Homogenization of Viscous Sublinear Hamilton--Jacobi Equations with $u/\epsilon$-Dependence

Guyu Jin

REVIEW 5 major objections 4 minor 27 references

Homogenization of Viscous Sublinear Hamilton--Jacobi Equations with $u/\epsilon$-Dependence

T0 review · 5 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper proves that viscous Hamilton–Jacobi equations with fast dependence on the phase $u^\epsilon/\epsilon$ and on $x/\epsilon$ homogenize to a first-order effective equation $u_t-c^\eta(Du)=0$ whenever the spatially oscillatory part…

desk verdict Genuinely new perturbative construction of periodic-parabolic correctors for viscous contact-type HJ homogenization; the main theorem is credible, but the uniform Schauder/Fredholm lemma on skew tori is under-proved and will need a fuller statement before Theorem 5 is sealed. read the letter →

arxiv 2608.00438 v1 pith:LMYMLQ2O submitted 2026-08-01 math.AP

Guyu Jin This is my paper · ORCID

classification math.AP MSC 35B2735F2135D4035K55

keywords viscousHamilton-Jacobiequationshomogenizationu/epsilon-periodicHamiltonianscontact-typeviscositysolutionsperiodic-paraboliccorrectorscellproblemlarge-timeaveraging

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves a homogenization result for a viscous Hamilton-Jacobi equation whose Hamiltonian depends on the unknown through the fast phase $u^\epsilon/\epsilon$, in addition to the fast space variable and the gradient. For Hamiltonians of the form $F(s,p)+\eta W(y,s,p)$ with $|\eta|$ small enough, the allowed size being set by the Lipschitz constant of the initial data, the solutions converge locally uniformly to the viscosity solution of the first-order effective equation $u_t-c^\eta(Du)=0$. In the unperturbed case $F(s,p)$, the effective speed comes from a one-dimensional monotone cell ODE, while in the perturbed case it comes from a periodic-parabolic corrector built by linearizing that one-dimensional profile. The paper also proves that the long-time average of the associated cell evolution equals the same cell speed. A reader should care because this is the viscous counterpart of contact-type $u/\epsilon$-periodic homogenization, combining first-order Hamilton-Jacobi structure with semilinear heat-equation structure.

What carries the argument

The central object is the periodic-parabolic corrector $\chi^\eta_p(y,z)=z+w^\eta_p(y,z)$ together with its cell speed $c^\eta(p)$, solving equation (3.2) on the skew torus $\mathbb{T}^{N+1}_p$, the torus obtained from the identifications $(y,r)\sim(y+k,r-p\cdot k)$ and $(y,r)\sim(y,r+1)$. In the unperturbed case the corrector reduces to a monotone one-dimensional profile $\chi_p$, whose slope $v(s)=\chi'_p$ solves the first-order periodic ODE $|p|^2vv'=cv+F(s,pv)$, with the speed fixed by the normalization $\int_0^1 v^{-1}\,ds=1$. For the perturbation, the paper linearizes at $\eta=0$ and obtains the periodic-parabolic operator $\tilde L_p\psi=c^0(p)\psi_r+F_s(\chi^0_p,pa_p)\psi+D_pF(\chi^0_p,pa_p)\cdot D_y\psi-\Delta_y\psi$; the strict positivity of $a_p=(\chi^0_p)'$ ensures the kernel is exactly the span of $a_p$, so the Fredholm alternative makes $\tilde A_p(\psi,\kappa)=\tilde L_p\psi+\kappa a_p$ an isomorphism, and a contraction mapping on a small ball gives the corrector branch with $\|\tilde\psi^\eta_p\|_{C^{2+\alpha,1+\alpha/2}}+|\kappa^\eta_p|\le C_K|\eta|$.

What would settle it

Take a smooth positive 1-periodic family $F_\delta(s,p)=\delta+\sqrt{\delta^2+|p|^2}$ and plot $T_p(\mu)=\int_0^1 v_\mu(s)^{-1}\,ds$ for the periodic ODE $|p|^2vv'=-\mu v+F_\delta(s,pv)$ as $\delta$ varies. If for some $\delta>0$ the map $\mu\mapsto T_p(\mu)$ is not strictly increasing, or two different $\mu$ satisfy $T_p(\mu)=1$, then the uniqueness of the cell speed in Theorem 3 fails; a single such numerical example would settle the claim.

