Analysis of Singularities for Partial Differential Equations

The following sections are included:

• Preface

• Contents

The following sections are included:

• The classical singularity propagation theorem

• Towards to modern theory

The following sections are included:

• Wave front set

• Singularity propagation theorem for equations of principal type

• Reflection of singularity on boundary

• Further discussions
  • Generalized reflection of singularity on boundary
  • The operators with multiple characteristics

The following sections are included:

• Theorem of propagation of 2s weak singularity

• Theorem on propagation of 3s weak singularity

• Singularity interaction and singularity index

• Propagation of conormal singularity

• Interaction of conormal singularities
  • Extension of the concept of conormal singularities
  • Pseudo-composition
  • Theorem on interaction of conormal singularities
  • Reflection of conormal singularities

The following sections are included:

• Theorem of propagation of singularities for principal type equations

• Propagation of conormal singularities for nonlinear equations

The following sections are included:

• Solutions with fan-shaped singularity structure of semilinear equations

• Solutions with flower-shaped singularity structure of semilinear equations

• Solutions with strong singularities of quasilinear equations (1-d case)

• Solutions with strong singularities of quasilinear equations (m-d case)
  • Fan-shaped singularity structure
  • Flower-shaped singularity structure

The following sections are included:

• The case of scalar equation
  • Two mechanism of blow-up of smooth solutions
  • Formation of a shock
  • Estimates of the solution in the neighborhood of the starting point of shock

• The case of system
  • Background and conclusion
  • The property of the first approximate solution
  • Estimates and convergence of the sequence of approximate solutions
  • The case for full Euler system

The following sections are included:

• Appendix A Brief review on paradifferential operators

• Bibliography