Singular limits in thermodynamics of viscous fluids
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[2017]
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| Ausgabe: | second edition |
| Schriftenreihe: | Advances in Mathematical Fluid Mechanics
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| 100 | 1 | |a Feireisl, Eduard |d 1957- |0 (DE-588)137457685 |4 aut | |
| 245 | 1 | 0 | |a Singular limits in thermodynamics of viscous fluids |c Eduard Feireisl ; Antonín Novotný |
| 250 | |a second edition | ||
| 264 | 1 | |a [Basel] |b Birkhäuser |c [2017] | |
| 264 | 4 | |c © 2017 | |
| 300 | |a xlii, 524 Seiten |c 23.5 cm x 15.5 cm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Advances in Mathematical Fluid Mechanics | |
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| 653 | |a Magnetohydrodynamics | ||
| 653 | |a Navier-Stokes-Fourier | ||
| 653 | |a Nonlinear Systems | ||
| 653 | |a Partial Differential Equations | ||
| 653 | |a Rhe | ||
| 653 | |a Single Limits | ||
| 653 | |a Thermodynamics | ||
| 653 | |a Viscous Fluids | ||
| 653 | |a fluid dynamics | ||
| 653 | |a fluid mechanics | ||
| 653 | |a partial differential equation | ||
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Datensatz im Suchindex
| _version_ | 1804178412356501504 |
|---|---|
| adam_text | Contents 1 Fluid Flow Modeling.................................................................................. 1.1 1.2 1.3 1.4 2 Fluids in Continuum Mechanics................................................. Balance Laws............................................................................... Field Equations ............................................................................ 1.3.1 Conservation of Mass..................................................... 1.3.2 Balance of Linear Momentum....................................... 1.3.3 Total Energy.................................................................... 1.3.4 Entropy........................................................................... Constitutive Relations.................................................................. 1.4.1 Molecular Energy and Transport Terms...................... 1.4.2 State Equations............................................................... 1.4.3 Effect of Thermal Radiation.......................................... 1.4.4 Typical Values of Some PhysicalCoefficients.............. Weak Solutions, A Priori Estimates........................................................ 2.1 2.2 Weak Formulation....................................................................... 2.1.1 Equation of Continuity.................................................. 2.1.2 Balance of Linear Momentum...................................... 2.1.3 Balance of Total Energy ................................................ 2.1.4 Entropy Production.........................................................
2.1.5 Constitutive Relations..................................................... A Priori Estimates ....................................................................... 2.2.1 Total Mass Conservation................................................ 2.2.2 Energy Estimates............................................................ 2.2.3 Estimates Based on Second Law of Thermodynamics........................................................... 2.2.4 Positivity of the Absolute Temperature........................ 2.2.5 Pressure Estimates ......................................................... 2.2.6 Pressure Estimates, an Alternative Approach............... 1 2 4 8 9 10 12 13 14 14 15 17 18 21 23 24 24 26 26 27 28 28 28 30 36 39 44 xxxvii
xxxviii Contents Existence Theory...................................................................................... 3.1 Hypotheses..................................................................................... 3.2 Structural Properties of Constitutive Functions............................ 3 Main Existence Result................................................................... 3.3.1 Approximation Scheme.................................................. 3.4 Solvability of the Approximate System......................................... 3.4.1 Approximate Continuity Equation ................................. 3.4.2 Approximate Internal Energy Equation..................... 3.4.3 Local Solvability of the Approximate Problem........... 3.4.4 Uniform Estimates and Global Existence.................... 3.5 Faedo-Galerkin Limit............................................................... 3.5.1 Estimates Independent of the Dimension of Faedo-Galerkin Approximations............. 79 3.5.2 Limit Passage in the Approximate Continuity Equation....................................................... 82 3.5.3 Strong Convergence of the Approximate Temperatures and the Limit in the Entropy Equation....................................................... 85 3.5.4 Limit in the Approximate Momentum Equation......... 3.5.5 The Limit System Resulting from the Faedo-Galerkin Approximation....................... 94 3.5.6 The Entropy Production Rate Represented by a Positive Measure....................................... 96 3.6 Artificial Diffusion
