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Equivalence of the weighted Galerkin energy with the H^s norm and its Gronwall inequality

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 let $J \subseteq I_{0}$ be an interval on which the Galerkin solution is defined. Let $A^{μ}, B$ be a periodic symmetric hyperbolic coefficient family on the open set $O \subseteq R^{m}$ , let $K \subseteq O$ be compact and $R \geq 0$ , and let $u_{N} : J \to X_{N}$ be a Galerkin solution (The Galerkin system is a smooth finite-dimensional ODE: local existence, smooth dependence and the continuation criterion) with $u_{N} (t)$ taking values in $K$ and $∥ u_{N} (t) ∥_{H^{s}} \leq R$ for all $t \in J$ . Define the weighted energy $e (t) : = \sum_{∣ α ∣ \leq s} \int_{T_{2 π}^{3}} (D^{α} u_{N} (t))^{T} A^{0} (t, x, u_{N} (t)) (D^{α} u_{N} (t)) d μ (x) .$ Then:

(a) there are $0 < a \leq b$ , depending on $A^{0}$ , $K$ , $s$ and $I_{0}$ , with $a ∥ u_{N} (t) ∥_{H^{s}}^{2} \leq e (t) \leq b ∥ u_{N} (t) ∥_{H^{s}}^{2}$ for all $t \in J$ ;

(b) $e$ is $C^{1}$ on $J$ and there is $C = C (A^{μ}, B, K, R, s, I_{0})$ with $∣ e^{'} (t) ∣ \leq C (1 + e (t)) (t \in J) .$ Consequently $e (t) \leq (1 + e (t_{0})) e^{C ∣ t - t_{0} ∣}$ for $t, t_{0} \in J$ by Gronwall's inequality.

Facts & Assumptions

Given: Countable Choice, $s \geq 5$ , 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]

Uniform bounds for $A^{0}$ : there are $0 < a \leq b$ with $a ∣ z ∣^{2} \leq z^{T} A^{0} (t, x, u) z \leq b ∣ z ∣^{2}$ 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)).

[L2]

The Galerkin equation $P_{N} [A^{0} (t, \cdot, u_{N}) \partial_{t} u_{N} + A^{i} (t, \cdot, u_{N}) \partial_{i} u_{N} + B (t, \cdot, u_{N})] = 0$ holds in $L^{2}$ ; $P_{N}$ is self-adjoint, commutes with $D^{α}$ and acts as the identity on $X_{N}$ (The Galerkin system is a smooth finite-dimensional ODE: local existence, smooth dependence and the continuation criterion, 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).

[L3]

The homogeneous order- $s$ commutator bounds give $∥ [D^{α}, A^{μ} (u_{N})] w ∥_{2} \leq C_{R} ∥ w ∥_{H^{s - 1}}$ for $∣ α ∣ \leq s$ ; the time derivative estimate gives $∥ \partial_{t} u_{N} ∥_{H^{s - 1}} \leq C_{R} (1 + ∥ u_{N} ∥_{H^{s}})$ , and composition gives $∥ B (u_{N}) ∥_{H^{s}} \leq C_{R} (1 + ∥ u_{N} ∥_{H^{s}})$ (Smooth composition, difference, coefficient product and commutator estimates on bounded H^s data, Uniform time-derivative bounds for Galerkin solutions in every H^j, j at most s minus one). The leading $A^{i} \partial_{i} D^{α} u_{N}$ term is integrated by parts, not bounded separately in $L^{2}$ .

[L4]

Write ${\dot{A}}^{0} = \partial_{t} A^{0} (t, x, u_{N}) + \partial_{u} A^{0} (t, x, u_{N}) [\partial_{t} u_{N}]$ , the total coefficient derivative. It is uniformly bounded in $L^{\infty}$ by the time derivative bound and embedding. The spatial coefficient derivatives have the same boundedness. Moreover $e$ is continuously differentiable: $u_{N} \in C^{1} (J; X_{N})$ , the coefficients are smooth and $t \mapsto A^{0} (t, x, u_{N} (t))$ is $C^{1}$ uniformly on the compact torus, so differentiation under the finite sum and the integral is legitimate, giving $e^{'} (t) = \sum_{∣ α ∣ \leq s} \int [2 (D^{α} \partial_{t} u_{N})^{T} A^{0} D^{α} u_{N} + (D^{α} u_{N})^{T} ({\dot{A}}^{0}) D^{α} u_{N}] d μ$ (The Galerkin system is a smooth finite-dimensional ODE: local existence, smooth dependence and the continuation criterion, The 2 pi-periodic three-torus, its normalized measure, test functions and the Sobolev spaces H^s).

[L5]

Young's inequality $2 ∣ a b ∣ \leq ∣ a ∣^{2} + ∣ b ∣^{2}$ and the elementary estimate $(1 + ∥ u_{N} ∥_{H^{s}})^{2} \leq 2 (1 + ∥ u_{N} ∥_{H^{s}}^{2})$ ; and $∥ u_{N} ∥_{H^{s}}^{2} \leq e / a$ by [L1].

