Index

A

A-Stokes approximation, 71–73, 86, 115, 117, 137
Applications, 3, 13, 15, 41, 65, 72, 76, 109, 132, 145, 170, 175, 183
Approximation, 29, 73, 89–92, 94, 173, 179
Arbitrariness, 50, 52, 65, 69
Aubin–Lions theorem, 110, 150, 156
Averaged Taylor polynomial, 63, 74, 76, 123

B

Banach function space, 9, 56, 102
Banach space, 4, 6, 10, 36, 101–103, 173, 175, 176, 184
reflexive, 102–104
Bi-Laplace equation, 191–193
Bochner spaces, 102, 103, 173
Bogovskiĭ operator, 28, 35, 49, 69, 71, 72, 92, 95, 111, 117, 132, 140–142
Boundary, 3, 7, 18, 20, 21, 106, 191, 193
Bounded domain, 32–34, 36, 43, 54, 69
Bounded linear operator, 36, 37, 173, 176
Boundedness, 38, 85, 128, 132, 133, 148, 156, 157, 191, 219
Bounds, uniform, 211, 213
Brownian motion, 171, 185, 186, 210

C

Chain, 45, 58, 67–69
Class, 30, 37, 150, 154, 165, 174, 178, 182, 217
Closure, 7, 11
Compactness, 103, 110, 146, 156, 183, 200, 203, 212, 213
Conditions, 23, 27, 29, 32–34, 36, 40, 49, 54, 59, 60, 64, 69, 70, 187, 188
fulfilling, 57
usual, 162, 165, 187
Constant, 10, 13, 15, 22, 33, 36–42, 44, 48, 49, 53–55, 57–59, 61–66, 69, 70, 105–107, 125, 126, 134
physical, 89, 185
Constitutive law, 22, 23, 30, 54, 89, 90, 108, 109
Construction, 13, 16, 18, 92, 116, 135, 182, 218
Continuity, 15, 35, 80, 92, 96, 102, 124, 132–134, 142, 182, 191, 211
Continuous function, 3, 4, 6, 13, 27, 61, 64, 66, 67, 92, 112, 146, 184, 204
Continuous local martingale, 167
Control, 107, 110, 113, 116, 118, 120, 121, 126
Convective term, 23, 24, 27, 54, 90, 109, 146, 147, 149, 152
Convergences, 75, 95, 97, 110–112, 132, 184, 189, 202, 205, 206, 211, 215–217
strong, 128, 214, 219
weak, 24, 128, 156, 207, 209, 212, 214
Cut-off function, 21, 106, 111, 219

D

De Rahm's theorem, 23, 32, 55, 93, 153
Decomposition, 72, 116, 128, 163, 167, 168, 193
Definition of Luxemburg norm, 39, 51, 52
Derivatives, 3, 19, 22, 108, 125
Difference, 111, 156, 167, 179, 217
Distributions, 31, 55, 102, 116–118, 127, 129, 154
Divergence, 26, 28, 46, 92
Divergence equation, 26, 35
Domains, 43–45, 86
Doob–Meyer decomposition, 163, 164
Drawbacks, 170, 186

E

Equivalence classes, 4, 165
Estimates, 12, 15–17, 19, 21, 72, 74–76, 82, 83, 87, 88, 106, 107, 123–125, 129, 130, 134, 135, 143, 157, 201
a priori, 149, 178, 180, 196, 200, 209
uniform, 94, 112, 204, 210
uniform a priori, 24, 25, 110, 111, 209
weak-type, 12, 83, 123
Existence, 7, 13, 14, 22, 23, 27, 31, 44, 54, 72, 94, 97, 108, 109, 146–149, 154, 155, 185, 186, 188
Existence of weak solutions, 76, 89, 145, 147, 186
Expectation values, 198, 218
Exponent, 14, 176
Extension, 12, 13, 19

F

Family, 20, 40, 44, 51, 183
Filtration, 161, 162, 165, 167–171, 176, 177, 187, 194, 202
Fluid, 7, 22, 23, 54, 89, 108, 145, 185
plastic, 89, 90, 92
Fluid mechanics, 29, 30
Framework, functional analytic, 3, 101
Fubini's theorem, 17, 123
Function spaces, 3, 7, 8, 71
borderline, 72
finite dimensional, 72, 80
Functions, 3–5, 7–14, 20–23, 26–32, 34–36, 47, 48, 54, 59, 63–66, 72–74, 89–92, 101–104, 111–113, 115–118, 121
bounded, 26, 27
differentiable, 58, 65
integrable, 25, 31, 37
maximal, 11, 12, 83, 85, 122
measurable, 9, 102, 116
non-negative, 13, 64
pressure, 31, 92, 113, 115, 145, 211, 216
sequence of, 47, 49, 52, 147, 194
smooth, 5, 29, 39, 61, 90
solenoidal, 85, 117
step, 101, 173
unique, 54, 190
vector-valued, 7, 36

