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The Galerkin system is a smooth finite-dimensional ODE: local existence, smooth dependence and the continuation criterion

Statement

Assume the Axiom of Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $s \geq 5$ be an integer, let $O \subseteq R^{m}$ be open, let $A^{μ}, B$ be a periodic symmetric hyperbolic coefficient family on $O$ over an open interval $I ∋ 0$ (Periodic first-order symmetric hyperbolic systems: coefficients, admissible values, data classes and solutions), let $u_{0} \in H^{s} (T_{2 π}^{3}; R^{m})$ take values in a compact $K \subseteq O$ with $∥ u_{0} ∥_{H^{s}} \leq R$ , and let $P_{N}$ , $X_{N}$ be the Galerkin projection and space (Fourier modes on the periodic three-torus and the Galerkin projection P_N). Fix $N \geq 0$ for which $P_{N} u_{0}$ takes values in $O$ . Then:

(a) there is an open interval $J_{N} ∋ 0$ and a $C^{1}$ map $u_{N} : J_{N} \to X_{N}$ such that for all $t \in J_{N}$ $P_{N} [A^{0} (t, \cdot, u_{N} (t)) \partial_{t} u_{N} (t) + \sum_{i = 1}^{3} A^{i} (t, \cdot, u_{N} (t)) \partial_{i} u_{N} (t) + B (t, \cdot, u_{N} (t))] = 0 in L^{2},$ with $u_{N} (0) = P_{N} u_{0}$ ;

(b) [continuation criterion] if $[0, T_{*})$ is the forward-maximal interval of existence of $u_{N}$ and there are a compact $C \subseteq O$ and a constant $R_{1} \geq 0$ with $u_{N} (t)$ taking values in $C$ and $∥ u_{N} (t) ∥_{H^{s}} \leq R_{1}$ for all $t \in [0, T_{*})$ , and $T_{*} \in I$ , then $u_{N}$ does not have $T_{*}$ as a finite forward endpoint: the solution extends beyond $T_{*}$ ;

(c) the solution is unique and its local flow depends smoothly on the finite-dimensional coordinates of the initial data.

Facts & Assumptions

Given: Countable Choice, integer $s \geq 5$ , the coefficient family on $O$ , the data $u_{0}$ with values in the compact $K$ and $∥ u_{0} ∥_{H^{s}} \leq R$ , and $N \geq 0$ with admissible $P_{N} u_{0}$ .

[L1]

$X_{N}$ is a finite-dimensional real vector space of dimension $d_{N} = m ∣ Λ_{N} ∣$ , and $P_{N}$ is linear, self-adjoint, idempotent with range $X_{N}$ and commutes with all spatial derivatives (Fourier modes on the periodic three-torus and the Galerkin projection P_N, The Galerkin projection is self-adjoint, commutes with derivatives, is a contraction in every H^r, converges to the identity and has the sharp Fourier-tail bound).

[L2]

Uniform positivity: for every compact $C \subseteq O$ and every compact time interval $I_{0}$ with $I_{0} \subseteq I$ there is $a > 0$ with $z^{T} A^{0} (t, x, u) z \geq a ∣ z ∣^{2}$ for all $z \in R^{m}$ , $(t, x, u) \in I_{0} \times T_{2 π}^{3} \times C$ (Uniform quadratic-form bounds for positive definite symmetric matrices and smoothness of the inverse clause (c)).

[L3]

Picard--Lindelöf: for $G$ continuous and locally Lipschitz on an open subset of $R \times R^{d}$ , every initial point has a unique local solution; the solution map is smooth when $G$ is smooth (Picard-Lindelöf local existence and uniqueness for first-order systems, Smooth dependence of ODE solutions on parameters, $C^{k}$ Euclidean maps and diffeomorphisms).

[L4]

The coefficient evaluations $t \mapsto A^{μ} (t, \cdot, π (z))$ and $B (t, \cdot, π (z))$ are smooth functions of $(t, z)$ for $z$ in the open set $V : = {z : π (z) takes values in O}$ , and the map $z \mapsto P_{N} [F (\cdot, π (z)) v]$ is smooth for fixed smooth $F$ and $v \in X_{N}$ , because all maps involved act between finite-dimensional spaces and $P_{N}$ is linear (Smooth composition, difference, coefficient product and commutator estimates on bounded H^s data, $C^{k}$ Euclidean maps and diffeomorphisms).

