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arxiv: 2604.00573 · v2 · submitted 2026-04-01 · 🧮 math.AP · cs.SY· eess.SY

Verifying Well-Posedness of Linear PDEs using Convex Optimization

Declan S. Jagt , Matthew M. Peet This is my paper

Pith reviewed 2026-05-13 22:38 UTC · model grok-4.3

classification 🧮 math.AP cs.SYeess.SY
keywords well-posednesslinear PDEsLumer-Phillips theoremconvex optimizationPartial Integral Equationexponential growth rateparabolic PDEshyperbolic PDEs
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The pith

Reformulating Lumer-Phillips conditions via PIE turns PDE well-posedness tests into convex optimization problems on L2.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that the Lumer-Phillips theorem gives necessary and sufficient conditions for well-posedness of dissipative linear PDEs, but these conditions are difficult to check because they apply only on the PDE domain, a proper subspace of L2. By switching to the Partial Integral Equation representation, which supplies a bijection between a fundamental state in L2 and the original PDE domain, the same conditions become operator inequalities defined everywhere on L2. These inequalities can be checked with convex optimization, which also returns the smallest number that bounds the exponential growth rate of solutions from above. The method is illustrated on standard parabolic and hyperbolic examples, confirming that well-posedness can be verified without working directly on the domain.

Core claim

Using the PIE representation, which introduces a fundamental state in the Hilbert space L2 and provides a bijection between this state space and the PDE domain, the Lumer-Phillips conditions are reformulated as operator inequalities on L2. These inequalities are then tested using convex optimization methods, establishing a least upper bound on the exponential growth rate of solutions. The approach is shown to work for several classical parabolic and hyperbolic PDEs.

What carries the argument

The Partial Integral Equation (PIE) representation, which supplies a bijection between a fundamental state in L2 and the PDE domain so that domain-restricted Lumer-Phillips conditions become L2 operator inequalities verifiable by convex optimization.

If this is right

  • Well-posedness and growth-rate bounds become computable for PDEs whose domains are hard to characterize directly.
  • The same convex program can be reused for families of PDEs that differ only by parameters, giving explicit stability margins.
  • Verification becomes available before any control design or numerical simulation step is attempted.
  • The approach covers both parabolic and hyperbolic linear PDEs without requiring separate analytic arguments for each case.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method could be embedded in automated modeling tools so that engineers obtain a well-posedness certificate as soon as a PDE is written down.
  • Because the test is phrased as a convex program, it may combine directly with linear matrix inequality techniques already used in finite-dimensional control.
  • If the bijection can be constructed for certain nonlinear or time-varying PDEs, the same optimization framework might extend beyond the linear dissipative setting treated here.

Load-bearing premise

The bijection between the fundamental state in L2 and the PDE domain preserves every property needed for the reformulated operator inequalities on L2 to be exactly equivalent to the original Lumer-Phillips conditions.

What would settle it

A specific linear PDE for which the convex optimization returns a finite growth-rate bound yet direct solution of the original PDE on its domain produces solutions that grow faster than that bound or fail to exist.

read the original abstract

Ensuring that a PDE model is well-posed is a necessary precursor to any form of analysis, control, or numerical simulation. Although the Lumer-Phillips theorem provides necessary and sufficient conditions for well-posedness of dissipative PDEs, these conditions must hold only on the domain of the PDE -- a proper subspace of $L_{2}$ -- which can make them difficult to verify in practice. In this paper, we show how the Lumer-Phillips conditions for PDEs can be tested more conveniently using the equivalent Partial Integral Equation (PIE) representation. This representation introduces a fundamental state in the Hilbert space $L_{2}$ and provides a bijection between this state space and the PDE domain. Using this bijection, we reformulate the Lumer-Phillips conditions as operator inequalities on $L_{2}$. We show how these inequalities can be tested using convex optimization methods, establishing a least upper bound on the exponential growth rate of solutions. We demonstrate the effectiveness of the proposed approach by verifying well-posedness for several classical examples of parabolic and hyperbolic PDEs.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper claims that the Lumer-Phillips conditions for well-posedness of dissipative linear PDEs can be equivalently reformulated as operator inequalities on the full L2 space by using the Partial Integral Equation (PIE) representation and its bijection between the PDE domain and a fundamental state in L2. These inequalities are then verified via convex optimization to compute a least upper bound on the exponential growth rate, with the approach illustrated on classical parabolic and hyperbolic PDE examples.

