Nonlinear partial differential equations: asymptotic behaviour of solutions and self-similar solutions
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| Main Authors: | , , |
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| Format: | Book |
| Language: | English |
| Published: |
Boston, Mass. [u.a.]
Birkhäuser
2010
|
| Series: | Progress in nonlinear differential equations and their applications
79 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVIII, 294 S. graph. Darst. |
| ISBN: | 9780817641733 |
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| 100 | 1 | |a Giga, Mi-Ho |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Nonlinear partial differential equations |b asymptotic behaviour of solutions and self-similar solutions |c Mi-Ho Giga ; Yoshikazu Giga ; Jürgen Saal |
| 264 | 1 | |a Boston, Mass. [u.a.] |b Birkhäuser |c 2010 | |
| 300 | |a XVIII, 294 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Progress in nonlinear differential equations and their applications |v 79 | |
| 650 | 0 | 7 | |a Nichtlineare partielle Differentialgleichung |0 (DE-588)4128900-6 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Nichtlineare partielle Differentialgleichung |0 (DE-588)4128900-6 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Giga, Yoshikazu |d 1955- |e Verfasser |0 (DE-588)121691195 |4 aut | |
| 700 | 1 | |a Saal, Jürgen |e Verfasser |4 aut | |
| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |z 978-0-8176-4651-6 |
| 830 | 0 | |a Progress in nonlinear differential equations and their applications |v 79 |w (DE-604)BV007934389 |9 79 | |
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| adam_text | Contents
Preface
........................................................xiii
Part I Asymptotic Behavior of Solutions of Partial Differential
Equations
1
Behavior Near Time Infinity of Solutions of the Heat
Equation
.................................................. 3
1.1
Asymptotic Behavior of Solutions Near Time Infinity
........ 3
1.1.1
Decay Estimate of Solutions
........................ 6
1.1.2
LP-L Estimates
.................................. 8
1.1.3
Derivative U>-Lq Estimates
......................... 8
1.1.4
Theorem on Asymptotic Behavior Near Time Infinity
.. 10
1.1.5
Proof Using Representation Formula of Solutions
...... 11
1.1.6
Integral Form of the Mean Value Theorem
............ 12
1.2
Structure of Equations and Self-Similar Solutions
............ 13
1.2.1
Invariance
Under Scaling
........................... 13
1.2.2
Conserved Quantity for the Heat Equation
........... 14
1.2.3
Scaling Transformation Preserving the Conserved
Quantity
.........................................
I·1)
1.2.4
Summary of Properties of a Scaling Transformation.
... 15
1.2.5
Self-Similar Solutions
.............................. 16
1.2.6
Expression of Asymptotic Formula Using Scaling
Transformations
................................... 16
1.2.7
Idea of the Proof Based on Scaling Transformation
.... 17
1.3
Compactness
............................................
IH
1.3.1
Family of Functions Consisting of Continuous Functions
19
1.3.2
Ascoli-Arzelà-type
Compactness Theorem
............ 22
1.3.3
Relative Compactness of a Family of Scaled Functions
. 22
1.3.4
Decay Estimates in Space Variables
.................. 25
Contents
1.3.5
Existence
of Convergent Subsequences
............... 26
1.3.6
Lemma
..........................................
