Random perturbation of PDEs and fluid dynamic models: Ecole d'Eté de Probabilités de Saint-Flour XL - 2010
Saved in:
| Main Author: | |
|---|---|
| Format: | Conference Proceeding Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2011
|
| Series: | Lecture notes in mathematics
2015 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | IX, 176 S. Ill., graph. Darst. |
| ISBN: | 3642182305 9783642182303 |
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Record in the Search Index
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IMAGE 1
CONTENTS
1 INTRODUCTION TO UNIQUENESS AND BLOW-UP 1
1.1 NON UNIQUENESS IN ORDINARY DIFFERENTIAL EQUATIONS 1
1.2 NON UNIQUENESS IN PARTIAL DIFFERENTIAL EQUATIONS 4
1.2.1 RANDOM PERTURBATIONS 6
1.3 DEFINITIONS OF UNIQUENESS 8
1.3.1 ABSTRACT SCHEME 8
1.3.2 DETERMINISTIC ODES 9
1.3.3 STOCHASTIC DIFFERENTIAL EQUATIONS 10
1.4 ZERO-NOISE AND SELECTION 13
1.5 EXAMPLES OF BLOW-UP 14
2 REGULARIZATION BY ADDITIVE NOISE 17
2.1 REGULARIZATION OF FUNCTIONS BY NOISE: OCCUPATION MEASURE 17 2.1.1
EXAMPLES 17
2.1.2 IS X I - HT,X+W(U) CONTINUOUS IN TOTAL VARIATION? 20 2.1.3 AN
ESTIMATE FOR HOLDER FUNCTIONS 21
2.1.4 AN ESTIMATE FOR L P FUNCTIONS 25
2.1.5 SUMMARY ON OCCUPATION MEASURE 29
2.2 REGULARIZATION OF SDE BY ADDITIVE NOISE 31
2.2.1 MAIN RESULT 31
2.2.2 PROOF OF THEOREM 2.7 34
2.2.3 STOCHASTIC FLOW OF DIFFEOMORPHISMS 37
2.3 INFINITE DIMENSIONAL EQUATIONS WITH ADDITIVE NOISE 40 2.3.1
INTRODUCTION 40
2.3.2 INFINITE DIMENSIONAL SET-UP 42
2.3.3 UNIQUENESS IN LAW 46
2.4 PATHWISE UNIQUENESS 53
2.4.1 FINITE DIMENSIONAL ORNSTEIN-UHLENBECK AND KOLMOGOROV EQUATIONS 56
2.4.2 PROOF OF THEOREM 2.10 62
BIBLIOGRAFISCHE INFORMATIONEN HTTP://D-NB.INFO/1008972584
DIGITALISIERT DURCH
IMAGE 2
VIII CONTENTS
3 DYADIC MODELS 71
3.1 INTRODUCTION: 3D EULER EQUATIONS 71
3.1.1 FOURIER FORMULATION OF EULER EQUATIONS 73
3.2 THE DYADIC MODEL 74
3.3 DETERMINISTIC RESULTS 77
3.3.1 PRELIMINARIES 77
3.3.2 ANOMALOUS ENERGY DISSIPATION 78
3.3.3 NUMERICAL PICTURE OF THE ANOMALOUS DISSIPATION 82 3.3.4 EXAMPLES
OF NON-UNIQUENESS 83
3.3.5 UNIQUENESS OF POSITIVE SOLUTIONS 84
3.3.6 SUMMARY AND OPEN QUESTIONS 86
3.4 RANDOM PERTURBATION 87
3.4.1 PRELIMINARY REMARKS 87
3.4.2 DEFINITIONS AND MAIN RESULTS 89
3.4.3 AUXILIARY LINEAR EQUATION 92
3.4.4 GIRSANOV TRANSFORM 96
3.4.5 PROOF OF THEOREM 3.4 96
3.4.6 PROOF OF THEOREM 3.5 99
4 TRANSPORT EQUATION 101
4.1 INTRODUCTION 101
4.1.1 LINEAR TRANSPORT: STRUCTURAL APPROACH 103
4.2 DETERMINISTIC TRANSPORT EQUATION 104
4.2.1 LIPSCHITZ CASE 104
4.2.2 WEAKLY DIFFERENTIABLE CASE 105
4.3 STOCHASTIC CASE 113
4.3.1 DEFINITIONS AND PRELIMINARIES 113
4.3.2 RENORMALIZED SOLUTIONS 115
4.3.3 MAIN THEOREM 117
4.3.4 REPRESENTATION OF THE LAW OF U (T, X) 121
4.3.5 ANALOGY WITH STABILIZATION BY NOISE 122
4.4 UNIQUENESS BY STOCHASTIC CHARACTERISTICS 123
4.4.1 STOCHASTIC FLOW OF SDE WITH NON REGULAR DRIFT 123 4.4.2 PROOF OF
UNIQUENESS USING THE FLOW: INTRODUCTION 123 4.4.3 DISTRIBUTIONAL
COMMUTATOR LEMMA 125