[L5]

Closed bounded Euclidean sets are compact, continuous real functions on compact metric spaces are bounded, and a $C^{1}$ finite-dimensional curve satisfies its norm-valued integral identity (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line, A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value, Fundamental theorem of calculus for Banach-valued continuous curves).

Proof

technique · direct

1.1L1L2L4algebra

(The finite-dimensional form of the Galerkin system.) Fix a basis of $X_{N}$ and let $π : R^{d_{N}} \to X_{N}$ be the coordinate isomorphism. Define $V : = {z \in R^{d_{N}} : π (z) (x) \in O \forall x}$ , an open set containing the coordinates of $P_{N} u_{0}$ , and for $z \in V$ and $v \in X_{N}$ define $M (t, z) v : = P_{N} [A^{0} (t, \cdot, π (z)) v]$ . For $v \neq 0$ , self-adjointness [L1] gives $⟨ v, M (t, z) v ⟩_{L^{2}} = \int v^{T} A^{0} (t, \cdot, π (z)) v d μ \geq a_{z} ∥ v ∥_{2}^{2}$ with $a_{z} > 0$ by [L2] applied to a compact set containing the values of $π (z)$ ; on compact subsets of $V$ the bound is uniform, so $M (t, z)$ is invertible with smooth inverse there by Uniform quadratic-form bounds for positive definite symmetric matrices and smoothness of the inverse clause (d). Define $G (t, z) : = π^{- 1} ⁣ (M (t, z)^{- 1} (- P_{N} [\sum_{i} A^{i} (t, \cdot, π (z)) \partial_{i} π (z) + B (t, \cdot, π (z))]))$ , which is smooth in $(t, z)$ on $I \times V$ by [L4] and the smoothness of the inverse; then $u_{N} (t) = π (z (t))$ solves the Galerkin system iff $z^{'} (t) = G (t, z (t))$ , and $z (0)$ is the coordinate of $P_{N} u_{0}$ .

2.1step 1.1L3

(Local existence and smooth dependence (a), (c).) Since $G$ is continuous and locally Lipschitz on the open set $I \times V$ and the initial point lies in it, Picard--Lindelöf [L3] gives a unique local solution through the initial point, defined on an open interval $J_{N} ∋ 0$ ; smoothness of $G$ and the smooth-dependence theorem [L3] give the smooth finite-dimensional flow, hence (c), and (a) follows by transporting through $π$ .

3.1step 1.1step 2.1L3L5algebra∎

(Continuation within the coefficient time domain.) Suppose $T_{*} \in I$ is finite and the stated value and norm bounds hold. The coordinate set $Q$ of fields in $X_{N}$ whose values lie in compact $C \subseteq O$ and whose $H^{s}$ norms are at most $R_{1}$ is closed and bounded in a finite-dimensional space, hence compact, and it lies inside $V$ . Choose $δ > 0$ with $[T_{*} - δ, T_{*} + δ] \subseteq I$ . The smooth field $G$ is bounded by some $L$ on this compact time interval times $Q$ . Thus for $T_{*} - δ < t < t^{'} < T_{*}$ the equation and the ordinary fundamental theorem of calculus give $∣ z (t^{'}) - z (t) ∣ \leq L ∣ t^{'} - t ∣$ . The coordinate curve has a limit $z_{*} \in Q$ as $t ↑ T_{*}$ , by completeness of Euclidean space and closedness of $Q$ . Extend it to $z (T_{*}) = z_{*}$ . Continuity of $G$ gives $G (t, z (t)) \to G (T_{*}, z_{*})$ , so its integral identity extends to $T_{*}$ and the extended curve is $C^{1}$ there from the left. Picard--Lindelöf [L3] at the interior point $(T_{*}, z_{*})$ supplies a local solution on both sides of $T_{*}$ . The original curve and this local solution agree on their overlap by uniqueness, hence glue to a solution beyond $T_{*}$ . This contradicts maximality and proves (b). The backward criterion follows by reversing time.

The Galerkin system is a smooth finite-dimensional ODE: local existence, smooth dependence and the continuation criterion

Statement

Facts & Assumptions

Proof

Depends on

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