Limit............................................................... 3.6.1 Uniform Estimates and Limit in the Approximate Continuity Equation........................................................ 3.6.2 Entropy Balance and Strong Convergence of the Approximate Temperatures................... 3.6.3 Uniform Pressure Estimates........................................... 3.6.4 Limit in the Approximate Momentum Equation and in the Energy Balance............................... 3.6.5 Strong Convergence of the Densities.............................. 3.6.6 Artificial Diffusion Asymptotic Limit............................. 3.7 Vanishing Artificial Pressure.................................................... 3.7.1 Uniform Estimates......................................................... 3.7.2 Asymptotic Limit for Vanishing Artificial Pressure ... 3.7.3 Entropy Balance and Pointwise Convergence of the Temperature.......................................... 125 3.7.4 Pointwise Convergence of the Densities......................... 3.7.5 Oscillations Defect Measure.......................................... 3.8 Regularity Properties of the Weak Solutions............................... 3.3 4 Asymptotic Analysis: An Introduction..................................................... 4.1 4.2 Scaling and Scaled Equations...................................................... Low Mach Number Limits.......................................................... 49 50 53 57 58 60 61 63 71 73 78 93 97 97 100 106 108 109 118 119 120 122 129 135 140 145 147 149
xxxix Contents Strongly Stratified Flows.............................................................. Acoustic Waves............................................................................. 4.4.1 Low Stratification.......................................................... 4.4.2 Strong Stratification....................................................... 4.4.3 Attenuation of Acoustic Waves..................................... Acoustic Analogies....................................................................... Initial Data..................................................................................... A General Approach to Singular Limits for the Full Navier-Stokes-Fourier System ...................................... 161 151 154 154 156 156 158 160 5 Singular Limits: Low Stratification...................................................... 5.1 Hypotheses and Global Existence for the Primitive System .... 5.1.1 Hypotheses.................................................................... 5.1.2 Global-in-Time Solutions............................................. 5.2 Dissipation Equation, Uniform Estimates................................... 5.2.1 Conservation of Total Mass........................................... 5.2.2 Total Dissipation Balance and Related Estimates ....... 5.2.3 Uniform Estimates......................................................... 5.3 Convergence................................................................................... 5.3.1 Equation of Continuity................................................... 5.3.2 Entropy
Balance............................................................. 5.3.3 Momentum Equation...................................................... 5.4 Convergence of the Convective Term.......................................... 5.4.1 Helmholtz Decomposition............................................. 5.4.2 Compactness of the Solenoidal Part.............................. 5.4.3 Acoustic Equation.......................................................... 5.4.4 Formal Analysis of Acoustic Equation......................... 5.4.5 Spectral Analysis of the Wave Operator...................... 5.4.6 Reduction to a Finite Number of Modes...................... 5.4.7 Weak Limit of the Convective Term: Time Lifting .... 5.5 Conclusion: Main Result.............................................................. 5.5.1 Weak Formulation of the Target Problem..................... 5.5.2 Main Result.................................................................... 5.5.3 Determining the Initial Temperature Distribution...... 5.5.4 Energy Inequality for the Limit System....................... 167 170 171 172 174 174 175 179 182 184 185 188 192 193 194 196 199 202 203 205 208 208 210 211 212 6 Stratified Fluids......................................................................................... 6.1 Motivation...................................................................................... 6.2 Primitive System............................................................................ 6.2.1 Field Equations............................................................... 6.2.2 Constitutive
Relations..................................................... 6.2.3 Scaling............................................................................. 6.3 Asymptotic Limit.......................................................................... 6.3.1 Static States..................................................................... 6.3.2 Solutions to the Primitive System................................. 6.3.3 Main Result..................................................................... 221 221 222 222 223 225 227 227 228 230 4.3 4.4 4.5 4.6 4.7
