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Embedding and Maximal Regular Differential Operators in Sobolev-Lions Spaces

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Acta Mathematica Sinica,English Series 2006年第5期22卷 1493-1508页

作者： veli B.SHAkHMUROV Istanbul University Engineering Faculty Department of Electric-Electronics EngineeringAvcilar 34850 Istanbul Turkey

This study focuses on vector-valued anisotropic Sobolev-Lions spaces associated with Banach spaces E0, E. Several conditions are found that ensure the continuity and compactness of embedding operators that are optimal regular in these spaces in terms of interpolations of spaces E0 and E. In particular, the most regular class of interpolation spaces Eα between E0, E depending on α and the order of space are found and the boundedness of differential operators D^α from this space to Eα-valued Lp,γ spaces is proved. These results are applied to partial differential-operator equations with parameters to obtain conditions that guarantee the maximal Lp,γ regularity and R-positivity uniformly with respect to these parameters.

关键词： embedding operators Banach-valued function spaces differential operator equations (DOE) maximal regularity operator-valued Fourier multipliers interpolation of Banach spaces

Maximal B-Regular Integro-Differential Equation

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Chinese Annals of Mathematics,Series B 2009年第1期30卷 39-50页

作者： veli SHAkHMUROV Rishad SHAHMUROV Department of Electronics Engineering and Communication Okan University Vocational High School Okan University

By using Fourier multiplier theorems, the maximal B-regularity of ordinary integro-differential operator equations is investigated. It is shown that the corresponding differential operator is positive and satisfies coercive estimate. Moreover, these results are used to establish maximal regularity for infinite systems of integro-differential equations.

关键词： Banach-valued Besov spaces Operator-valued multipliers,Boundary value problems Integro-differential equations

Impact of Zika virus non-structural protein mutations on hippocampal damage

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Neural Regeneration Research 2025年第8期20卷 2307-2308页

作者： Larissa M.g.Cassiano Roney S.Coimbra Neurogenômica/Imunopatologia Instituto RenéRachouFiocruzBelo HorizonteBrazil Departamento de Bioquímica e Imunologia Instituto de Ciências BiológicasUniversidade Federal de Minas GeraisBelo HorizonteBrazil

The Zika virus(ZIkV),a member of the Flaviviridae family,attracted worldwide attention for its connection to severe neurological effects,notably microcephaly in newborns,first reported during the 2015 epidemic in ***,its impact goes beyond fetal and neonatal abnormalities,also affecting the central nervous system(CNS)in both children and adults,leading to enduring cognitive and behavioral impairments.

关键词： damage fetal neonatal

Carleman estimates and unique continuation property for the anisotropic differential-operator equations

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Science China Mathematics 2008年第7期51卷 1215-1231页

作者： veli B SHAkHMUROV Department of Mathematics Okan UniversityAkfirat34959 TuzlaIstanbulTurkey

The unique continuation theorems for the anisotropic partial differential-operator equations with variable coefficients in Banach-valued L p -spaces are studied. To obtain the uniform maximal regularity and the Carleman type estimates for parameter depended differential-operator equations, the sufficient conditions are founded. By using these facts, the unique continuation properties are established. In the application part, the unique continuation properties and Carleman estimates for finite or infinite systems of quasielliptic partial differential equations are studied.

关键词： Carleman estimates unique continuation embedding theorems Banach-valued function spaces differential operator equations maximal L p -regularity operator-valued Fourier multipliers interpolation of Banach spaces 34g10 35J25 35J70

Human endogenous retrovirus type-W and multiple sclerosis–related smoldering neuroinflammation

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Neural Regeneration Research 2025年第3期20卷 813-814页

作者： Joel gruchot Laura Reiche Andrew Chan Robert Hoepner Patrick küry Department of Neurology Medical Faculty and University Hospital DüsseldorfHeinrich Heine University DüsseldorfDüsseldorfGermany Department of Neurology InselspitalBern University Hospital and University of BernBernSwitzerland

Introduction to human endogenous retrovirus type-W(HERV-W): genomic inheritance from the past includes retroviral sequences that have been stably incorporated into our genomes and account for up to 8% of human DNA.

关键词： endogenous inflammation Human

Nucleoside modified mRNA-lipid nanoparticles as a new delivery platform for the repair of the injured spinal cord

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Neural Regeneration Research 2025年第8期20卷 2311-2312页

作者： krisztián Pajer Tamás Bellák Antal Nógrádi Department of Anatomy Histology and EmbryologyAlbert Szent-Györgyi Medical SchoolUniversity of SzegedSzegedHungary

Spinal cord injury and treatment opportunities:The adult mammalian spinal cord has a very limited capacity for spontaneous regeneration due to various intrinsic molecular and cellular *** the spinal cord neurons have the capacity to regenerate their axons,the expression of growth inhibitory factors,lack or suppression of proper guidance cues,and profound inflammatory responses do not permit successful regeneration(khyeam et al.,2021).

