NLSCritical.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

4 declarations, 1 contain sorry, 0 declared as axioms

Energy-critical NLS in three dimensions.

This file states an a priori estimate for the energy-critical nonlinear Schrödinger equation on $ℝ^3$. By 'soft' arguments, this bound implies global well-posedness in the energy space, scattering, asymptotic completeness, and uniform regularity (all of which are less elementary to state). This is remarked on p. 770 and p. 793 in the referenced article which first proved the a priori bound.

The main theorem is:

NLS_apriori: a bound on the space-time $L^10$ norm in terms of the initial energy for local smooth solutions to the NLS equation.

References

Colliander, Keel, Staffilani, Takaoka, Tao, Annals of Mathematics 167 (2008) proves the a priori estimate, and thereby, global well-posedness.

Imports (1)

FormalConjecturesUtil

structure NLSCritical.LocalSchwartzSolution A smooth solution to the NLS equation on the time interval $[-l, l]$ line 48 source
Source documentation
A smooth solution to the NLS equation on the time interval $[-l, l]$. We additionally require that $x ↦ u(t,x)$ is Schwartz for all $t$ in the interval.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real
```
structure LocalSchwartzSolution (l : ℝ) where
…
```
def NLSCritical.energy The Hamiltonian of the energy-critical NLS in $ℝ^3$ line 60 source
Source documentation
The Hamiltonian of the energy-critical NLS in $ℝ^3$. The kinetic term is $\frac12\sum_{j=1}^3 |\partial_j u|^2$, as in (1.2) of the reference.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: EuclideanSpace, Real, Fin, OfNat.ofNat, Nat, Complex
Proof uses: MeasureTheory.integral, ENNReal, MeasureTheory.MeasureSpace.volume, HAdd.hAdd, HMul.hMul, HDiv.hDiv, Finset.sum, Finset.univ, HPow.hPow, Norm.norm, ContinuousLinearMap, RingHom.id, fderiv, EuclideanSpace.single
78 instances, coercions and auxiliary constants left out.
```
noncomputable def energy (u : ℝ^3 → ℂ) : ℝ :=
…
```
def NLSCritical.L10_local The space-time $L^10$ norm, restricted to the time interval $[-l,l]$ line 67 source
Source documentation
The space-time $L^10$ norm, restricted to the time interval $[-l,l]$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, EuclideanSpace, Fin, OfNat.ofNat, Nat, Complex
Proof uses: MeasureTheory.integral, MeasureTheory.Measure.restrict, MeasureTheory.MeasureSpace.volume, Set.Icc, Neg.neg, ENNReal, HPow.hPow, Norm.norm
31 instances, coercions and auxiliary constants left out.
```
noncomputable def L10_local (l : ℝ) (u : ℝ → ℝ^3 → ℂ) :=
…
```
theorem NLSCritical.NLS_apriori The $L^10_{t,x}$ a priori estimate for local schwartz solutions sorry line 74 source
Source documentation
The $L^10_{t,x}$ a priori estimate for local schwartz solutions.

depends on sorry what this means
Statement uses: Exists, Real, GT.gt, OfNat.ofNat, NLSCritical.LocalSchwartzSolution, LE.le, NLSCritical.L10_local, NLSCritical.LocalSchwartzSolution.u, NLSCritical.energy
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, Nat
9 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem NLS_apriori :
    ∃ f : ℝ → ℝ, ∀ l > 0, ∀ s : LocalSchwartzSolution l,
    L10_local l s.u ≤ f (energy (s.u 0)) := by
…
```
Mentions: NLSCritical.LocalSchwartzSolution, NLSCritical.L10_local, NLSCritical.energy