xl Contents 6.4 6.5 6.6 6.7 7 Uniform Estimates........................................................................ 6.4.1 Dissipation Equation, Energy Estimates...................... 6.4.2 Pressure Estimates ......................................................... Convergence Towards the Target System.................................... 6.5.1 Anelastic Constraint........................................................ 6.5.2 Determining the Pressure............................................... 6.5.3 Driving Force................................................................... 6.5.4 Momentum Equation....................................................... Analysis of the Acoustic Waves..................................................... 6.6.1 Acoustic Equation........................................................... 6.6.2 Spectral Analysis of the Wave Operator....................... 6.6.3 Convergence of the Convective Term........................... Asymptotic Limit in the Entropy Balance................................... Interaction of Acoustic Waves with Boundary..................................... 7.1 Problem Formulation..................................................................... 7.1.1 Field Equations............................................................... 7.1.2 Physical Domain and Boundary Conditions............... 7.2 Main Result: The No-Slip Boundary Conditions....................... 7.2.1 Preliminaries: Global Existence................................... 7.2.2 Compactness of the Family of Velocities.................... 7.3 Uniform
Estimates......................................................................... 7.4 Analysis of Acoustic Waves.......................................................... 7.4.1 Acoustic Equation.......................................................... 7.4.2 Spectral Analysis of the Acoustic Operator................. 7.5 Strong Convergence of the Velocity Field................................. 7.5.1 Compactness of the Solenoidal Component................ 7.5.2 Reduction to a Finite Number of Modes...................... 7.5.3 Strong Convergence....................................................... 7.6 Asymptotic Limit on Domains with Oscillatory Boundaries and Complete Slip Boundary Conditions. 297 7.7 Uniform Bounds............................................................................ 7.8 Convergence of the Velocity Trace on Oscillatory Boundary... 7.9 Strong Convergence of the Velocity Field Revisited ................. 7.9.1 Solenoidal Component.................................................. 7.9.2 Acoustic Waves.............................................................. 7.9.3 Strong Convergence of the Gradient Component........ 7.10 Concluding Remarks...................................................................... 8 Problems on Large Domains.................................................................... 8.1 8.2 8.3 Primitive System........................................................................... Oberbeck-Boussinesq Approximation in Exterior Domains ... Uniform
Estimates........................................................................ 8.3.1 Static Solutions.............................................................. 8.3.2 Estimates Based on the Hypothesis of Thermodynamic Stability............................... 319 233 234 240 242 243 243 246 248 249 250 252 254 259 263 266 266 267 269 270 272 273 275 275 278 288 289 290 291 300 301 304 304 305 308 311 313 313 317 318 318
xli Contents Estimates Based on the Specific Form of Constitutive Relations..................................... 322 Convergence, Part I....................................................................... Acoustic Equation ......................................................................... 8.5.1 Boundedness of the Data................................................ 8.5.2 Acoustic Equation Revisited.......................................... Regularization and Extension to Ω ............................................. 8.6.1 Regularization................................................................. 8.6.2 Reduction to Smooth Data............................................. Dispersive Estimates and Time Decay of Acoustic Waves ....... 8.7.1 Compactness of the Solenoidal Components............... 8.7.2 Analysis of Acoustic Waves.......................................... 8.7.3 Decay Estimates via RAGE Theorem........................... Convergence to the Target System............................................... Dispersive Estimates Revisited..................................................... 8.9.1 RAGE Theorem via Spectral Measures........................ 8.9.2 Decay Estimates via Kato’s Theorem........................... Conclusion.................................................................................... 8.3.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 9 Vanishing Dissipation Limits..................................................................... 9.1 9.2 9.3 9.4 9.5 10 Problem Formulation...................................................................
