Nonlinear partial differential equations with applications:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Basel [u.a.]
Birkhäuser
2013
|
| Edition: | 2. ed. |
| Series: | International series of numerical mathematics
153 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | XX, 476 S. graph. Darst. |
| ISBN: | 3034805128 9783034805124 |
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MARC
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| 100 | 1 | |a Roubíček, Tomáš |d 1956- |e Verfasser |0 (DE-588)112701434 |4 aut | |
| 245 | 1 | 0 | |a Nonlinear partial differential equations with applications |c Tomáš Roubíček |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a Basel [u.a.] |b Birkhäuser |c 2013 | |
| 300 | |a XX, 476 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a International series of numerical mathematics |v 153 | |
| 650 | 0 | 7 | |a Nichtlineare partielle Differentialgleichung |0 (DE-588)4128900-6 |2 gnd |9 rswk-swf |
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Record in the Search Index
| _version_ | 1807955055216689152 |
|---|---|
| adam_text |
Contents
Preface
xi
Preface
to the
2nd
edition
xv
Notational conventions
xvii
1
Preliminary general material
1
1.1
Functional analysis
. 1
1.1.1
Normed spaces, Banach spaces, locally convex spaces
. 1
1.1.2
Junctions and mappings on Banach spaces, dual spaces
. . 3
1.1.3
Convex sets
. 6
1.1.4
Compactness
. 7
1.1.5
Fixed-point theorems
. 8
1.2
Function spaces
. 8
1.2.1
Continuous and smooth functions
. 9
1.2.2
Lebesgue
integrable
functions
. 10
1.2.3
Sobolev spaces
. 14
1.3
Nemytskii mappings
. 19
1.4
Green formula and some inequalities
. 20
1.5
Bochner spaces
. 22
1.6
Some ordinary differential equations
. 25
1 STEADY-STATE PROBLEMS
29
2
Pseudomonotone or
weakJy continuous mappings
31
2.1
Abstract theory, basic definitions, Galerkin method
. 31
2.2
Some facts about pseudomonotone mappings
. 35
2.3
Equations with monotone mappings
. 37
2.4 Quasilinear
elliptic equations
. 42
2.4.1
Boundary-value problems for 2nd-order equations
. 43
2.4.2
Weak formulation
. 44
2.4.3
Pseudomonotomcity, coercivity, existence of solutions
. 48
vin
Contents
2
AA
Higher-order equations
. 56
2.5
Weakly continuous .mappings,
semilinear
equations
. 61
2.6
Examples and exercises
. 64
2.6.1
General tools
. 64
2.6.2 Semilinear
heat equation of type
div (A (z, u)
Vu)
=
g
. 68
2.6.3 Quasilinear
equations of type
—
div(|Vu|p~2Vi¿)-|-c(u, Vu)--g
75
2.7
Excursion to regularity for
semilinear
equations
. 85
2.8
Bibliographical remarks
. 92
3
Accretive mappings
95
3.1
Abstract theory
. 95
3.2
Applications to boundary-value problems
. 99
3.2.1
Duality mappings in Lebesgue and Sobolev spaces
. 99
3.2.2
Accretivity of monotone
quasilinear
mappings
. 101
3.2.3
Accretivity of heat equation
. 105
3.2.4
Accretivity of some other boundary-value problems
. 108
3.2.5
Excursion to equations with measures in right-hand sides
. 109
3.3
Exercises
. 112
3.4
Bibliographical remarks
. 114
4
Potential problems: smooth case
115
4.1
Abstract theory
. 115
4.2
Application to boundary-value problems
. 120
4.3
Examples and exercises
. 126
4.4
Bibliographical remarks
. 130
5
Nonsmooth problems; variational inequalities
133
5.1
Abstract inclusions with a potential
. 133
5.2
Application to elliptic variational inequalities
. 137
5.3
Some abstract non-potential inclusions
. 145
5.4
Excursion to quasivariational inequalities
. 154
5.5
Exercises
. 157
5.6
Some applications to free-boundary problems
. 163
5.6.1
Porous media now: a potential variational inequality
. 163
5.6.2
Continuous casting: a non-potential variational inequality
. 166
5.7
Bibliographical remarks
. 169
Systems of equations: particular examples
171
6.1
Minimization-type variational method: polyconvex functionals
.
171
6
6.1
Minimization-type variational method: polyconvex functionals
. . . 171
6.2
Buoyancy-driven viscous flow
. 178
6.3
Reaction-diffusion system
. 186
6.4
Thermistor
. 188
6.5
Semiconductors
. 192
Contents
їх
II EVOLUTION PROBLEMS
199
7
Special auxiliary tools
201
7.1
Sobolev-Bochner space W1>p-q(r,V1,V2)
. 201
7.2
Gelfand triple, embedding
W1 w'
(1-у
,V)
С С
{I; H)
.