4.4.4 CONVERGENCE OF THE COMMUTATOR COMPOSED WITH THE FLOW 127
4.4.5 FINAL STATEMENT 130
5 OTHER MODELS: UNIQUENESS AND SINGULARITIES 133
5.1 NEGATIVE EXAMPLES 133
5.1.1 LINEAR TRANSPORT EQUATION WITH RANDOM COEFFICIENTS. 133 5.1.2
EULER EQUATION WITH TOO SIMPLE NOISE 134
5.1.3 INVISCID BURGERS EQUATION: NON UNIQUENESS 136 5.1.4 INVISCID
BURGERS EQUATION: BLOW-UP 137
IMAGE 3
CONTENTS IX
5.2 POSITIVE EXAMPLES 139
5.2.1 STOCHASTIC CONTINUITY EQUATION 139
5.2.2 POINT VORTEX MOTION 143
5.3 SINGULARITIES OF STOCHASTIC SCHROEDINGER EQUATION 149
5.4 ADDITIVE NOISE IN 3D NAVIER-STOKES EQUATIONS 152
5.4.1 ABOUT SINGULARITIES 153
5.4.2 MARKOV SELECTIONS AND STRONG FELLER PROPERTY 154 5.5 CONCLUSIONS
AND OPEN PROBLEMS 155
5.5.1 SPDES: THE EFFECTS OF NOISE 155
5.5.2 WHAT WE HAVE LEARNED ON THE EFFECT OF NOISE ON UNIQUENESS 156
5.5.3 OTHER OPEN QUESTIONS 158
REFERENCES 161 |
| any_adam_object | 1 |
| author | Flandoli, Franco |
| author_GND | (DE-588)135622360 |
| author_facet | Flandoli, Franco |
| author_role | aut |
| author_sort | Flandoli, Franco |
| author_variant | f f ff |
| building | Verbundindex |
| bvnumber | BV037332351 |
| classification_rvk | SI 850 |
| classification_tum | PHY 220f MAT 606f MAT 350f |
| ctrlnum | (OCoLC)712777040 (DE-599)DNB1008972584 |
| dewey-full | 515.353 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.353 |
| dewey-search | 515.353 |
| dewey-sort | 3515.353 |
| dewey-tens | 510 - Mathematics |
| discipline | Physik Mathematik |
| format | Conference Proceeding Book |
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| genre_facet | Konferenzschrift |
| id | DE-604.BV037332351 |
| illustrated | Illustrated |
| indexdate | 2024-07-20T11:05:16Z |
| institution | BVB |
| institution_GND | (DE-588)16152110-1 |
| isbn | 3642182305 9783642182303 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-022486230 |
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| physical | IX, 176 S. Ill., graph. Darst. |
| publishDate | 2011 |
| publishDateSearch | 2011 |
| publishDateSort | 2011 |
| publisher | Springer |
| record_format | marc |
| series | Lecture notes in mathematics |
| series2 | Lecture notes in mathematics / École d'Été Probabilités de Saint-Flour |
| spelling | Flandoli, Franco Verfasser (DE-588)135622360 aut Random perturbation of PDEs and fluid dynamic models Ecole d'Eté de Probabilités de Saint-Flour XL - 2010 Franco Flandoli Berlin [u.a.] Springer 2011 IX, 176 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Lecture notes in mathematics / École d'Été Probabilités de Saint-Flour 2015 Störungstheorie (DE-588)4128420-3 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Stochastische Differentialgleichung (DE-588)4057621-8 gnd rswk-swf Stochastischer Prozess (DE-588)4057630-9 gnd rswk-swf Strömungsmechanik (DE-588)4077970-1 gnd rswk-swf (DE-588)1071861417 Konferenzschrift gnd-content Partielle Differentialgleichung (DE-588)4044779-0 s Störungstheorie (DE-588)4128420-3 s Stochastischer Prozess (DE-588)4057630-9 s Stochastische Differentialgleichung (DE-588)4057621-8 s Strömungsmechanik (DE-588)4077970-1 s 1\p DE-604 DE-604 Ecole d'Eté de Probabilités 40 2010 Saint-Flour Sonstige (DE-588)16152110-1 oth Erscheint auch als Online-Ausgabe 978-3-642-18231-0 Lecture notes in mathematics 2015 (DE-604)BV000676446 2015 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3631580&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022486230&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 | Flandoli, Franco Random perturbation of PDEs and fluid dynamic models Ecole d'Eté de Probabilités de Saint-Flour XL - 2010 Lecture notes in mathematics Störungstheorie (DE-588)4128420-3 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd Stochastische Differentialgleichung (DE-588)4057621-8 gnd Stochastischer Prozess (DE-588)4057630-9 gnd Strömungsmechanik (DE-588)4077970-1 gnd |
| subject_GND | (DE-588)4128420-3 (DE-588)4044779-0 (DE-588)4057621-8 (DE-588)4057630-9 (DE-588)4077970-1 (DE-588)1071861417 |
| title | Random perturbation of PDEs and fluid dynamic models Ecole d'Eté de Probabilités de Saint-Flour XL - 2010 |
| title_auth | Random perturbation of PDEs and fluid dynamic models Ecole d'Eté de Probabilités de Saint-Flour XL - 2010 |
| title_exact_search | Random perturbation of PDEs and fluid dynamic models Ecole d'Eté de Probabilités de Saint-Flour XL - 2010 |
| title_full | Random perturbation of PDEs and fluid dynamic models Ecole d'Eté de Probabilités de Saint-Flour XL - 2010 Franco Flandoli |
| title_fullStr | Random perturbation of PDEs and fluid dynamic models Ecole d'Eté de Probabilités de Saint-Flour XL - 2010 Franco Flandoli |
| title_full_unstemmed | Random perturbation of PDEs and fluid dynamic models Ecole d'Eté de Probabilités de Saint-Flour XL - 2010 Franco Flandoli |
| title_short | Random perturbation of PDEs and fluid dynamic models |
| title_sort | random perturbation of pdes and fluid dynamic models ecole d ete de probabilites de saint flour xl 2010 |
| title_sub | Ecole d'Eté de Probabilités de Saint-Flour XL - 2010 |
| topic | Störungstheorie (DE-588)4128420-3 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd Stochastische Differentialgleichung (DE-588)4057621-8 gnd Stochastischer Prozess (DE-588)4057630-9 gnd Strömungsmechanik (DE-588)4077970-1 gnd |
| topic_facet | Störungstheorie Partielle Differentialgleichung Stochastische Differentialgleichung Stochastischer Prozess Strömungsmechanik Konferenzschrift |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=3631580&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=022486230&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000676446 |
| work_keys_str_mv | AT flandolifranco randomperturbationofpdesandfluiddynamicmodelsecoledetedeprobabilitesdesaintflourxl2010 AT ecoledetedeprobabilitessaintflour randomperturbationofpdesandfluiddynamicmodelsecoledetedeprobabilitesdesaintflourxl2010 |