9.1.1 Physical Space and Boundary Conditions.................... 9.1.2 Initial Data........................................................................ 9.1.3 Target Problem................................................................ 9.1.4 Strategy of the Proof of Stability of Smooth Solutions to the TargetProblem..................... 374 Relative Energy Inequality........................................................... Uniform Estimates......................................................................... Well-Prepared Initial Data............................................................ Ill-Prepared Initial Data................................................................. 9.5.1 Acoustic Equation........................................................... 9.5.2 Transport Equation.......................................................... 9.5.3 Stability via the Relative Energy Inequality................. 9.5.4 Conclusion....................................................................... Acoustic Analogies....................................................................................... 10.1 10.2 10.3 10.4 10.5 Asymptotic Analysis and the Limit System................................ Acoustic Equation Revisited........................................................ Two-Scale Convergence............................................................... 10.3.1 Approximate Methods.................................................. Lighthill’s Acoustic Analogy in the Low Mach Number
Regime............................................................................ 422 10.4.1 Ill-Prepared Data........................................................... 10.4.2 Well-Prepared Data ....................................................... Concluding Remarks..................................................................... 325 326 329 333 335 335 338 349 350 350 355 358 361 361 363 367 369 370 371 372 372 374 378 381 389 389 391 392 406 409 410 412 416 421 422 423 426
xiii Contents Appendix ........................................................................................................ 11 Spectral Theory of Self-Adjoint Operators................................. Mollifiers........................................................................................ Basic Properties of Some Elliptic Operators .............................. 11.3.1 A Priori Estimates.......................................................... 11.3.2 Fredholm Alternative.................................................... 11.3.3 Spectrum of a Generalized Laplacian.......................... 11.3.4 Neumann Laplacian on Unbounded Domains............ 11.4 Normal Traces.............................................................................. 11.5 Singular and Weakly Singular Operators.................................. 11.6 The Inverse of the div-Operator (Bogovskii Formula)............ 11.7 Helmholtz Decomposition.......................................................... 11.8 Function Spaces of Hydrodynamics.......................................... 11.9 Poincaré Type Inequalities ......................................................... 11.10 Korn Type Inequalities................................................................. 11.11 Estimating Vu by Means of di vru and curicu............................ 11.12 Weak Convergence and Monotone Functions........................... 11.13 Weak Convergence and Convex Functions............................... 11.14 Div-Curl Lemma.......................................................................... 11.15 Maximal
Regularity for Parabolic Equations............................. 11.16 Quasilinear Parabolic Equations................................................. 11.17 Basic Properties of the Riesz Transform and Related Operators.......................................................................... 486 11.18 Commutators Involving Riesz Operators................................... 11.19 Renormalized Solutions to the Equation of Continuity............ 11.20 Transport Equation and the Euler System .................................. 11.1 11.2 11.3 429 429 432 433 434 437 438 440 443 447 448 458 460 462 464 469 471 476 480 482 484 489 492 499 Fluid Flow Modeling................................................................... Mathematical Theory of Weak Solutions................................... Existence Theory.......................................................................... Analysis of Singular Limits........................................................ Propagation of Acoustic Waves.................................................. Relative Energy, Inviscid Limits................................................. 501 501 502 503 503 504 505 Bibliography.......................................................................................................... 507 Index........................................................................................................................ 519 12 Bibliographical Remarks........................................................................... 12.1 12.2 12.3 12.4 12.5 12.6
