Source-linked AI summary
An Inverse Problem for Determining the Piston Speed from a Given Lipschitz Leading Shock
Gui-Qiang G. Chen, Qianfeng Li, Yun Pu, Yongqian Zhang
TL;DR
Given initial shock-tube data and a prescribed Lipschitz leading shock, the paper seeks the piston speed and the intervening flow. It constructs approximate solutions under an Oleinik-type condition and establishes an entropy solution from which the piston speed is determined.
Problem
The paper asks how to determine piston speed and the associated flow field from initial data and a prescribed leading shock.
Source: §1 · pp. 1–2Method
It uses a forward-backward Riemann-type construction with an Oleinik-type condition on the prescribed shock to build approximate solutions.
Source: §1 · p. 4, §3.1 · p. 11Results
Under the stated shock-speed condition, the constructed flow forms an entropy solution between the piston and leading shock, with piston speed given by the boundary velocity.
Source: §1 · p. 3Takeaways & Limitations
The construction provides a way to recover piston speed from the flow associated with a prescribed leading shock.
Source: §1 · p. 2The entropy construction relies on an Oleinik-type condition requiring s + q to increase monotonically for a suitably slowly increasing q.
Source: §1 · p. 3
Abstract
from arXiv · showhide
We analyze an inverse problem for determining the piston speed and the associated flow field from a prescribed leading shock and the initial data in a shock tube. The gas flow is described by the isentropic Euler equations (i.e., the $p$-system), while the trajectory of the leading shock is prescribed as a given Lipschitz curve. Under an Ole\u{i}nik-type entropy condition on the leading shock, we develop a modified wavefront tracking scheme to construct the flow field behind the shock. This construction enables us to determine the corresponding piston speed and the associated flow field.
1. Introduction
The paper asks how to recover piston speed and the flow between piston and shock from initial data and a prescribed leading shock. It develops a construction for entropy solutions under an Oleinik-type condition on the shock.
1. Introduction
- The main theorem gives a unique state U behind the initial leading shock such that the piecewise profile with U0 ahead of the shock forms an entropy solution.The profile uses U for 0 < x < s0t and the initial state U0 for x > s0t.Source: §1 · p. 2
- For leading-shock speeds satisfying the stated hypotheses, the theorem also constructs an entropy solution between the piston and shock and identifies piston speed as the boundary velocity u(0, ·).The flow is U within the shock curve and U0 beyond it.Source: §1 · p. 3
- A modified wavefront tracking algorithm constructs approximate solutions to the boundary while determining the approximate piston velocity through the non-reflection boundary condition.The method assumes sufficiently small total variation of the prescribed leading shock speed and solves forward-backward Riemann-type problems along lines orthogonal to the time-marching direction.Source: §1 · p. 3
- The inverse problem has mixed characteristic directions, and backward waves from convex shock portions may converge, threatening entropy admissibility of the limit.Unlike in, the Rankine-Hugoniot conditions couple and fix the Riemann invariants behind the shock, restricting the available degrees of freedom.Source: §1 · pp. 3–4
- The Oleinik-type condition that s + q be monotonically increasing prevents same-family waves from intersecting and keeps backward rarefaction strengths small enough for the limiting solution to satisfy the Lax entropy inequality.Here q is suitably slowly increasing, and the condition restricts the curvature of convex portions of the leading shock profile.Source: §1 · pp. 3–4
2. Wave Curves and Forward-Backward Riemann-Type Problems
Section 2 reviews the system’s wave curves, studies forward-backward Riemann-type problems, and derives wave-interaction estimates for use throughout the paper.
2.1. Wave curves.
- Because p'(v)<0, system (1.1) is strictly hyperbolic; the section recalls rarefaction curves and defines shock curves through Rankine–Hugoniot conditions.Source: §2.1 · pp. 4–5
2.2. Strong shocks and the background solutions.
- For the specified shock speed s0, a unique state U is connected to U0 by a 2-shock.Source: §2.2 · p. 6
2.3. Weak waves and forward-backward Riemann-type problems.
- A forward-backward Riemann-type problem prescribes Ub and Ua on opposite time sides of a line in the λ♮ direction.Source: §2.3 · p. 7
- For nearby states Ua and Ub, Lemma 2.3 establishes solvability of the weak-wave equation using the implicit function theorem.Source: §2.3 · p. 8
2.4. Wave interaction estimates.
- The interaction analysis states that the state behind the leading shock is controlled by shock speed and gives a local result for nearby speeds.Source: §2.4 · p. 9
- When s1<s2, β1>0 and β2<0; when s1>s2, β1<0 and β2>0.Source: §2.4 · p. 10
3. A Modified Wavefront Tracking Algorithm
The section constructs approximate solutions with modified wavefront tracking, solving forward-backward Riemann-type problems to determine both the flow and piston speed. It approximates rarefactions with piecewise-constant states and the prescribed shock with piecewise-linear trajectories.
