LEADER 01963nam 2200385 n 450 001 996391798303316 005 20200824121808.0 035 $a(CKB)4940000000108105 035 $a(EEBO)2240960247 035 $a(UnM)99862804e 035 $a(UnM)99862804 035 $a(EXLCZ)994940000000108105 100 $a19930222d1648 uh | 101 0 $aeng 135 $aurbn||||a|bb| 200 10$aHis Majesties declaration to the Army concerning their seizing and securing of his royall person, and his resolution thereupon$b[electronic resource] $eTo be communicated to the L: Gen: Fairfax, Lieu. Gen. Crumwell, and all other collonels, lieut. collonels, majors, and captains, throughout the Army; to be published at the head of each regiment: and dispersed throughout the cities of London, York, Lincoln, and all other cities, corporations, market towns within the kingdom of England; and dominion of Wales: likewise, his message to both Houses of Parliament, and his resolution thereupon. Signed, Charles Rex 210 $aLondon $cPrinted for R. Rishton$d1648 215 $a[5], 2-3, [1] p 300 $aThe third page is numbered 1. 300 $aWith letters from the Convention of Estates of Scotland and the Parliamentary Army. 300 $aAnnotation on Thomason copy: "Nou: 21". 300 $aReproduction of the original in the British Library. 330 $aeebo-0018 607 $aGreat Britain$xHistory$yCivil War, 1642-1649$vEarly works to 1800 701 $aCharles$cKing of England,$f1600-1649.$0793295 712 02$aScotland.$bConvention of Estates. 712 02$aEngland and Wales.$bArmy. Miscellaneous Documents. 1648. 801 0$bCu-RivES 801 1$bCu-RivES 801 2$bCStRLIN 801 2$bWaOLN 906 $aBOOK 912 $a996391798303316 996 $aHis Majesties declaration to the Army concerning their seizing and securing of his royall person, and his resolution thereupon$92310042 997 $aUNISA LEADER 07277nam 22006615 450 001 9910739446303321 005 20251008151953.0 010 $a3-031-29670-2 024 7 $a10.1007/978-3-031-29670-3 035 $a(MiAaPQ)EBC30684875 035 $a(Au-PeEL)EBL30684875 035 $a(DE-He213)978-3-031-29670-3 035 $a(PPN)272259381 035 $a(CKB)27973052800041 035 $a(EXLCZ)9927973052800041 100 $a20230811d2023 u| 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aPseudo-Monotone Operator Theory for Unsteady Problems with Variable Exponents /$fby Alex Kaltenbach 205 $a1st ed. 2023. 210 1$aCham :$cSpringer International Publishing :$cImprint: Springer,$d2023. 215 $a1 online resource (364 pages) 225 1 $aLecture Notes in Mathematics,$x1617-9692 ;$v2329 311 08$aPrint version: Kaltenbach, Alex Pseudo-Monotone Operator Theory for Unsteady Problems with Variable Exponents Cham : Springer International Publishing AG,c2023 9783031296697 320 $aIncludes bibliographical references. 327 $aIntro -- Preface -- Contents -- Notation-Related Comments -- 1 Introduction -- 1.1 Fluid Mechanics and Pseudo-Monotone Operator Theory: A Never-Ending Love Affair or Merely an On-Off Relationship? -- 1.2 Electro-Rheological Fluids -- 1.3 Aims and Outline of This Manuscript -- 1.3.1 Main Part -- 1.3.2 Extensions -- 2 Preliminaries -- 2.1 Theory of Pseudo-Monotone Operators -- 2.2 Variable Exponent Spaces -- 2.2.1 Classical Function Spaces -- 2.2.2 Variable Lebesgue Spaces -- 2.2.3 Duality in Variable Lebesgue Spaces -- 2.2.4 Variable Sobolev Spaces -- 2.2.5 The Hardy-Littlewood Maximal Operator and log-Hölder Continuity -- 2.2.6 Mollification in Lp(·)(Rn) -- 2.3 Banach-Valued Function Spaces -- 2.3.1 Banach-Valued Classical Function Spaces -- 2.3.2 Bochner-Lebesgue Spaces -- 2.3.3 Bochner-Sobolev Spaces -- 2.3.4 Advanced Theory of Pseudo-Monotone Operators for Evolution Equations -- Part I Main Part -- 3 Variable Bochner-Lebesgue Spaces -- 3.1 The Spaces Xq,p(QT) and Xq,p(QT) -- 3.2 Duality in Xq,p(QT) -- 3.3 Embedding Theorems for Xq,p(QT) -- 3.4 Smoothing in Xq,p(QT) -- 3.5 The Spaces q,p(QT) and q,p(QT) -- 3.6 Generalized Time Derivative in q,p(QT)* -- 3.7 Formula of Integration-by-Parts for Wq,p(QT) -- 3.8 Abstract Existence Result for Lipschitz Domains -- 3.9 Application to Model Problem -- 4 Solenoidal Variable Bochner-Lebesgue Spaces -- 4.1 The Spaces Vq,p(QT) and Vq,p(QT) -- 4.2 Duality in Vq,p(QT) -- 4.3 Smoothing in Vq,p(QT) -- 4.3.1 Failure of Smoothing via Bogovski? Correction -- 4.3.2 Smoothing via Transversal Expansion of LipschitzDomains -- 4.4 The Spaces q,p(QT) and q,p(QT) -- 4.5 Generalized Time Derivative in q,p(QT)* -- 4.6 Formula of Integration-by-Parts for Wq,p,?