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| any_adam_object | 1 |
| author | Feireisl, Eduard 1957- Novotný, Antonín 1959-2021 |
| author_GND | (DE-588)137457685 (DE-588)143304194 |
| author_facet | Feireisl, Eduard 1957- Novotný, Antonín 1959-2021 |
| author_role | aut aut |
| author_sort | Feireisl, Eduard 1957- |
| author_variant | e f ef a n an |
| building | Verbundindex |
| bvnumber | BV044877036 |
| classification_rvk | SK 540 UF 4000 |
| ctrlnum | (OCoLC)1030903869 (DE-599)DNB1135582483 |
| discipline | Physik Mathematik |
| edition | second edition |
| format | Book |
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| id | DE-604.BV044877036 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T08:03:34Z |
| institution | BVB |
| institution_GND | (DE-588)1064344704 |
| isbn | 9783319637808 3319637800 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-030271353 |
| oclc_num | 1030903869 |
| open_access_boolean | |
| owner | DE-29T DE-20 DE-83 DE-355 DE-BY-UBR |
| owner_facet | DE-29T DE-20 DE-83 DE-355 DE-BY-UBR |
| physical | xlii, 524 Seiten 23.5 cm x 15.5 cm |
| publishDate | 2017 |
| publishDateSearch | 2017 |
| publishDateSort | 2017 |
| publisher | Birkhäuser |
| record_format | marc |
| series2 | Advances in Mathematical Fluid Mechanics |
| spelling | Feireisl, Eduard 1957- (DE-588)137457685 aut Singular limits in thermodynamics of viscous fluids Eduard Feireisl ; Antonín Novotný second edition [Basel] Birkhäuser [2017] © 2017 xlii, 524 Seiten 23.5 cm x 15.5 cm txt rdacontent n rdamedia nc rdacarrier Advances in Mathematical Fluid Mechanics Schwache Lösung (DE-588)4131068-8 gnd rswk-swf Viskose Strömung (DE-588)4226965-9 gnd rswk-swf System von partiellen Differentialgleichungen (DE-588)4116672-3 gnd rswk-swf Hydromechanik (DE-588)4026312-5 gnd rswk-swf Kompressible Strömung (DE-588)4032018-2 gnd rswk-swf Mathematik (DE-588)4037944-9 gnd rswk-swf Thermodynamik (DE-588)4059827-5 gnd rswk-swf Wärmeleitung (DE-588)4064192-2 gnd rswk-swf PHD Dissipation Magnetohydrodynamics Navier-Stokes-Fourier Nonlinear Systems Partial Differential Equations Rhe Single Limits Thermodynamics Viscous Fluids fluid dynamics fluid mechanics partial differential equation Viskose Strömung (DE-588)4226965-9 s Kompressible Strömung (DE-588)4032018-2 s Wärmeleitung (DE-588)4064192-2 s System von partiellen Differentialgleichungen (DE-588)4116672-3 s Schwache Lösung (DE-588)4131068-8 s DE-604 Hydromechanik (DE-588)4026312-5 s Thermodynamik (DE-588)4059827-5 s Mathematik (DE-588)4037944-9 s 1\p DE-604 Novotný, Antonín 1959-2021 (DE-588)143304194 aut Springer International Publishing (DE-588)1064344704 pbl Elektronische Reproduktion 9783319637815 Erscheint auch als Online-Ausgabe 9783319637815 Vorangegangen ist 9783764388423 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=5cd9442ad86148fea084f1d679c10291&prov=M&dok_var=1&dok_ext=htm Inhaltstext X:MVB http://www.springer.com/ Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030271353&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Feireisl, Eduard 1957- Novotný, Antonín 1959-2021 Singular limits in thermodynamics of viscous fluids Schwache Lösung (DE-588)4131068-8 gnd Viskose Strömung (DE-588)4226965-9 gnd System von partiellen Differentialgleichungen (DE-588)4116672-3 gnd Hydromechanik (DE-588)4026312-5 gnd Kompressible Strömung (DE-588)4032018-2 gnd Mathematik (DE-588)4037944-9 gnd Thermodynamik (DE-588)4059827-5 gnd Wärmeleitung (DE-588)4064192-2 gnd |
| subject_GND | (DE-588)4131068-8 (DE-588)4226965-9 (DE-588)4116672-3 (DE-588)4026312-5 (DE-588)4032018-2 (DE-588)4037944-9 (DE-588)4059827-5 (DE-588)4064192-2 |
| title | Singular limits in thermodynamics of viscous fluids |
| title_auth | Singular limits in thermodynamics of viscous fluids |
| title_exact_search | Singular limits in thermodynamics of viscous fluids |
| title_full | Singular limits in thermodynamics of viscous fluids Eduard Feireisl ; Antonín Novotný |
| title_fullStr | Singular limits in thermodynamics of viscous fluids Eduard Feireisl ; Antonín Novotný |
| title_full_unstemmed | Singular limits in thermodynamics of viscous fluids Eduard Feireisl ; Antonín Novotný |
| title_short | Singular limits in thermodynamics of viscous fluids |
| title_sort | singular limits in thermodynamics of viscous fluids |
| topic | Schwache Lösung (DE-588)4131068-8 gnd Viskose Strömung (DE-588)4226965-9 gnd System von partiellen Differentialgleichungen (DE-588)4116672-3 gnd Hydromechanik (DE-588)4026312-5 gnd Kompressible Strömung (DE-588)4032018-2 gnd Mathematik (DE-588)4037944-9 gnd Thermodynamik (DE-588)4059827-5 gnd Wärmeleitung (DE-588)4064192-2 gnd |
| topic_facet | Schwache Lösung Viskose Strömung System von partiellen Differentialgleichungen Hydromechanik Kompressible Strömung Mathematik Thermodynamik Wärmeleitung |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=5cd9442ad86148fea084f1d679c10291&prov=M&dok_var=1&dok_ext=htm http://www.springer.com/ http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030271353&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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