G

Galerkin-Ansatz, 109, 177, 178, 194
Generality, 45, 64, 66, 123
Generalized Newtonian fluids, 3, 22, 29, 30, 72, 101
Gradients, 11, 12, 19, 21, 30, 31, 45, 71, 75, 81, 106, 115, 125, 186, 191
bounded, 92, 112
distributional, 10, 31, 32
level of, 71, 73, 115
Growth, 33, 34
linear, 169, 170, 196

H

Harmonic functions, 75, 85, 116, 131, 211, 213
Hilbert spaces, 166, 173–175, 178, 186
separable, 174, 175
Hilbert–Schmidt operators, 173, 178, 186
Hölder's inequality, 19, 96, 107, 119, 140, 142, 157, 180
Homogeneous, 22, 61, 89, 108, 145

I

Initial datum, 108, 145, 170, 178, 187, 188
Integrands, 96, 111, 165
Itô's formula, 161, 175, 180, 182, 188, 189, 197, 218
Itô's lemma, 167, 168

J

Jakubowski–Skorokhod theorem, 212, 214

K

Kernel, 37, 38, 45–47, 49, 57, 61
Kondrachov's theorem, 24, 27
Korn's inequality, 24, 29, 30, 32, 35, 78, 79, 93, 155, 197

L

Laminates, 64, 66
Laws, 169, 183, 184, 188, 194, 201–204, 213, 214
joint, 201, 204, 213
Lebesgue measure, 56, 64
Lipschitz boundary, 6, 7, 13, 18, 29, 62
Lipschitz constant, 12, 13, 21, 113
Lipschitz domain, 11, 44, 61, 69
open bounded, 57, 58
Lipschitz truncation, 11–18, 27, 28, 71, 75–77, 83, 89, 92, 104, 105, 112, 113, 115, 188
parabolic, 112
Lipschitz truncation method, 3, 101, 112, 146, 188
Local finiteness, 81, 82, 107, 127
Luxemburg norm, 9, 39, 51, 52

M

Martingale solutions, 169, 170, 186, 188
weak, 185, 187, 188, 199, 207, 209
Martingales, 163, 164, 189
continuous, 163, 175
Maximal operator, 12, 72
Measurability, 173, 191, 192
Metric, 163, 165, 166
Metric spaces, 183, 184
Monotone operator theory, 23–25, 109, 110, 146, 151, 189, 206, 209, 217, 218
Monotonicity, 26, 152, 157, 196, 207, 221, 222
Motion, 30, 60, 77, 89, 185, 186

N

Navier–Stokes equations, 22, 31, 108, 175, 186
Negative norm theorem, 31–33, 35
Norm, 6, 9, 10, 25, 31–33, 36, 47, 55, 56, 62, 81–83, 102–104, 111, 124, 126
Norm equivalence, 74, 82, 124, 126
Norm topology, 103, 182
Null sequence, 77, 83, 85, 96
strong, 127, 129, 132
weak, 76, 127, 129

O

Open sets, 10, 14, 20, 40, 44, 55, 56, 62, 183
Operator, 29, 40, 45, 47, 49, 50, 63, 72, 93, 128, 191, 193, 202
linear projection, 60, 61
Orlicz space, 8–10, 29–32, 35, 38, 54–56
Orlicz–Sobolev space, 10, 32

P

Parabolic interpolation, 109–111, 146, 150, 210
Perturbation, 22, 185, 186
Poincaré's inequality, 22, 119, 120, 126, 139, 141, 142
Pointwise, 47, 74, 75, 80, 116–118, 124, 130, 184
Polynomials, 63, 124, 126
linear, 74, 76, 120, 124
Power law fluids, 3, 23, 27, 91, 101, 109, 145, 177
Power law model, 23, 54, 109, 145, 154, 185
Prandtl–Eyring fluids, 30, 71, 89, 92
Pressure, 22, 23, 30, 31, 54, 89, 91–93, 97, 108, 111, 145, 146, 155, 157, 185, 188, 189, 192, 210
Pressure recovery, 29, 32, 35
Probabilistic sense, 169, 188
Probability, 164, 177, 206
Probability laws, 183, 184
family of, 183, 184
Probability space, 161, 162, 168–171, 174–176, 178, 183, 184, 202–205, 207, 210, 212, 214, 215
Process, 162–166, 173, 175, 203, 215
adapted, 165, 167, 169, 171, 177, 189
increasing, 163–165
measurable, 165, 176–178
valued, 170, 173
Wiener, 175–178, 203
Projection, orthogonal, 148, 149, 179, 195
Prokhorov's theorem, 202
Properties, 8, 20, 37–39, 49, 50, 63, 71, 72, 74, 76, 77, 83, 105, 106, 115–117, 122–124, 127–129, 138, 139, 161, 162
basic, 3, 12, 13, 75
boundedness, 45, 47
cone, 32–34, 36, 43, 44, 54
continuity, 77, 83, 139
convergence, 28, 128