(QT) -- 5 Existence Theory for Lipschitz Domains -- 5.1 Bochner Pseudo-Monotonicity, Bochner Condition (M) and Bochner Coercivity -- 5.2 The Hirano-Landes Approach. 327 $a5.3 C0-Bochner Pseudo-Monotonicity, C0-Bochner Condition (M) and C0-Bochner Coercivity -- 5.4 Abstract Existence Theorem for Lipschitz Domains and p-?2 -- 5.5 Application to Model Problems -- 5.5.1 Unsteady p(·,·)-Stokes Equations in a Lipschitz Domain with p-?2 -- 5.5.2 Unsteady p(·,·)-Navier-Stokes Equations in a Lipschitz Domain with p- ?3d+2d+2 -- Part II Extensions -- 6 Pressure Reconstruction -- 6.1 Pressure Reconstruction -- 6.2 Application to Model Problems -- 6.3 Applicability of Parabolic L?- and Lipschitz Truncation -- 6.3.1 Parabolic L?- and Lipschitz Truncation -- 6.3.2 Parabolic Solenoidal Lipschitz Truncation -- 7 Existence Theory for Irregular Domains -- 7.1 Bochner-Sobolev Condition (M) -- 7.2 L1-Monotonicity -- 7.2.1 Finite Radon Measures -- 7.2.2 Minty-Trick Like Argument for L1-Monotone Operators -- 7.3 Anisotropic Variable Exponent Bochner-Lebesgue Spaces -- 7.3.1 The Space Xq,p,s,div(QT) -- 7.3.2 Duality in Xq,p,s,div(QT) -- 7.3.3 Smoothing in Xq,p,s,div(QT) -- 7.3.4 Generalized Time Derivative in Xq,p,s,div(QT)* and Formula of Integration-by-Parts for Wq,p,s,div(QT) -- 7.4 First Parabolic Compensated Compactness Principle -- 7.5 Abstract Existence Result for Irregular Domains and p-?2 -- 7.6 Application to Model Problems -- 7.6.1 Unsteady p(·,·)-Stokes Equations in an Irregular Domain with p-?2 -- 7.6.2 Unsteady p(·,·)-Navier-Stokes Equations in an Irregular Domain with p- > -- 3d+2d+2 -- 8 Existence Theory for p-< -- 2 -- 8.1 Second Parabolic Compensated Compactness Principle -- 8.2 Unsteady p(·,·)-Navier-Stokes Equations in an Irregular Domain with p-> -- 3dd+2 -- 8.3 Unsteady p(·,·)-Stokes Equations in an Irregular Domain with p-> -- 2dd+2 -- 9 Appendix -- 9.1 Point-Wise Poincaré Inequality for the Symmetric Gradient -- 9.2 Generalized Lebesgue Differentiation Theorem. 327 $a9.3 Portemanteau Theorem for Weak-* Convergence -- References. 330 $aThis book provides a comprehensive analysis of the existence of weak solutions of unsteady problems with variable exponents. The central motivation is the weak solvability of the unsteady p(.,.)-Navier?Stokes equations describing the motion of an incompressible electro-rheological fluid. Due to the variable dependence of the power-law index p(.,.) in this system, the classical weak existence analysis based on the pseudo-monotone operator theory in the framework of Bochner?Lebesgue spaces is not applicable. As a substitute for Bochner?Lebesgue spaces, variable Bochner?Lebesgue spaces are introduced and analyzed. In the mathematical framework of this substitute, the theory of pseudo-monotone operators is extended to unsteady problems with variable exponents, leading to the weak solvability of the unsteady p(.,.)-Navier?Stokes equations under general assumptions. Aimed primarily at graduate readers, the book develops the material step-by-step, starting with the basics of PDE theory andnon-linear functional analysis. The concise introductions at the beginning of each chapter, together with illustrative examples, graphics, detailed derivations of all results and a short summary of the functional analytic prerequisites, will ease newcomers into the subject. 410 0$aLecture Notes in Mathematics,$x1617-9692 ;$v2329 606 $aFunctional analysis 606 $aFluid mechanics 606 $aOperator theory 606 $aDifferential equations 606 $aFunctional Analysis 606 $aEngineering Fluid Dynamics 606 $aOperator Theory 606 $aDifferential Equations 615 0$aFunctional analysis. 615 0$aFluid mechanics. 615 0$aOperator theory. 615 0$aDifferential equations. 615 14$aFunctional Analysis. 615 24$aEngineering Fluid Dynamics. 615 24$aOperator Theory. 615 24$aDifferential Equations. 676 $a515.353 700 $aKaltenbach$b Alex$01423850 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910739446303321 996 $aPseudo-Monotone Operator Theory for Unsteady Problems with Variable Exponents$93552518 997 $aUNINA