Proof

technique · direct

1.1L1algebra

(Equivalence (a).) For each $∣ α ∣ \leq s$ the integrand $(D^{α} u_{N})^{T} A^{0} D^{α} u_{N}$ lies between $a ∣ D^{α} u_{N} ∣^{2}$ and $b ∣ D^{α} u_{N} ∣^{2}$ pointwise by [L1]; integrating and summing over $α$ gives $a ∥ u_{N} ∥_{H^{s}}^{2} \leq e \leq b ∥ u_{N} ∥_{H^{s}}^{2}$ .

1.2L2L4algebra

(Substituting the Galerkin equation into $e^{'}$ .) For each $∣ α ∣ \leq s$ use $D^{α} u_{N} \in X_{N}$ and self-adjoint commutativity [L2]: $\int (D^{α} u_{N})^{T} A^{0} D^{α} \partial_{t} u_{N} = \int (D^{α} u_{N})^{T} D^{α} (A^{0} \partial_{t} u_{N}) - \int (D^{α} u_{N})^{T} [D^{α}, A^{0}] \partial_{t} u_{N}$ and $\int (D^{α} u_{N})^{T} P_{N} D^{α} (A^{0} \partial_{t} u_{N}) = - \int (D^{α} u_{N})^{T} D^{α} (A^{i} \partial_{i} u_{N} + B)$ by the Galerkin equation and self-adjointness. Substituting into [L4] and splitting $D^{α} (A^{i} \partial_{i} u_{N}) = A^{i} D^{α} \partial_{i} u_{N} + [D^{α}, A^{i}] \partial_{i} u_{N}$ , the symmetric term integrates by parts on the torus: $\int (D^{α} u_{N})^{T} A^{i} D^{α} \partial_{i} u_{N} = - \frac{1}{2} \int (D^{α} u_{N})^{T} (\partial_{i} A^{i}) (D^{α} u_{N})$ . Hence $e^{'} (t) = \sum_{∣ α ∣ \leq s} [- 2 \int (D^{α} u_{N})^{T} [D^{α}, A^{0}] \partial_{t} u_{N} - 2 \int (D^{α} u_{N})^{T} [D^{α}, A^{i}] \partial_{i} u_{N} + \int (D^{α} u_{N})^{T} (\partial_{i} A^{i}) D^{α} u_{N} - 2 \int (D^{α} u_{N})^{T} D^{α} B + \int (D^{α} u_{N})^{T} ({\dot{A}}^{0}) D^{α} u_{N}]$ .

2.1step 1.1step 1.2L3L5algebra

(Bounding the terms by $1 + e$ .) Each of the five sums is bounded using Cauchy--Schwarz and [L3]: $∣ \int (D^{α} u_{N})^{T} [D^{α}, A^{0}] \partial_{t} u_{N} ∣ \leq ∥ D^{α} u_{N} ∥_{2} C_{R} ∥ \partial_{t} u_{N} ∥_{H^{s - 1}} \leq C_{R} (1 + ∥ u_{N} ∥_{H^{s}}) ∥ D^{α} u_{N} ∥_{2}$ ; similarly for the spatial commutators with $∥ \partial_{i} u_{N} ∥_{H^{s - 1}} \leq ∥ u_{N} ∥_{H^{s}}$ ; the $\partial_{i} A^{i}$ and ${\dot{A}}^{0}$ terms are bounded by a constant times $∥ D^{α} u_{N} ∥_{2}^{2}$ ; and $∣ \int (D^{α} u_{N})^{T} D^{α} B ∣ \leq C_{R} (1 + ∥ u_{N} ∥_{H^{s}}) ∥ D^{α} u_{N} ∥_{2}$ . Summing over $∣ α ∣ \leq s$ , using Young's inequality [L5] and $∥ u_{N} ∥_{H^{s}}^{2} \leq a^{- 1} e$ from step 1.1, gives $∣ e^{'} ∣ \leq C (1 + e)$ with $C = C (A^{μ}, B, K, R, s, I_{0})$ .

3.1step 2.1L4∎

(Gronwall consequence.) By step 1.1 and step 2.1, $e (t) \leq (1 + e (t_{0})) e^{C ∣ t - t_{0} ∣}$ on every interval where the bounds hold, by Gronwall's integral inequality (Gronwall's integral inequality with variable and constant coefficients); this uses only $∣ e^{'} ∣ \leq C (1 + e)$ and the differentiability of $e$ from [L4].

Remarks

Role. The equivalence (a) turns the a priori bound on $e$ into the $H^{s}$ bound assumed in Compactness of H^r-bounded families in H^{r-1} and locally uniform H^{s-1} subsequences of Galerkin solutions and in the uniform-existence clause of Local existence of Sobolev and classical solutions of a periodic symmetric hyperbolic system, uniform on bounded H^s data.

Equivalence of the weighted Galerkin energy with the H^s norm and its Gronwall inequality

Statement

Facts & Assumptions

Proof

Remarks

Depends on

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Sources