feat(Navier-Stokes): formalization of Navier–Stokes existence and smoothness

FormalConjectures/Millenium/NavierStokes.lean

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+/-
+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