Analysis of Singularities for Partial Differential Equations

Preface......Page 8
1.1 The classical singularity propagation theorem......Page 10
1.2 Towards to modern theory......Page 18
2.1 Wave front set......Page 22
2.2 Singularity propagation theorem for equations of principal type......Page 32
2.3 Reection of singularity on boundary......Page 39
2.4.1 Generalized reection of singularity on boundary......Page 52
2.4.2 The operators with multiple characteristics......Page 55
3. Singularity analysis for semilinear equations......Page 58
3.1 Theorem of propagation of 2s weak singularity......Page 59
3.2 Theorem on propagation of 3s weak singularity......Page 66
3.3 Singularity interaction and singularity index......Page 71
3.4 Propagation of conormal singularity......Page 82
3.5.1 Extension of the concept of conormal singularities......Page 89
3.5.2 Pseudo-composition......Page 95
3.5.3 Theorem on interaction of conormal singularities......Page 96
3.5.4 Reection of conormal singularities......Page 99
4.1 Theorem of propagation of singularities for principal type equations......Page 102
4.2 Propagation of conormal singularities for nonlinear equations......Page 110
5. Propagation of strong singularities for nonlinear equations......Page 120
5.1 Solutions with fan-shaped singularity structure of semilinear equations......Page 121
5.2 Solutions with ower-shaped singularity structure of semilinear equations......Page 131
5.3 Solutions with strong singularities of quasilinear equations (1-d case)......Page 140
5.4.1 Fan-shaped singularity structure......Page 146
5.4.2 Flower-shaped singularity structure......Page 151
6.1.1 Two mechanism of blow-up of smooth solutions......Page 156
6.1.2 Formation of a shock......Page 158
6.1.3 Estimates of the solution in the neighborhood of the starting point of shock......Page 165
6.2 The case of system......Page 168
6.2.1 Background and conclusion......Page 169
6.2.2 The property of the first approximate solution......Page 172
6.2.3 Estimates and convergence of the sequence of approximate solutions......Page 178
6.2.4 The case for full Euler system......Page 186
A.1 Diadic decomposition......Page 190
A.2 Paradifferential operators and paralinearzation......Page 194
A.3 Paracomposition......Page 198
Bibliography......Page 200