3. A Modified Wavefront Tracking Algorithm
- The modified wavefront tracking construction solves forward-backward Riemann-type problems and applies the boundary condition iteratively to approximate the flow and piston speed.Wavefronts propagate in the nL direction and generate new problems when they interact with another front or the piston boundary.Source: §3 · p. 11
3.1. Accurate Riemann solver.
- The accurate Riemann solver approximates rarefaction waves by piecewise-constant states separated by rarefaction fronts.The solver accommodates both shock waves and rarefaction fronts.Source: §3.1 · p. 11, §3 · p. 11
3.2. Construction of the approximate solutions.
- The prescribed leading shock is approximated by a piecewise-linear trajectory, and its approximate speed converges to the prescribed speed almost everywhere as Δt tends to zero.The construction connects sampled shock positions with line segments; the passage states almost-everywhere convergence of the approximating shock speed.Source: §3.2 · pp. 12, 14
- Waves of the same family do not interact, so the construction needs neither wave-speed adjustments nor non-physical waves.At the piston boundary, impinging waves are absorbed without generating reflected wavefronts.Source: §3.2 · p. 16
4. Validity of the Extension
The section establishes that approximate solutions can be extended by controlling their total variation and preventing same-family wavefront intersections. A non-increasing Glimm-type functional supports the bounds used in this construction.
4. Validity of the Extension
- A non-increasing Glimm-type functional yields a uniform total-variation bound, while separation of same-family wavefronts permits extension of the approximate solution.The extension argument also shows that same-family wavefronts do not interact in the region under consideration.Source: §4 · p. 16
4.1. Glimm-type functionals.
- When there are no irregular interaction points, the number of wavefronts remains finite and the functional satisfies F(τ+) ≤ F(0), keeping the approximate solution in Oε0(U).The stated bound assumes F(0) < εg and a sufficiently small ϵinv.Source: §4.1 · p. 18
4.2. Wavefronts of the same family are separated.
- For suitably small Λ and ϵinv, both 1-wavefronts and 2-wavefronts remain separated, so same-family intersection points are absent.The section establishes nonintersection for the two families in Propositions 4.11 and 4.12.Source: §4.2 · pp. 19, 21, 24
5. Proof of the Main Theorem
This section assembles wave-strength and piston-speed bounds with subsequential convergence results. It uses these results to prove the main theorem, with details omitted.
5. Proof of the Main Theorem
- Every rarefaction front in the approximate solution satisfies a bound with a constant Cra independent of δ.The section also states that the approximate piston speeds are bounded.Source: §5 · p. 27
- The approximate solutions and piston speeds have convergent subsequences, with the piston speeds converging to up ∈ BV(R+).The passage identifies this convergence as part of Proposition 5.3.Source: §5 · p. 28
- The proof of the main theorem is obtained using Lemmas 5.1 and 5.2 and Proposition 5.3, together with arguments like those in [7, §7.4].The authors omit the proof details.Source: §5 · p. 28
Appendix A. A lemma for interaction estimates
Appendix A gives an interaction-estimate lemma for a twice-differentiable function with Lipschitz-continuous second derivatives. It expresses the function through its restrictions to the coordinate axes and bounds the remainder.
Appendix A. A lemma for interaction estimates
- The lemma assumes Ψ is twice differentiable and has Lipschitz-continuous second derivatives.These are the stated regularity conditions for the estimate.Source: Appendix A · p. 28
- Ψ(x, y) is decomposed as Ψ(0, y) + Ψ(x, 0) − Ψ(0, 0), plus a remainder of order O(1)(|x| + |y|)|x||y|.The expansion separates the contributions along the two coordinate axes and subtracts the shared origin value.Source: Appendix A · p. 28
- The O(1) remainder constant depends only on the Lipschitz constant for Ψ's second derivatives.This dependence is specified alongside the expansion.Source: Appendix A · p. 28