diff --git a/FormalConjectures/Millenium/NavierStokes.lean b/FormalConjectures/Millenium/NavierStokes.lean new file mode 100644 index 000000000..1bef6b715 --- /dev/null +++ b/FormalConjectures/Millenium/NavierStokes.lean @@ -0,0 +1,239 @@ +/- +Copyright 2025 The Formal Conjectures Authors. + +Licensed under the Apache License, Version 2.0 (the "License"); +you may not use this file except in compliance with the License. +You may obtain a copy of the License at + + https://www.apache.org/licenses/LICENSE-2.0 + +Unless required by applicable law or agreed to in writing, software +distributed under the License is distributed on an "AS IS" BASIS, +WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. +See the License for the specific language governing permissions and +limitations under the License. +-/ + +import FormalConjectures.Util.ProblemImports + +/-! +# Existence And Smoothness Of The Navier–Stokes Equation + +This file formalizes the Clay Mathematics Institute millennium problem concerning +the existence and smoothness of solutions to the Navier-Stokes equations in three +spatial dimensions. While the definitions are generalized to arbitrary dimension n, +the millennium problem specifically concerns the case n = 3. + +## References +- [Wikipedia](https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_existence_and_smoothness) +- [Clay Mathematics Institute](https://www.claymath.org/wp-content/uploads/2022/06/navierstokes.pdf) + +## Main Theorems (Clay Millennium Problem for n = 3) + +The Clay Millennium Problem asks for a proof of one of the following four statements: + +- `navier_stokes_existence_and_smoothness_R3`: (A) Global existence on ℝ³ +- `navier_stokes_existence_and_smoothness_periodic`: (B) Global existence on ℝ³/ℤ³ +- `navier_stokes_breakdown_R3`: (C) Existence of breakdown scenario on ℝ³ +- `navier_stokes_breakdown_periodic`: (D) Existence of breakdown scenario on ℝ³/ℤ³ +-/ + +open ContDiff Set InnerProductSpace MeasureTheory EuclideanGeometry + +/-- +The divergence of a vector field `v : ℝⁿ → ℝⁿ` at point `x`, +computed as the trace of the Jacobian matrix. + +In coordinates: div v = Σᵢ ∂vᵢ/∂xᵢ +-/ +noncomputable +def divergence {n : ℕ} (v : ℝ^n → ℝ^n) (x : ℝ^n) : ℝ := (fderiv ℝ v x).trace ℝ (ℝ^n) + +notation "∇⬝" => divergence + +/-- +A function `f : ℝⁿ → α` is periodic if it is periodic in each coordinate +with period 1, i.e., `f(x + eᵢ) = f(x)` for each unit vector `eᵢ`. +This captures functions on the n-torus ℝⁿ/ℤⁿ. +-/ +def IsPeriodic {α : Sort*} {n : ℕ} (f : ℝ^n → α) : Prop := + ∀ x i, f (x + EuclideanSpace.single i 1) = f x + +/-- +Basic conditions on initial velocity field for the Navier-Stokes equations +in n-dimensional space. + +The initial velocity must be: +- Divergence-free (incompressibility condition: ∇·v₀ = 0) +- Smooth (C^∞) + +These conditions apply regardless of spatial dimension. +-/ +structure InitialVelocityCondition {n : ℕ} (v₀ : ℝ^n → ℝ^n) : Prop where + /-- The initial velocity field is divergence-free (equation 2). + This is the incompressibility constraint for the fluid. -/ + div_free : ∀ x, divergence v₀ x = 0 + /-- The initial velocity field is smooth (C^∞ in all variables) -/ + smooth : ContDiff ℝ ∞ v₀ + +/-- +Initial velocity conditions for the Navier-Stokes problem on all of ℝⁿ. + +In addition to being smooth and divergence-free, the velocity must decay +faster than