Mathematics / research
Nonlinear evolution equations
Examine local existence, dispersive estimates, invariant quantities, critical scaling, and obstructions to global control.
Your route through mathematics
Follow the path or begin with a question. Every lesson is open.
Your place, at your pace. Progress is optional and saved only in this browser when available.
Read / explore / reason
Inside this module
Module 01 research
Nonlinear evolution equations
Examine local existence, dispersive estimates, invariant quantities, critical scaling, and obstructions to global control.
- 01
Fixed-point local well-posedness
Construct a nonlinear evolution through its integral equation, choosing a complete solution space and proving a quantitative contraction.
35 min - 02
Scaling criticality and invariant norms
Derive the natural rescaling of nonlinear Schrödinger and Navier–Stokes equations, then identify which norms can detect concentration.
35 min - 03
Dispersive decay and stationary phase
Explain decay through oscillatory cancellation, compute the free Schrödinger kernel bound, and isolate the nondegeneracy behind stationary phase.
35 min - 04
Strichartz estimates and admissible exponents
Match spacetime integrability to free Schrödinger scaling, respect the excluded endpoint, and close a cubic nonlinear estimate with a small time factor.
36 min - 05
Nonlinear Schrödinger evolution and conservation
Derive mass and Hamiltonian invariants with a fixed sign convention, then explain when their bounds extend a local energy-space solution.
36 min - 06
Solitons and orbital stability
Construct a focusing cubic standing wave, identify its phase-and-translation orbit, and state the exact one-dimensional energy-space stability setting.
36 min - 07
KdV evolution and dispersive balancing
Derive the Korteweg–de Vries traveling profile, fix its dispersion sign, and distinguish nonlinear balance from diffusion or linear superposition.
35 min - 08
Incompressible Euler evolution and vorticity
Derive pressure and vorticity constraints, compare two-dimensional transport with three-dimensional stretching, and separate energy control from derivative control.
36 min - 09
Navier–Stokes energy and weak solutions
Derive viscous dissipation, construct the logic of Leray–Hopf compactness, and distinguish global energy-class existence from strong regularity.
37 min - 10
Continuation criteria and regularity questions
Use conditional norm bounds to extend classical solutions, compare energy and critical exponents, and read dated research claims with their exact hypotheses.
37 min