204
7.3
Aubin-Lions
lemma
. 207
8
Evolution by
pseudomonotone or
weakly continuous mappings
213
8.1
Abstract initial-value problems
. 213
8.2
Rothe method
. 215
8.3
Further estimates
. 230
8.4
Galerkm method
. 240
8.5
Uniqueness and continuous dependence on data
. 247
8.6
Application to
quasilinear
parabolic equations
. 251
8.7
Application to
semilinear
parabolic equations
. 261
8.8
Examples and exercises
. 264
8.8.1
General tools
. 264
8.8.2
Parabolic equation of type
^u-div(]Vu|p~2Vu)+c(u)=cř
. 266
8.8.3 Semilinear
heat equation c(u)
-Lu
—
div(/t(i¿)Vu)
=
д
. 277
8.8.4
Navier-Stokes equation
щи+{и-Ч)и-аи+Х7тг=д,
divu=0
. 279
8.8.5
Some more exercises
. 282
8.9
Global monotonicity approach, periodic, problems
. 288
8.10
Problems with a convex potential: direct method
. 294
8.11
Bibliographical remarks
. 300
9
Evolution governed by accretive mappings
303
9.1
Strong solutions
. 303
9.2
Integral solutions
. 308
9.3
Excursion to nonlinear semigroups
. 314
9.4
Applications to initial-boundary-value problems
. 319
9.5
Applications to some systems
. 326
9.6
Bibliographical remarks
. 332
10
Evolution governed by certain set-valued mappings
335
10.1
Abstract problems: strong solutions
. 335
10.2
Abstract problems: weak solutions
. 339
10.3
Examples of unilateral parabolic problems
. 343
10.4
Bibliographical remarks
. 349
11
Doubly-nonlinear problems
351
11.1
Inclusions of the type 3*(^u)
+
<ЭФ(и) Э
ƒ. 351
11.1.1
Potential
Φ
valued in
R U
{+00}. 351
11.1.2
Potential
Φ
valued in
K U
{+00}. 358
11.1.3
Uniqueness and continuous dependence on data
. 366
χ
Contents
11-2
Inclusions of the type
^Е(и) І
дФ(и) Э
ƒ. 367
11.2.1
The case
E
:=--
<9Ф.
368
11.2.2
The case
E
non-potential
. 372
11.2.3
Uniqueness
. 375
11.3
2ncUorder equations
. 377
11.4
Exercises
. 385
11.5
Bibliographical remarks
. 390
12
Systems of equations: particular examples
393
12.1
Thermo-vis
co-
elasticity
. 393
12.2
Buoyancy-driven viscous flow
. 405
12.3
Predator-prey system
. 408
12.4
Semiconductors
. 412
12.5
Phase-field model
. 416
12.6
Navier-Stokes-Nernst-Planck-Poisson-type system
. 420
12.7
Thermistor with eddy currents
. 426
12.8
Thermodynamics of magnetic materials
. 432
12.9
Thermo-visco-elasticity: fully nonlinear theory
. 438
Bibliography
449
Index
469 |
| any_adam_object | 1 |
| author | Roubíček, Tomáš 1956- |
| author_GND | (DE-588)112701434 |
| author_facet | Roubíček, Tomáš 1956- |
| author_role | aut |
| author_sort | Roubíček, Tomáš 1956- |
| author_variant | t r tr |
| building | Verbundindex |
| bvnumber | BV040730132 |
| classification_rvk | SK 500 SK 540 |
| classification_tum | MAT 356f MAT 355f |
| ctrlnum | (OCoLC)827086555 (DE-599)DNB1023815761 |
| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV040730132 |
| illustrated | Illustrated |
| indexdate | 2024-08-21T00:31:40Z |
| institution | BVB |
| isbn | 3034805128 9783034805124 |
| language | English |
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| physical | XX, 476 S. graph. Darst. |
| publishDate | 2013 |
| publishDateSearch | 2013 |
| publishDateSort | 2013 |
| publisher | Birkhäuser |
| record_format | marc |
| series | International series of numerical mathematics |
| series2 | International series of numerical mathematics |
| spelling | Roubíček, Tomáš 1956- Verfasser (DE-588)112701434 aut Nonlinear partial differential equations with applications Tomáš Roubíček 2. ed. Basel [u.a.] Birkhäuser 2013 XX, 476 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier International series of numerical mathematics 153 Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 gnd rswk-swf Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 s DE-604 Erscheint auch als Online-Ausgabe 978-3-0348-0513-1 International series of numerical mathematics 153 (DE-604)BV035415862 153 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=4068575&prov=M&dok_var=1&dok_ext=htm Inhaltstext Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025710195&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Roubíček, Tomáš 1956- Nonlinear partial differential equations with applications International series of numerical mathematics Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 gnd |
| subject_GND | (DE-588)4128900-6 |
| title | Nonlinear partial differential equations with applications |
| title_auth | Nonlinear partial differential equations with applications |
| title_exact_search | Nonlinear partial differential equations with applications |
| title_full | Nonlinear partial differential equations with applications Tomáš Roubíček |
| title_fullStr | Nonlinear partial differential equations with applications Tomáš Roubíček |
| title_full_unstemmed | Nonlinear partial differential equations with applications Tomáš Roubíček |
| title_short | Nonlinear partial differential equations with applications |
| title_sort | nonlinear partial differential equations with applications |
| topic | Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 gnd |
| topic_facet | Nichtlineare partielle Differentialgleichung |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=4068575&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=025710195&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV035415862 |
| work_keys_str_mv | AT roubicektomas nonlinearpartialdifferentialequationswithapplications |