any polynomial at spatial infinity (condition 4 in Fefferman's paper). + +This condition ensures the velocity field has finite energy and reasonable +behavior as |x| → ∞. +-/ +structure InitialVelocityConditionRn {n : ℕ} (v₀ : ℝ^n → ℝ^n) : Prop + extends InitialVelocityCondition v₀ where + /-- All derivatives of v₀ decay faster than any polynomial (condition 4). + + For any derivative order m and any decay rate K, there exists a constant C + such that |∂ᵐv₀(x)| ≤ C/(1+|x|)^K. -/ + decay : ∀ m : ℕ, ∀ K : ℝ, ∃ C : ℝ, ∀ x, ‖iteratedFDeriv ℝ m v₀ x‖ ≤ C / (1 + ‖x‖)^K + +/-- +Initial velocity conditions for the periodic Navier-Stokes problem on ℝⁿ/ℤⁿ. + +The velocity must be smooth, divergence-free, and periodic with period 1 +in each coordinate (condition 8 in Fefferman's paper). +-/ +structure InitialVelocityConditionPeriodic {n : ℕ} (v₀ : ℝ^n → ℝ^n) : Prop + extends InitialVelocityCondition v₀ where + /-- The initial velocity is periodic with period 1 in each direction (condition 8) -/ + periodic : IsPeriodic v₀ + +/-- +Basic smoothness condition on external forcing term. + +The force f(x,t) must be smooth (C^∞) in both space and time variables for t ≥ 0. +-/ +structure ForceCondition {n : ℕ} (f : ℝ^n → ℝ → ℝ^n) : Prop where + /-- The force is smooth on ℝⁿ × [0,∞) -/ + smooth : ContDiffOn ℝ ∞ (↿f) (Set.univ ×ˢ Set.Ici 0) + +/-- +Force conditions for the Navier-Stokes problem on all of ℝⁿ. + +The force must be smooth and decay faster than any polynomial +in both space and time (condition 5 in Fefferman's paper). +-/ +structure ForceConditionRn {n : ℕ} (f : ℝ^n → ℝ → ℝ^n) : Prop + extends ForceCondition f where + /-- All derivatives of f decay faster than any polynomial in space and time (condition 5). + + For any derivative orders m and decay rate K, there exists C such that + |∂ᵐ_{x,t} f(x,t)| ≤ C/(1+|x|+t)^K for t > 0. + -/ + decay : ∀ m : ℕ, ∀ K : ℝ, ∃ C : ℝ, ∀ x, ∀ t > 0, + ‖iteratedFDeriv ℝ m (↿f) (x,t)‖ ≤ C / (1 + ‖x‖ + t)^K + +/-- +Force conditions for the periodic Navier-Stokes problem on ℝⁿ/ℤⁿ. + +The force must be smooth, periodic in space, and decay in time +(conditions 8-9 in Fefferman's paper). +-/ +structure ForceConditionPeriodic {n : ℕ} (f : ℝ^n → ℝ → ℝ^n) : Prop + extends ForceCondition f where + /-- The force is periodic in space with period 1 for all times (condition 8) -/ + periodic : ∀ t ≥ 0, IsPeriodic (f · t) + /-- All derivatives of f decay faster than any polynomial in time (condition 9). -/ + decay : ∀ m : ℕ, ∀ K : ℝ, ∃ C : ℝ, ∀ x, ∀ t > 0, + ‖iteratedFDeriv ℝ m (↿f) (x,t)‖ ≤ C / (1 + t)^K + +/-- +A solution (v, p) to the Navier-Stokes equations in n-dimensional space +with viscosity nu, initial velocity v₀, and external force f. + +This structure captures the core requirements for a solution: +1. The velocity and pressure satisfy the Navier-Stokes PDE (equation 1) +2. The velocity remains divergence-free for all time (equation 2) +3. The initial condition is satisfied (equation 3) +4. The solution is smooth (C^∞) for all time t ≥ 0 (equations 6, 11) +-/ +structure NavierStokesExistenceAndSmoothness {n : ℕ} + (nu : ℝ) (v₀ : ℝ^n → ℝ^n) (f : ℝ^n → ℝ → ℝ^n) + (v : ℝ^n → ℝ → ℝ^n) (p : ℝ^n → ℝ → ℝ) : Prop where + + /-- The Navier-Stokes equation (equation 1): + ∂v/∂t + (v·∇)v = ν∆v - ∇p + f -/ + navier_stokes : ∀ x, ∀ t > 0, + deriv (v x ·) t + fderiv ℝ (v · t) x (v x t) = nu • Δ (v · t) x - gradient (p · t) x + f x t + + /-- Incompressibility constraint (equation 2): ∇·v = 0 for all x and t ≥ 0. -/ + div_free : ∀ x, ∀ t ≥ 0, ∇⬝ (v · t) x = 0 + + /-- Initial condition (equation 3): v(x,0) = v₀(x) for all x. -/ + initial_condition : ∀ x, v x 0 = v₀ x + + /-- The velocity field is smooth (C^∞) on ℝⁿ × [0,∞) (conditions 6, 11). -/ + velocity_smooth : ContDiffOn ℝ ∞ (↿v) (Set.univ ×ˢ Set.Ici 0) + + /-- The pressure field is smooth (C^∞) on ℝⁿ × [0,∞) (conditions 6, 11) -/ + pressure_smooth : ContDiffOn ℝ ∞ (↿p) (Set.univ ×ˢ Set.Ici 0) + +/-- +A solution to the Navier-Stokes equations on all of ℝⁿ with appropriate +decay and energy bounds. + +In addition to the basic solution properties, we require: +- The velocity is square-integrable at each time (finite kinetic energy) +- The total energy remains bounded for all time (condition 7) +-/ +structure NavierStokesExistenceAndSmoothnessRn {n : ℕ} + (nu : ℝ) (v₀ : ℝ^n → ℝ^n) (f : ℝ^n → ℝ → ℝ^n) + (v : ℝ^n → ℝ → ℝ^n) (p : ℝ^n → ℝ → ℝ) : Prop + extends NavierStokesExistenceAndSmoothness nu v₀ f v p where + + /-- The velocity is square-integrable at each time t ≥ 0. -/ + integrable : ∀ t ≥ 0, Integrable (‖v · t‖^2) + + /-- The kinetic energy ∫|v(x,t)|²dx remains uniformly bounded for all time (condition 7). -/ + globally_bounded_energy : ∃ E, ∀ t ≥ 0, (∫ x : ℝ^n, ‖v x t‖^2) < E + + +/-- +A solution to the Navier-Stokes equations on the n-torus ℝⁿ/ℤⁿ. + +The velocity and pressure must be periodic with period 1 in each +spatial direction for all times (condition 10). +-/ +structure NavierStokesExistenceAndSmoothnessPeriodic {n : ℕ} + (nu : ℝ) (v₀ : ℝ^n → ℝ^n) (f : ℝ^n → ℝ → ℝ^n) + (v : ℝ^n → ℝ → ℝ^n) (p : ℝ^n → ℝ → ℝ) : Prop + extends NavierStokesExistenceAndSmoothness nu v₀ f v p where + + /-- The velocity is periodic in space with period 1 for all times t ≥ 0 (condition 10). -/ + velocity_periodic : ∀ t ≥ 0, IsPeriodic (v · t) + + /-- The pressure is periodic in space with period 1 for all times t ≥ 0 (condition 10). -/ + pressure_periodic : ∀ t ≥ 0, IsPeriodic (p · t) + + +/-- (A) Existence and smoothness of Navier–Stokes solutions on ℝ³. -/ +@[category research open, AMS 35] +theorem navier_stokes_existence_and_smoothness_R3 (nu : ℝ) (hnu : nu > 0) + (v₀ : ℝ³ → ℝ³) (hv₀ : InitialVelocityConditionRn v₀) : + ∃ v p, NavierStokesExistenceAndSmoothnessRn (n:=3) nu v₀ (f:=0) v p := sorry + +/-- (B) Existence and smoothness of Navier–Stokes solutions in ℝ³/ℤ³. -/ +@[category research open, AMS 35] +theorem navier_stokes_existence_and_smoothness_periodic (nu : ℝ) (hnu : nu > 0) + (v₀ : ℝ³ → ℝ³) (hv₀ : InitialVelocityConditionPeriodic v₀) : + ∃ v p, NavierStokesExistenceAndSmoothnessPeriodic (n:=3) nu v₀ (f:=0) v p := sorry + +/-- (C) Breakdown of Navier–Stokes solutions on ℝ³. -/ +@[category research open, AMS 35] +theorem navier_stokes_breakdown_R3 (nu : ℝ) (hnu : nu > 0) : + ∃ (v₀ : ℝ³ → ℝ³) (f : ℝ³ → ℝ → ℝ³), + InitialVelocityConditionRn v₀ ∧ ForceConditionRn f ∧ + ¬(∃ v p, NavierStokesExistenceAndSmoothnessRn (n:=3) nu v₀ f v p) := sorry + +/-- (D) Breakdown of Navier–Stokes Solutions on ℝ³/ℤ³. -/ +@[category research open, AMS 35] +theorem navier_stokes_breakdown_periodic (nu : ℝ) (hnu : nu > 0) : + ∃ (v₀ : ℝ³ → ℝ³) (f : ℝ³ → ℝ → ℝ³), + InitialVelocityConditionPeriodic v₀ ∧ ForceConditionPeriodic f ∧ + ¬(∃ v p, NavierStokesExistenceAndSmoothnessPeriodic (n:=3) nu v₀ f v p) := sorry