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Uniform time-derivative bounds for Galerkin solutions in every H^j, j at most s minus one

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, fix a compact interval $I_{0}$ inside the coefficient time domain, and assume $J \subseteq I_{0}$ . Let $A^{μ}, B$ be a periodic symmetric hyperbolic coefficient family on the open set $O \subseteq R^{m}$ (Periodic first-order symmetric hyperbolic systems: coefficients, admissible values, data classes and solutions), let $K \subseteq O$ be compact and $R \geq 0$ , and let $u_{N} : J \to X_{N}$ be a Galerkin solution of The Galerkin system is a smooth finite-dimensional ODE: local existence, smooth dependence and the continuation criterion for some $N \geq 0$ . Suppose that $u_{N} (t)$ takes values in $K$ and satisfies $∥ u_{N} (t) ∥_{H^{s}} \leq R$ for every $t \in J$ . Then there is $C = C (A^{μ}, B, K, R, s, I_{0})$ such that for every integer $0 \leq j \leq s - 1$ and every $t \in J$ $∥ \partial_{t} u_{N} (t) ∥_{H^{j}} \leq C (1 + ∥ u_{N} (t) ∥_{H^{s}}) .$

Facts & Assumptions

Given: Countable Choice, integer $s \geq 5$ , the coefficient family on $O$ , the compact $K$ , the bound $R$ , a Galerkin solution $u_{N}$ on $J$ with values in $K$ and $∥ u_{N} ∥_{H^{s}} \leq R$ .

[L1]

$P_{N}$ is linear and self-adjoint on $L^{2}$ , commutes with every spatial derivative $D^{α}$ , and is the identity on $X_{N}$ (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, Fourier modes on the periodic three-torus and the Galerkin projection P_N).

[L2]

The Galerkin solution satisfies, at every $t \in J$ , $P_{N} [A^{0} (t, \cdot, u_{N}) \partial_{t} u_{N} + A^{i} (t, \cdot, u_{N}) \partial_{i} u_{N} + B (t, \cdot, u_{N})] = 0$ in $L^{2}$ , $\partial_{t} u_{N} (t) \in X_{N}$ , and $u_{N} \in C^{1} (J; X_{N})$ (The Galerkin system is a smooth finite-dimensional ODE: local existence, smooth dependence and the continuation criterion, Periodic first-order symmetric hyperbolic systems: coefficients, admissible values, data classes and solutions).

[L3]

Uniform positivity of $A^{0}$ on the coefficient set: there is $a > 0$ with $z^{T} A^{0} (t, x, u) z \geq a ∣ z ∣^{2}$ for all $(t, x, u) \in J \times T_{2 π}^{3} \times K$ (Uniform quadratic-form bounds for positive definite symmetric matrices and smoothness of the inverse clause (c)).

[L4]

Composition, product and commutator estimates: for the stated coefficient maps and the bound $∥ u_{N} ∥_{H^{s}} \leq R$ and values in $K$ , $∥ F (t, \cdot, u_{N}) w ∥_{H^{j}} \leq C_{R} ∥ w ∥_{H^{j}}$ for $0 \leq j \leq s - 1$ , and $∥ [D^{α}, A (t, \cdot, u_{N})] w ∥_{2} \leq C_{R} ∥ w ∥_{H^{∣ α ∣ - 1}}$ for $1 \leq ∣ α ∣ \leq s - 1$ (Smooth composition, difference, coefficient product and commutator estimates on bounded H^s data clauses (c),(d)); and composition into $H^{j}$ is bounded by $C_{R} (1 + ∥ u_{N} ∥_{H^{s}})$ for $0 \leq j \leq s$ (clause (a)).

[L5]

Multi-index Leibniz expansion $D^{α} (F w) = \sum_{β \leq α} (\binom{α}{β}) D^{α - β} F D^{β} w$ and the fact that $D^{α} u_{N} \in X_{N}$ for $u_{N} \in X_{N}$ ( $C^{k}$ maps and multi-index derivative notation in Euclidean space, 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 clause (b)).