关键词： injured capacity modified

ABSTRACT CAPACITY OF REgIONS AND COMPACT EMBEDDINg WITH APPLICATIONS

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Acta Mathematica Scientia 2011年第1期31卷 49-67页

作者： veli Shakhmurov Department of Electronics Engineering and Communication Okan University

The weighted Sobolev-Lions type spaces W pl,γ（Ω; E0, E） = W pl,γ（Ω; E） ∩ Lp,γ （Ω; E0） are studied, where E0, E are two Banach spaces and E0 is continuously and densely embedded on E. A new concept of capacity of region Ω ∈ Rn in W pl,γ（; E0, E） is introduced. Several conditions in terms of capacity of region Ω and interpolations of E0 and E are found such that ensure the continuity and compactness of embedding operators. In particular, the most regular class of interpolation spaces Eα between E0 and E, depending of α and l, are found such that mixed differential operators Dα are bounded and compact from W pl,γ（Ω; E0, E） to Eα-valued Lp,γ spaces. In applications, the maximal regularity for differential-operator equations with parameters are studied.

关键词： Capacity of regions embedding theorems Banach-valued function spaces differential-operator equations Semigroups of operators operator-valued Fourier multipliers interpolation of Banach spaces

Compact embedding in Besov spaces and B-separable elleptic operators

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Science China Mathematics 2010年第4期53卷 1067-1084页

作者： SHAkHMUROV veli B. Department of Electronics Engineering and Communication Okan University

Necessary and sufficient conditions for compactness of sets in Banach space valued Besov class Bp,q s(Ω;E) is *** embedding theorems in Besov-Lions type spaces B l,s p,q(Ω;E0,E) are studied,where E0,E are two Banach spaces and E 0  *** most regular class of interpolation space E α,between E 0 and E are found such that the mixed differential operator D α is bounded and compact from B p,q l,s (Ω;E 0,E) to B p,q s (Ω;E α) and Ehrling-Nirenberg-gagliardo type sharp estimates *** using these results the separability of differential operators with variable coefficients and the maximal B-regularity of parabolic Cauchy problem are *** applications,the infinite systems of the elliptic partial differential equations and parabolic Cauchy problems are studied.

关键词： embedding theorems Banach-valued function spaces differential-operator equations maximal B-regularity operator-valued Fourier multipliers interpolation of Banach spaces abstract parabolic Cauchy problem

Regression analysis of squeezing-induced hybrid nanofluid flow in Darcy-Forchheimer porous medium

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Applied Mathematics and Mechanics(English Edition) 2025年第1期46卷 193-208页

作者： k.MUHAMMAD M.SARFRAZ Department of Humanities and Sciences School of Electrical Engineering and Computer Science National University of Sciences and Technology Department of Mathematics Quaid-i-Azam University

This article presents a mathematical model addressing a scenario involving a hybrid nanofluid flow between two infinite parallel plates. One plate remains stationary, while the other moves downward at a squeezing velocity. The space between these plates contains a Darcy-Forchheimer porous medium. A mixture of water-based fluid with gold(Au) and silicon dioxide(Si O2) nanoparticles is formulated. In contrast to the conventional Fourier's heat flux equation, this study employs the Cattaneo-Christov heat flux equation. A uniform magnetic field is applied perpendicular to the flow direction, invoking magnetohydrodynamic(MHD) effects. Further, the model accounts for Joule heating, which is the heat generated when an electric current passes through the fluid. The problem is solved via NDSolve in MATHEMATICA. Numerical and statistical analyses are conducted to provide insights into the behavior of the nanomaterials between the parallel plates with respect to the flow, energy transport, and skin *** findings of this study have potential applications in enhancing cooling systems and optimizing thermal management strategies. It is observed that the squeezing motion generates additional pressure gradients within the fluid, which enhances the flow rate but reduces the frictional drag. Consequently, the fluid is pushed more vigorously between the plates, increasing the flow velocity. As the fluid experiences higher flow rates due to the increased squeezing effect, it spends less time in the region between the plates. The thermal relaxation, however, abruptly changes the temperature, leading to a decrease in the temperature fluctuations.

关键词： Nanofluidics