Significance. If the central equivalence is rigorously established, the work provides a computationally practical bridge between abstract semigroup theory and convex optimization, enabling more accessible verification of well-posedness for PDE models that is relevant to analysis, control design, and simulation.

major comments (1)
  1. [Reformulation via PIE bijection (abstract and main derivation sections)] The load-bearing claim is the exact equivalence of Lumer-Phillips dissipativity (Re<Ax,x> ≤ 0 on D(A)) and range condition under the PIE bijection to operator inequalities on all of L2. The manuscript must explicitly define the induced operator on L2, prove that the bijection is isometric (preserves the inner product), maps D(A) onto a dense subspace whose closure is L2, and conjugates the generator action so that the numerical range and surjectivity transfer without loss or gain; otherwise the convex program may certify an incorrect growth bound.
minor comments (2)
  1. Provide explicit operator definitions and the precise form of the convex program (e.g., the semidefinite or sum-of-squares constraints used to enforce the inequalities).
  2. Clarify how boundary conditions are encoded in the PIE representation and whether they remain strongly or weakly enforced after the mapping.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful and constructive review. The central concern regarding the rigor of the equivalence proof under the PIE bijection is well-taken, and we address it directly below by committing to explicit additions in the revised manuscript.

read point-by-point responses

  1. Referee: The load-bearing claim is the exact equivalence of Lumer-Phillips dissipativity (Re<Ax,x> ≤ 0 on D(A)) and range condition under the PIE bijection to operator inequalities on all of L2. The manuscript must explicitly define the induced operator on L2, prove that the bijection is isometric (preserves the inner product), maps D(A) onto a dense subspace whose closure is L2, and conjugates the generator action so that the numerical range and surjectivity transfer without loss or gain; otherwise the convex program may certify an incorrect growth bound.

    Authors: We agree that the equivalence requires a fully explicit proof to ensure the convex program yields a correct growth bound. In the revised version we will add a dedicated subsection that (i) defines the induced operator on the full L2 space via the PIE representation, (ii) proves the bijection is isometric by direct verification of inner-product preservation, (iii) shows that the image of D(A) is dense in L2, and (iv) demonstrates conjugation of the generator action so that the numerical range and range condition transfer exactly. These additions will be placed immediately after the statement of the main theorem in the derivation section. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation applies established theorems via new reformulation

full rationale

The paper invokes the Lumer-Phillips theorem as an external necessary-and-sufficient condition for well-posedness and treats the PIE representation (including its bijection to L2) as an established equivalent form drawn from prior literature. The central step is the subsequent translation of those conditions into operator inequalities on the full L2 space that admit convex-optimization certificates; this translation is presented as a convenience mapping rather than a re-derivation that reduces to the inputs by construction. No self-definitional loops, fitted parameters renamed as predictions, or load-bearing self-citations that close the argument appear in the provided derivation chain. The approach therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on two background results: the Lumer-Phillips theorem and the existence of an equivalent PIE representation with a bijection to the PDE domain. No free parameters or new invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Lumer-Phillips theorem supplies necessary and sufficient conditions for well-posedness of dissipative linear PDEs on the PDE domain.
    Invoked directly in the abstract as the starting point whose verification is being made easier.
  • domain assumption The PIE representation provides a bijection between a fundamental state in L2 and the original PDE domain that preserves the relevant operator properties.
    This bijection is the key step that allows moving the conditions from the PDE domain to L2 operator inequalities.

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