged equations describing the perturbed oscillations of the exo-Moon spin axis. These equations contain parameters changing slowly over time. Using the smallness of the planets' masses relative to the mass of the star, we have obtained simplified equations of oscillations of the exo-Earth spin axis by the small parameter method. Time integration of simplified equations gives the axial tilt of exo-Moon as a function of time. It is shown that the torques from the exo-Jupiter create a secular, long-period oscillation mode in axial tilt with a frequency equals to frequency of unperturbed spin axis precession of the exo-Earth. The impact of the exo-Moon on the evolution of the exo-Earth spin axis is that short-period harmonics appear in the oscillations of the axial tilt. The frequency of such oscillations is close to the precession frequency of the ascending node longitude of the exo-Moon orbit.\u0000We have calculated the evolution of exo-Earth axial tilt for two exo-planetary systems, i.e., for a system similar to the solar system, and for a planetary exo-system 7 Canis Majoris. The effect of destabilization (stabilization) of the exo-Earth tilt oscillations due to the torques exerted by exo-Moon and exo-Jupiter is described.","PeriodicalId":43239,"journal":{"name":"Vestnik Udmurtskogo Universiteta-Matematika Mekhanika Kompyuternye Nauki","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76405494","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Vector fields with zero flux through circles of fixed radius on $mathbb{H}^2$","authors":"N. Volchkova, V. Volchkov","doi":"10.35634/vm220101","DOIUrl":"https://doi.org/10.35634/vm220101","url":null,"abstract":"","PeriodicalId":43239,"journal":{"name":"Vestnik Udmurtskogo Universiteta-Matematika Mekhanika Kompyuternye Nauki","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81257566","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On local extension of the group of parallel translations in three-dimensional space","authors":"V. Kyrov","doi":"10.35634/vm220105","DOIUrl":"https://doi.org/10.35634/vm220105","url":null,"abstract":"","PeriodicalId":43239,"journal":{"name":"Vestnik Udmurtskogo Universiteta-Matematika Mekhanika Kompyuternye Nauki","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90072780","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On uniform convergence of approximations of the double layer potential near the boundary of a two-dimensional domain","authors":"D. Ivanov","doi":"10.35634/vm220103","DOIUrl":"https://doi.org/10.35634/vm220103","url":null,"abstract":"","PeriodicalId":43239,"journal":{"name":"Vestnik Udmurtskogo Universiteta-Matematika Mekhanika Kompyuternye Nauki","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74103190","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
x +Ax = f [u](x), x(0) = a ∈ H; x ∈ W = {x ∈ X : x ∈ X}, где u ∈ U — управление, f [u] : C(0, T ;H) → X∗ — вольтерров оператор (W ⊂ C(0, T ;H)), доказана тотально (по множеству допустимых управлений) глобальная разрешимость при условии глобальной разрешимости некоторого функционально-интегрального неравенства в пространстве R. Во многих частных случаях указанное неравенство может быть конкретизировано как задача Коши для обыкновенного дифференциального уравнения. Фактически, развивается аналогичный результат, доказанный автором ранее для случая линейного оператора A и V = H = V ∗. Отдельно рассматриваются случаи компактного вложения пространств, усиления условия монотонности и совпадения тройки пространств V = H = H∗. В последних двух случаях доказывается также единственность решения. В первом случае применяется теорема Шаудера, в остальных — технология продолжения решения по времени (то есть продолжения вдоль вольтерровой цепочки). Приводятся конкретные примеры задания оператора A.
Ax = f (u) (x + x), x (0) = a _arg H;x _arg W = {x _arg x: x _arg x}, u _arg u - f (u)管理:C (0, T; H)→x∗沃尔泰拉算子(W⊂C (0, T; H)),证明全面(许多容许)全球管理解决的全球解决的条件有些泛函积分不等式空间r .许多个案柯西不等式可以充实任务指定为常微分方程。事实上，作者之前在A和V = H = V的情况下证明了类似的结果。分别考虑了紧凑空间投资、强化单调条件和三组V = H = H空间的重叠。在过去的两个案例中，也证明了解决方案的独特性。第一个例子是肖德定理，另一个例子是继续时间解决方案的技术(即沿着伏尔泰链继续)。以下是运营商A任务的具体例子。
{"title":"On totally global solvability of evolutionary equation with monotone nonlinear operator","authors":"A. Chernov","doi":"10.35634/vm220109","DOIUrl":"https://doi.org/10.35634/vm220109","url":null,"abstract":"x +Ax = f [u](x), x(0) = a ∈ H; x ∈ W = {x ∈ X : x ∈ X}, где u ∈ U — управление, f [u] : C(0, T ;H) → X∗ — вольтерров оператор (W ⊂ C(0, T ;H)), доказана тотально (по множеству допустимых управлений) глобальная разрешимость при условии глобальной разрешимости некоторого функционально-интегрального неравенства в пространстве R. Во многих частных случаях указанное неравенство может быть конкретизировано как задача Коши для обыкновенного дифференциального уравнения. Фактически, развивается аналогичный результат, доказанный автором ранее для случая линейного оператора A и V = H = V ∗. Отдельно рассматриваются случаи компактного вложения пространств, усиления условия монотонности и совпадения тройки пространств V = H = H∗. В последних двух случаях доказывается также единственность решения. В первом случае применяется теорема Шаудера, в остальных — технология продолжения решения по времени (то есть продолжения вдоль вольтерровой цепочки). Приводятся конкретные примеры задания оператора A.","PeriodicalId":43239,"journal":{"name":"Vestnik Udmurtskogo Universiteta-Matematika Mekhanika Kompyuternye Nauki","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87363682","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Group pursuit in a problem with fractional derivatives in the class of positional strategies with a guide","authors":"N. Petrov, A. I. Machtakova","doi":"10.35634/vm220107","DOIUrl":"https://doi.org/10.35634/vm220107","url":null,"abstract":"","PeriodicalId":43239,"journal":{"name":"Vestnik Udmurtskogo Universiteta-Matematika Mekhanika Kompyuternye Nauki","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80928346","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Pauli's theorem in Clifford algebras of odd dimension","authors":"S. P. Kuznetsov, V. Mochalov, V. Chuev","doi":"10.35634/vm220104","DOIUrl":"https://doi.org/10.35634/vm220104","url":null,"abstract":"","PeriodicalId":43239,"journal":{"name":"Vestnik Udmurtskogo Universiteta-Matematika Mekhanika Kompyuternye Nauki","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80302136","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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