27
1.4
Characterization of Limit Functions
........................ 27
1.4.1
Limit of the Initial Data
........................... 28
1.4.2
Weak Form of the Initial Value Problem for the Heat
Equation
......................................... 29
1.4.3
Weak Solutions for the Initial Value Problem
......... 30
1.4.4
Limit of a Sequence of Solutions to the Heat Equation
. 31
1.4.5
Characterization of the Limit of a Family of Scaled
Functions
........................................ 32
1.4.0
Uniqueness Theorem When Initial Data is the Delta
Function
......................................... 33
1.4.7
Completion of the Proof of Asymptotic Formula
(1.9)
Based on Scaling Transformation
.................... 34
1.4.8
Remark on Uniqueness Theorem
.................... 34
Behavior Near Time Infinity of Solutions
of the Vorticity Equations
................................. 37
2.1
Navier
-Stokes Equations and Vorticity Equations
............ 38
2.1.1
Vorticity
.................. ...................... 39
2.1.2
Vorticity and Velocity
.............................. 40
2.1.3
Biot-Savart Law
.................................. 41
2.1.4
Derivation of the Vorticity Equations
................ 42
2.2
Asymptotic Behavior Near Time Infinity
................... 42
2.2.1
Unique Existence Theorem
......................... 43
2.2.2
Theorem for Asymptotic Behavior of the Vorticity
..... 44
2.2.3
Scaling
Invariance
................................. 44
2.2.4
Conservation of the Total Circulation
................ 45
2.2.5
Rotationally Symmetric Self-Similar Solutions
......... 46
2.3
Global Lq~Ll Estimates for Solutions of the Heat Equation
with a Transport Term
................................... 47
2.3.1
Fundamental Lq-Lr Estimates
...................... 47
2.3.2
Change Ratio of 17-Norm per Time: Integral Identities
. 48
2.3.3
Nonincrease of I^-Norm
............................ 49
2.3.4
Application of the Nash Inequality
.................. 50
2.3.5
Proof of
Fundamental
Lq-Ll Estimates
............... 53
2.3.6
Extension of Fundamental Lq-Lx Estimates
........... 55
2.3.7
Maximum Principle
................................ 55
2.3.8
Preservation of Noniiegativity
....................... 56
2.4
Estimates for Solutions of Vorticity Equations
............... 58
2.4.1
Estimates for Vorticity and Velocity
................. 58
2.4.2
Estimates for Derivatives of the Vorticity
............. 62
2.4.3
Decay Estimates for the Vorticity in Spatial Variables
.. 68
Contents ix
2.5
Proof of the Asymptotic Formula
.......................... 72
2.5.1
Characterization of the Limit Function as a Weak
Solution
.......................................... 73
2.5.2
Estimates for the Limit Function
.................... 76
2.5.3
Integral Equation Satisfied by Weak Solutions
......... 80
2.5.4
Uniqueness of Solutions of Limit Equations
........... 81
2.5.5
Completion of the Proof of the Asymptotic Formula
... 83
2.6
Formation of the Burgers Vortex
.......................... 84
2.6.1
Convergence to the Burgers Vortex
.................. 85
2.6.2
Asymmetric Burgers Vortices
....................... 87
2.7
Self-Similar Solutions of the Navier-Stokes Equations and
Related Topics
.......................................... 88
2.7.1
Short History of Research on Asymptotic Behavior
of Vorticity
....................................... 89
2.7.2
Problems of Existence of Solutions
................... 91
2.7.3
Self-Similar Solutions
.............................. 93
2.8
Uniqueness of the Limit Equation for Large Circulation
...... 97
2.8.1
Uniqueness of Weak Solutions
....................... 97
2.8.2
Relative Entropy
.................................. 98
2.8.3
Boundedness of the Entropy
........................100
2.8.4
Rescaling
.........................................100
2.8.5
Proof of the Uniqueness Theorem
...................101
2.8.6
Remark on Asymptotic Behavior of the Vorticity
......102
Self-Similar Solutions for Various Equations
...............105
3.1
Porous Medium Equation
.................................105
3.1.1
Self-Similar Solutions Preserving Total Mass
..........107
3.1.2
Weak Solutions
...................................108
3.1.3
Asymptotic Formula
...............................109
3.2
Roles of Backward Self-Similar Solutions
...................109
3.2.1
Axisymmetric Mean Curvature Flow Equation
........110
3.2.2
Backward Self-Similar Solutions
and Similarity Variables
............................