Proof

technique · direct

1.1L1L2L3L4algebra

(Order zero.) Pair the Galerkin equation [L2] with $\partial_{t} u_{N}$ : since $\partial_{t} u_{N} \in X_{N}$ , self-adjointness [L1] gives $\int (\partial_{t} u_{N})^{T} P_{N} [A^{0} \partial_{t} u_{N}] = \int (\partial_{t} u_{N})^{T} A^{0} \partial_{t} u_{N} \geq a ∥ \partial_{t} u_{N} ∥_{2}^{2}$ by [L3]; the right-hand side is $- \int (\partial_{t} u_{N})^{T} P_{N} [A^{i} \partial_{i} u_{N} + B] \leq ∥ \partial_{t} u_{N} ∥_{2} ∥ A^{i} \partial_{i} u_{N} + B ∥_{2}$ , and $∥ A^{i} \partial_{i} u_{N} + B ∥_{L^{2}} \leq C_{R} (1 + ∥ u_{N} ∥_{H^{s}})$ by [L4] (the composition $A^{i} (t, \cdot, u_{N})$ lies in $H^{s} \subseteq C^{0}$ , so its sup norm is at most $C_{R}$ , and $∥ \partial_{i} u_{N} ∥_{2} \leq ∥ u_{N} ∥_{H^{s}}$ ; the term $B$ is similar). Hence $a ∥ \partial_{t} u_{N} ∥_{2} \leq C_{R} (1 + ∥ u_{N} ∥_{H^{s}})$ .

2.1step 1.1L1L2L3L4L5algebra

(Induction on the order: the estimate at order $j$ .) Fix an integer $j$ with $1 \leq j \leq s - 1$ and suppose the bound holds for all orders below $j$ ; fix $∣ α ∣ = j$ . Applying $P_{N} D^{α}$ to the Galerkin equation [L2] (legitimate by commutation [L1]) and pairing with $D^{α} \partial_{t} u_{N} \in X_{N}$ gives, after moving the leading term and using self-adjointness [L1], $a ∥ D^{α} \partial_{t} u_{N} ∥_{2} \leq ∥ [D^{α}, A^{0} (t, \cdot, u_{N})] \partial_{t} u_{N} ∥_{2} + ∥ D^{α} (A^{i} (t, \cdot, u_{N}) \partial_{i} u_{N} + B (t, \cdot, u_{N})) ∥_{2}$ : indeed $D^{α} (A^{0} \partial_{t} u_{N}) = A^{0} D^{α} \partial_{t} u_{N} + [D^{α}, A^{0}] \partial_{t} u_{N}$ by [L5], and the pairing of $P_{N} D^{α} (A^{0} \partial_{t} u_{N})$ with $D^{α} \partial_{t} u_{N}$ removes $P_{N}$ . The first term is at most $C_{R} ∥ \partial_{t} u_{N} ∥_{H^{j - 1}}$ by [L4], hence at most $C_{R}^{'} (1 + ∥ u_{N} ∥_{H^{s}})$ by the induction hypothesis; the second is at most $C_{R} (1 + ∥ u_{N} ∥_{H^{s}})$ by [L4] applied to the compositions $A^{i} (t, \cdot, u_{N})$ and the product bound, since $∥ \partial_{i} u_{N} ∥_{H^{j}} \leq ∥ u_{N} ∥_{H^{s}}$ . Summing over the finitely many $∣ α ∣ \leq j$ gives the estimate at order $j$ .

3.1step 1.1step 2.1discharge-induction∎

(Conclusion.) The induction of step 2.1, starting from step 1.1, gives the stated bound for every $0 \leq j \leq s - 1$ after enlarging the constant to be independent of $j$ .

Remarks

Role. The bound at order $s - 1$ controls the time commutator in Equivalence of the weighted Galerkin energy with the H^s norm and its Gronwall inequality and gives the equicontinuity of the Galerkin family in Compactness of H^r-bounded families in H^{r-1} and locally uniform H^{s-1} subsequences of Galerkin solutions; all constants are independent of $N$ .

Depends on

Dependency tree · 2 levels

Sources

H. Friedrich and A. D. Rendall, The Cauchy Problem for the Einstein Equations, Living Reviews in Relativity 3 (2000), article 1 (arXiv:gr-qc/0002074), Section 3.1 (Symmetric Hyperbolic Systems): the definition of a quasilinear symmetric hyperbolic system, the finite-dimensional mollifier approximation (3.1) solved by ODE theory, the energy and Moser product/composition estimates, the higher-order energy inequality, uniqueness, the Rellich/Ascoli compactness step and the passage to a classical solution, with Remarks 1-6 on smooth persistence, the continuation criterion, Cauchy stability and the non-identity time matrix