Preface ........................................................ xi
I. Preliminaries
1. Regular and Singular Perturbations ........................... 3
2. Evolution Equations in Hilbert Spaces ....................... 17
II. Singularly Perturbed Hyperbolic Problems
3. Presentation of the Problems ................................ 37
4. Hyperbolic Systems with Algebraic Boundary Conditions
4.1. A zeroth order asymptotic expansion .................... 44
4.2. Existence, uniqueness and regularity of the solutions
of problems Pε and P0 .................................. 46
4.3. Estimates for the remainder components ................. 59
5. Hyperbolic Systems with Dynamic Boundary Conditions
5.1. A first order asymptotic expansion for the solution of
problem (LS), (IС), (BC.l) ............................. 66
5.1.1. Formal expansion ................................ 67
5.1.2. Existence, uniqueness and regularity of the
solutions of problems Pε, P0 and P1 ............. 70
5.1.3. Estimates for the remainder components .......... 79
5.2. A zeroth order asymptotic expansion for the solution of
problem (NS), (IС), (BC.l) ............................. 83
5.2.1. Formal expansion ................................ 84
5.2.2. Existence, uniqueness and regularity of the
solutions of problems Pε and P0 ................. 85
5.2.3. Estimates for the remainder components .......... 91
5.3. A zeroth order asymptotic expansion for the solution of
problem (NS), (IС), (ВС.2) ............................. 96
5.3.1. Formal expansion ................................ 96
5.3.2. Existence, uniqueness and regularity of the
solutions of problems Pε and P0 ................. 97
5.3.3. Estimates for the remainder components ......... 102
5.4. A zeroth order asymptotic expansion for the solution
of problem (LS)', (IС), (BC.l) ........................ 105
5.4.1. Formal expansion ............................... 106
5.4.2. Existence, uniqueness and regularity of the
solutions of problems Pε and P0 ................ 107
5.4.3. Estimates for the remainder components ......... 109
III. Singularly Perturbed Coupled Boundary Value Problems
6. Presentation of the Problems ............................... 113
7. The Stationary Case
7.1. Asymptotic analysis of problem (P.1)ε ................. 119
7.1.1. Higher order asymptotic expansion .............. 122
7.1.2. Existence, uniqueness and regularity of the
solutions of problems (P.1)ε and (P.l)k ........ 124
7.1.3. Estimates for the remainder components ......... 127
7.2. Asymptotic analysis of problem (P.2)ε ................. 130
7.2.1. First order asymptotic expansion ............... 130
7.2.2. Existence, uniqueness and regularity of the
solutions of problems (P.2)ε, (P.2)0
and (P.2)1 ..................................... 132
7.2.3. Estimates for the remainder components ......... 134
7.3. Asymptotic analysis of problem (P.3)ε ................. 137
7.3.1. Formal expansion ............................... 137
7.3.2. Existence, uniqueness and regularity of the
solutions of problems (P.3)ε and (P.3)0 ........ 138
7.3.3. Estimates for the remainder components ......... 141
8. The Evolutionary Case
8.1. A first order asymptotic expansion for the solution of
problem (P.1)ε ........................................ 145
8.1.1. Formal expansion ............................... 145
8.1.2. Existence, uniqueness and regularity of the
solutions of problems (P.1)ε, (P.1)0
and (P.1)1 ..................................... 147
8.1.3. Estimates for the remainder components ......... 156
8.2. A first order asymptotic expansion for the solution of
problem (P.2)ε ........................................ 161
8.2.1. Formal expansion ............................... 161
8.2.2. Existence, uniqueness and regularity of the
solutions of problems (P.2)ε, (P.2)1
and (P.2)0 ..................................... 163
8.2.3. Estimates for the remainder components ......... 169
8.3. A zeroth order asymptotic expansion for the solution of
problem (P.3)e ........................................ 173
8.3.1. Formal expansion ............................... 173
8.3.2. Existence, uniqueness and regularity of the
solutions of problems (P.3)ε and (P3)0 ......... 174
8.3.3. Estimates for the remainder components ......... 177
IV. Elliptic and Hyperbolic Regularizations of Parabolic Problems
9. Presentation of the Problems ............................... 181
10. The Linear Case
10.1. Asymptotic analysis of problem (P.l)ε ............... 187
10.1.1. Nth order asymptotic expansion .............. 187
10.1.2. Existence, uniqueness and regularity of the
solutions of problems (P.1)ε and (P.1)k ..... 189
10.1.3. Estimates for the remainder ................. 192
10.2. Asymptotic analysis of problem (P.2)ε ............... 195
10.2.1. Nth order asymptotic expansion .............. 196
10.2.2. Existence, uniqueness and regularity of the
solutions of problems (P.2)ε and (P2)k ...... 197
10.2.3. Estimates for the remainder ................. 197
10.3. Asymptotic analysis of problem (P.3)ε ............... 199
10.3.1. Nth order asymptotic expansion .............. 199
10.3.2. Existence, uniqueness and regularity of the
solutions of problems (P.3)ε and (P.3)k ..... 200
10.3.3. Estimates for the remainder ................. 203
10.4. An Example .......................................... 204
11. The Nonlinear Case
11.1. Asymptotic analysis of problem (P.1)ε ............... 211
11.1.1. A zeroth order asymptotic expansion for the
solution of problem (P.1)ε .................. 211
11.1.2. Existence, uniqueness and regularity of the
solutions of problems (P.1)ε and P0 ......... 211
11.1.3. Estimates for the remainder ................. 214
11.2. Asymptotic analysis of problem (P.2)ε ............... 216
11.2.1. A first order asymptotic expansion for the
solution of problem (P.2)ε .................. 216
11.2.2. Existence, uniqueness and regularity of the
solutions of problems (P.2)ε, P0
and (P.2)1 .................................. 217
11.2.3. Estimates for the remainder ................. 219
11.3. Asymptotic analysis of problem (P.3)ε ............... 221
11.3.1. A first order asymptotic expansion .......... 221
11.3.2. Existence, uniqueness and regularity of the
solutions of problems (P3)ε, P0 and (P3)1 ... 222
11.3.3. Estimates for the remainder ................. 224
Bibliography .................................................. 227
Index ......................................................... 231
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