Ill
3.2.3
Nonexistence of
Nontrivial
Self-Similar Solutions
......114
3.2.4
Asymptotic Behavior of Solutions Near Pinching Points
116
3.2.5
Monotonicity
Formula
.............................121
3.2.6
The Cases of
a
Semilinear
Heat Equation
and a Harmonic Map Flow Equation
.................125
3.3
Nondiffusion-Type Equations
.............................129
3.3.1
Nonlinear
Schrödinger
Equations
....................130
3.3.2
KdV Equation
....................................132
3.4
Notes and Comments
....................................134
3.4.1
A Priori Upper Bound
.............................134
3.4.2
Related Results on Forward Self-Similar Solutions
.....135
Contents
Part II Useful Analytic Tools
Various Properties of Solutions of the Heat Equation
......141
4.1
Convolution, the Young Inequality, and Lp-Lg Estimates
.....141
4.1.1
The Young Inequality
..............................142
4.1.2
Proof of LP-LO Estimates
...........................145
4.1.3
Algebraic Properties of Convolution
.................145
4.1.4
Interchange of Differentiation and Convolution
........146
4.1.5
Interchange of Limit and Differentiation
..............149
4.1.6
Smoothness of the Solution of the Heat Equation
......150
4.2
Initial Values of the Heat Equation
........................150
4.2.1
Convergence to the Initial Value
.....................150
4.2.2
Uniform Continuity
................................151
4.2.3
Convergence Theorem
.............................151
4.2.4
Corollary
.........................................153
4.2.5
Applications of the Convergence Theorem
4.2.3.......153
4.3
Inhoinogeneous Heat Equations
...........................154
4.3.1
Representation of Solutions
.........................155
4.3.2
Solutions of the Inhomogeneous Equation:
Case of Zero Initial Value
..........................150
4.3.3
Solutions of Iuhomogeueous Equations: General Case
.. 160
4.3.4
Singular Inhomogeneous Term at
t
= 0...............160
4.4
Uniqueness of Solutions of the Heat Equation
...............164
4.4.1
Proof of the Uniqueness Theorem 1.4.G
...............164
4.4.2
Fundamental Uniqueness Theorem
...................164
4.4.3
Inhomogeneous Equation
...........................167
4.4.4
Unique Solvability for Heat Equations
with Transport Term
..............................168
4.4.5
Fundamental
Solutions and Their Properties
..........174
4.5
Integration by Parts
.....................................177
4.5.1
An Example for Integration by Parts
in the Whole Space
................................178
4.5.2
A Whole Space Divergence Theorem
.................179
4.5.3
Integration by Parts on Bounded Domains
............179
Compactness Theorems
....................................181
5.1
Compact Domains of Definition
...........................181
5.1.1
Ascoli-Arzelà
Theorem
.............................181
5.1.2
Compact Embeddings
..............................184
5.2
Xoncompact Domains of Definition
........................185
5.2.1
Ascoli--Arzelà-Type
Compactness Theorem
...........185
5.2.2
Construction of Subsequences
.......................186
5.2.3
Equidecay and Uniform Convergence
.................186
5.2.4
Proof of Lemma
1.3.0..............................187
5.2.5
Convergence of Higher Derivatives
...................187
Contents xi
6
Calculus Inequalities
.......................................189
6.1
The Gagliardo-Nirenberg Inequality and the Nash Inequality
. 189
6.1.1
The Gagliardo-Nirenberg Inequality
.................190
6.1.2
The Nash Inequality
...............................191
6.1.3
Proof of the Nash Inequality
........................191
6.1.4
Proof of the Gagliardo-Nirenberg Inequality
(Case of
σ
< 1) ...................................194
6.1.5
Remarks on the Proofs
.............................199
6.1.6
A Remark on Assumption
(6.3).....................199
6.2
Boundedness of the Riesz Potential
........................200
6.2.1
The Hardy-Littlewood-Sobolev Inequality
............200
6.2.2
The Distribution Function and Lp-Integrability
.......201
6.2.3
Lorentz
Spaces
....................................203
6.2.4
The Marcinkiewicz Interpolation Theorem
............203
6.2.5
Gauss Kernel Representation of the Riesz Potential
.... 209
6.2.6
Proof of the Hardy-Littlewood-Sobolev Inequality
.....210
6.2.7
Completion of the Proof
............................212
6.3
The Sobolev Inequality
...................................212
6.3.1
The Inverse of the Laplarian (if
> 3).................212
0.3.2
The Inverse of the Laplaciau (n
= 2).................214
6.3.3
Proof of the Sobolev Inequality (r
> 1)...............216
6.3.4
Aii
Elementary Proof of the Sobolev Inequality
(?■ = 1). 217
6.3.5
The Newton Potential
.............................218
6.3.6
Remark on Differentiation Under the Integral Sign
.....221
6.4