Q

Quadratic variation, 164, 174, 203
Quantities, 31, 33, 154, 164, 169, 185, 220

R

Random variables, 161, 162, 166, 183, 184, 204
Regularity theory, 12, 30, 85, 211
local, 192, 193, 213, 215
standard interior, 119
Representation formula, 8, 61, 63, 66
Restriction, 112, 135, 146, 186, 202, 204

S

Semi-continuity, lower, 13, 14, 17, 207
Semi-martingale, 167, 168, 174
Sequence, 6, 7, 11, 16, 24, 27, 52, 61, 66, 67, 77, 91, 92, 101, 103, 128, 166, 167, 177
diagonal, 26, 84
double, 16, 83, 95, 127, 129, 217
Sequence of approximated solutions, 25, 109
Simplicity, 5, 90, 146, 178, 184, 185, 209
Singular integral operator, 32, 37, 38, 45–47, 49, 61, 71–73, 116
Situations, 22, 30, 33, 34, 91, 92, 111, 122, 212
Sobolev function, 7, 11, 71, 115
global, 75, 80
Solenoidal Lipschitz truncation, 71–74, 89, 92, 95, 115, 132, 135, 145, 146, 154
Solenoidal test functions, 92, 117
Space, 5, 7, 11, 36, 47, 54, 60, 63, 77, 91, 102, 117, 146, 147, 165, 166, 194
path, 201, 203, 204, 213
quasi-Polish, 184, 212
topological, 183, 184
Space of test functions, 94, 147
Space time cylinder, 104, 115
Stationary case, 23, 27, 71, 73, 85, 105, 110–113, 146
Stationary flows, 3, 22, 89
Stationary problems, 71, 110
Stochastic analysis, 161, 167
Stochastic basis, 187, 189, 199
Stochastic Fubini theorem, 176
Stochastic heat equation, 173, 177
Stochastic integrals, 164, 167, 175, 177
Stochastic integration, 161
Stochastic ODEs, 170, 178
Stochastic PDEs, 103, 161, 170, 173, 183
Stochastic power law fluids, 186
Stochastic processes, 161, 162, 164, 165, 167, 174
adapted, 163, 165, 169
valued, 170, 173
Subsequence, 24, 26, 52, 61, 77, 83, 84, 91, 94, 95, 110, 128, 131, 149, 152, 155, 156, 209–211
Subspace, 9, 11, 36, 56, 104, 204
Symmetric gradient, 19, 22, 30, 91, 108
Symmetric part, 65, 66
System, 14, 17, 22, 94, 108, 118, 145, 179, 194, 195, 203
approximated, 189, 209

T

Terms, 24–26, 28, 47, 49, 67, 90, 105, 110–113, 120, 135, 139, 140, 182, 188, 189
Test functions, 18, 23, 25–27, 71, 94, 95, 107, 109, 111, 112, 116, 118, 126, 135–137, 146, 147, 150, 178
admissible, 24, 25, 110, 130, 151, 194
divergence-free, 31, 115
Trajectories, 161, 169, 171, 174
Transfers, 14, 128, 130, 188
Truncation, 15, 21, 23, 25, 27, 28, 72, 106, 107, 111–113, 115–117, 124–126, 146, 185, 186, 188, 189, 209, 217

U

Unique function, 136
Unity, partition of, 20, 22, 74, 77, 105, 106, 122, 126

V

Variations, 11, 203
total, 164
Velocity, 22, 108, 110, 185, 194
Version, 31, 32, 35, 36, 55, 103, 104, 175
Viscosity, 22, 23, 89, 108
Viscous stress tensor, 22, 108
Vitali's convergence theorem, 202, 214
Vitali's theorem, 95

W

Weak solutions, 14, 17, 22, 23, 25, 27, 31, 54, 76, 89–92, 94, 97, 108, 109, 118, 119, 145–147, 169
mathematical theory of, 3, 101
Weak topologies, 103, 184, 212, 213
Wiener process, 162, 165, 167–169, 186, 203

Y

Young functions, 8–11, 29, 30, 32–34, 36–38, 50, 54–59, 61, 62, 64, 69
Young inequality, 96, 135, 139, 142, 181, 198, 208
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