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Variable-coefficient parabolic residual and boundary stability

Statement

Let $U \subset R^{d}$ be nonempty, bounded, and open, $T > 0$ , and $P [w] = - w_{t} + \sum_{i, j = 1}^{d} a_{i j} (x, t) D_{i} D_{j} w + \sum_{j = 1}^{d} {\tilde{b}}_{j} (x, t) D_{j} w + \tilde{c} (x, t) w$ on $Q = U \times (0, T)$ . Suppose $a (x, t)$ is symmetric, $ξ^{T} a (x, t) ξ \geq λ_{0} ∣ ξ ∣^{2}$ with $λ_{0} > 0$ , $\tilde{b}, \tilde{c}$ are bounded and continuous, and $\tilde{c} \leq 0$ . There is $C_{P} < \infty$ , depending only on $U, T, λ_{0}$ and the coefficient bounds, such that every $w \in C^{2, 1} (Q) \cap C^{0} (\overset{ˉ}{U} \times [0, T])$ with $P [w] \in C^{0} (Q)$ satisfies $∥ w ∥_{C^{0} (Q)} \leq C_{P} (∥ P [w] ∥_{C^{0} (Q)} + ∥ w ∥_{C^{0} (Γ_{T})}),$ where $Γ_{T} = (\overset{ˉ}{U} \times {0}) \cup (\partial U \times [0, T])$ excludes the final-time face.

Statement references

Shin et al., Appendix F, Lemma F.2: parabolic residual stability estimate

Shin et al., Appendix F, proof of Lemma F.2

Facts & Assumptions

Given: The bounded cylinder, uniformly elliptic coefficient matrix, bounded drift and zero-order coefficients with $\tilde{c} \leq 0$ , and the regularity and data of $w$ in the Statement.

[F1]

The parabolic operator has the sign convention $P = - \partial_{t} + L_{t}$ and its prescribed boundary is initial plus lateral data (Shin's parabolic operator, data, and initial-lateral boundary).

[F2]

On the final-time face the time derivative is the left derivative; at a final-time maximum it is nonnegative (Parabolic cylinder and parabolic boundary).

Proof

Proof strategy: use a spatial exponential barrier and the weak parabolic maximum principle.

1.1givenF1algebrachoose

Choose $a_{1} < b_{1}$ with $a_{1} \leq x_{1} \leq b_{1}$ on $\overset{ˉ}{U}$ . Let $B = ∥ {\tilde{b}}_{1} ∥_{\infty}$ , $C_{0} = ∥ \tilde{c} ∥_{\infty}$ , and choose $κ > 0$ so that $κ^{2} λ_{0} - κ B - C_{0} \geq 1$ . For $ψ (x, t) = e^{κ x_{1}}$ , [F1] gives $P [ψ] = e^{κ x_{1}} (κ^{2} a_{11} + κ {\tilde{b}}_{1} + \tilde{c}) \geq η : = e^{κ a_{1}} > 0$ . Choose $M > e^{κ b_{1}}$ and put $Φ = M - ψ$ . Then $Φ > 0$ and $P [Φ] = M \tilde{c} - P [ψ] \leq - η$ .

2.1givenstep 1.1algebra

Let $R = ∥ P [w] ∥_{\infty}$ , $Q_{0} = ∥ w ∥_{C^{0} (Γ_{T})}$ , and $z = w - Q_{0} - (R / η) Φ$ . On $Γ_{T}$ , $z \leq 0$ , and in $Q$ , $P [z] = P [w] - Q_{0} \tilde{c} - (R / η) P [Φ] \geq - R + R = 0.$

3.1step 1.1step 2.1F2contradiction

Fix any $0 < τ < T$ and work on $\overset{ˉ}{U} \times [0, τ]$ , where the time derivatives at spatial interior points at $t = τ$ exist because $τ < T$ . Suppose $z > 0$ somewhere on this stopped cylinder. Since $z \leq 0$ on $Γ_{T}$ , a positive maximum occurs at a spatial interior point and at a time in $(0, τ]$ . For sufficiently small $δ > 0$ , $z + δ ψ$ still has a positive maximum away from $Γ_{T}$ . At this maximum, the spatial gradient is zero and the spatial Hessian is negative semidefinite; the time derivative is zero for $t < τ$ and nonnegative from the left if $t = τ$ . Thus [F2] and $\tilde{c} \leq 0$ imply $P [z + δ ψ] \leq 0$ , while step 2.1 and step 1.1 give $P [z + δ ψ] > 0$ . This contradiction shows $z \leq 0$ on every stopped cylinder; let $τ ↑ T$ and use continuity at the final face. Hence $w \leq Q_{0} + (R / η) ∥ Φ ∥_{\infty}$ .

4.1step 3.1algebra∎

Apply the same argument to $- w$ , for which $P [- w] = - P [w]$ and the boundary norm is still $Q_{0}$ . Combining the upper and lower bounds gives the asserted estimate with $C_{P} = \max {1, ∥ Φ ∥_{\infty} / η}$ .

Depends on

Used by

Parabolic PINN minimizers converge to the initial-boundary solution Theorem

Dependency tree · two levels

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Sources

Shin, Darbon, and Karniadakis, On the Convergence of Physics-Informed Neural Networks for Linear Second-Order Elliptic and Parabolic Type PDEs (standard reference, not scraped)
John K. Hunter, Notes on Partial Differential Equations (standard reference, not scraped)