Boundedness of Singular Integral Operators
.................222
6.4.1
Cube Decomposition
...............................222
6.4.2
The
Calderón-Zygmuiid
Inequality
..................225
6.4.3
L 2 Boundedness
...................................227
6.4.4
Weak L1 Estimate
.................................228
6.4.5
Completion of the Proof
............................234
6.5
Notes and Comments
....................................234
7
Convergence Theorems in the Theory of Integration
.......239
7.1
Interchange of Integration and Limit Operations
.............239
7.1.1
Dominated Convergence Theorem
...................240
7.1.2
Fatou s Lemma
...................................242
7.1.3
Monotone Convergence Theorem
....................242
7.1.4
Convergence for Riemami Integrals
..................243
7.2
Commutativity of Integration and Differentiation
............244
7.2.1
Differentiation Under the Integral Sign
...............244
7.2.2
Cominutativity of the Order of Integration
...........245
7.3
Bounded Extension
......................................246
Answers to Exercises
..........................................249
xii Contents
Comments on Further References
..............................273
References
.....................................................275
Glossary
.......................................................289
Index
..........................................................293
|
| any_adam_object | 1 |
| author | Giga, Mi-Ho Giga, Yoshikazu 1955- Saal, Jürgen |
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| id | DE-604.BV025600163 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T22:37:16Z |
| institution | BVB |
| isbn | 9780817641733 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-020195695 |
| oclc_num | 699719031 |
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| owner | DE-11 DE-20 DE-824 DE-355 DE-BY-UBR DE-188 |
| owner_facet | DE-11 DE-20 DE-824 DE-355 DE-BY-UBR DE-188 |
| physical | XVIII, 294 S. graph. Darst. |
| publishDate | 2010 |
| publishDateSearch | 2010 |
| publishDateSort | 2010 |
| publisher | Birkhäuser |
| record_format | marc |
| series | Progress in nonlinear differential equations and their applications |
| series2 | Progress in nonlinear differential equations and their applications |
| spelling | Giga, Mi-Ho Verfasser aut Nonlinear partial differential equations asymptotic behaviour of solutions and self-similar solutions Mi-Ho Giga ; Yoshikazu Giga ; Jürgen Saal Boston, Mass. [u.a.] Birkhäuser 2010 XVIII, 294 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Progress in nonlinear differential equations and their applications 79 Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 gnd rswk-swf Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 s DE-604 Giga, Yoshikazu 1955- Verfasser (DE-588)121691195 aut Saal, Jürgen Verfasser aut Erscheint auch als Online-Ausgabe 978-0-8176-4651-6 Progress in nonlinear differential equations and their applications 79 (DE-604)BV007934389 79 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020195695&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Giga, Mi-Ho Giga, Yoshikazu 1955- Saal, Jürgen Nonlinear partial differential equations asymptotic behaviour of solutions and self-similar solutions Progress in nonlinear differential equations and their applications Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 gnd |
| subject_GND | (DE-588)4128900-6 |
| title | Nonlinear partial differential equations asymptotic behaviour of solutions and self-similar solutions |
| title_auth | Nonlinear partial differential equations asymptotic behaviour of solutions and self-similar solutions |
| title_exact_search | Nonlinear partial differential equations asymptotic behaviour of solutions and self-similar solutions |
| title_full | Nonlinear partial differential equations asymptotic behaviour of solutions and self-similar solutions Mi-Ho Giga ; Yoshikazu Giga ; Jürgen Saal |
| title_fullStr | Nonlinear partial differential equations asymptotic behaviour of solutions and self-similar solutions Mi-Ho Giga ; Yoshikazu Giga ; Jürgen Saal |
| title_full_unstemmed | Nonlinear partial differential equations asymptotic behaviour of solutions and self-similar solutions Mi-Ho Giga ; Yoshikazu Giga ; Jürgen Saal |
| title_short | Nonlinear partial differential equations |
| title_sort | nonlinear partial differential equations asymptotic behaviour of solutions and self similar solutions |
| title_sub | asymptotic behaviour of solutions and self-similar solutions |
| topic | Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 gnd |
| topic_facet | Nichtlineare partielle Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020195695&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV007934389 |
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