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Extended reading notes

Core claim

The central claim is that, under positivity, strict subhomogeneity along rays, recession-at-infinity, and regularity assumptions on $F$, and a natural Lipschitz/periodicity condition on $W$, the equation $u^\epsilon_t+F(u^\epsilon/\epsilon,Du^\epsilon)+\eta W(x/\epsilon,u^\epsilon/\epsilon,Du^\epsilon)=\epsilon\Delta u^\epsilon$ homogenizes to $u_t-c^\eta(Du)=0$ for $|\eta|$ below a threshold determined by the compact gradient set $K_0=B_{\mathrm{Lip}(u_0)}(0)$ (Theorem 5). The speed $c^\eta(p)$ is defined through a corrector $\chi^\eta_p(y,z)=z+w^\eta_p(y,z)$ solving the periodic-parabolic cell equation (3.2), with $\chi^\eta_p(y,z+1)=\chi^\eta_p(y,z)+1$ and $(\chi^\eta_p)_z>0$. The construction linearizes the cell operator at the unperturbed one-dimensional profile; the linearized operator is a periodic-parabolic operator on a skew torus whose kernel is spanned by the strictly positive slope $a_p=(\chi^0_p)'$, and the Fredholm alternative turns this into an isomorphism, after which a contraction argument yields a unique small corrector branch. For general periodic Hamiltonians satisfying only (H1)-(H2), the paper shows $w(\zeta,\tau)/\tau\to\lambda(p)$ for the cell evolution $w_\tau+H(\zeta,p\cdot\zeta+w,p+Dw)=\Delta w$, and in the perturbative class this averaged constant coincides with the corrector speed $c^\eta(p)$.

Load-bearing premise

The strict decrease of the ratio $F(s,\rho p)/\rho$ as $\rho$ grows is the load-bearing premise; without it, the one-dimensional cell problem may have several speeds or a non-positive corrector slope, and the later perturbative step would lose the unique linearized solution it needs.

Editorial extensions

If this is right

The unperturbed Hamiltonian $F(s,p)$ alone yields homogenization to $u_t-c(Du)=0$, so a viscous equation with fast phase dependence and no spatial oscillation has a first-order effective equation whose speed is selected by a one-dimensional travelling-wave profile.
When $|\eta|$ is below the threshold controlled by $\mathrm{Lip}(u_0)$, the effective speed $p\mapsto c^\eta(p)$ is continuous on compact gradient sets, and the corrector admits uniform $C^2$ bounds, which is exactly what Evans' perturbed test function argument needs.
The long-time average of the cell evolution $w(\zeta,\tau)/\tau$ converges to the same speed as the corrector, so in the perturbative class the homogenized Hamiltonian can be read off either from the static cell problem or from the evolutionary cell problem.
The spatial Lipschitz estimate for the half-relaxed limits keeps every gradient appearing in the viscosity test inside the compact set $K_0$ on which the corrector was built, making the smallness condition on $\eta$ quantitative and datum-dependent.
The explicit example with $\eta_0=1/10$ shows the perturbative class is non-empty even with a nontrivial spatial oscillation, and in that example the corrector and speed are known in closed form.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper, Example 5.3 reveals a gauge-type recipe: any smooth periodic $\varphi(y)$ produces an admissible perturbation $W_\eta$ by the quotient $[\Delta_y\varphi+F(s-\eta\varphi,q-\eta D_y\varphi)-F(s,q)]/\eta$, so the perturbative class contains many Hamiltonians with exact correctors $\chi^0_p(z)+\eta\varphi(y)$ and unchanged speed $c^0(p)$.
Beyond the paper, the contraction argument suggests that the smallness threshold $\eta_K$ can be quantified in terms of the Holder norms of $F$ and $W$ on an enlarged phase-gradient region, which would turn the qualitative homogenization theorem into a rate-bearing construction.
Beyond the paper, the large-time averaging theorem holds under only (H1)-(H2), before any corrector is available; this opens a numerical route to approximate $\lambda(p)$ by simulating the cell evolution and comparing its long-time average to the corrector speed in the perturbative regime.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper proves homogenization for viscous Hamilton-Jacobi equations of contact type H(y,s,p)=F(s,p)+ηW(y,s,p), when F is strictly monotone along rays and the perturbation is small. The main novelty is real: the linearization at the one-dimensional ODE profile gives a periodic-parabolic operator on a p-dependent skew torus, and correctors are constructed by a Fredholm alternative plus contraction. That technique is new, and the unperturbed theorem (with H=F depending on s and gradient) is a useful extension of both the semilinear heat equation setting and the first-order contact theory.

The unperturbed cell problem is solved completely, with explicit ODE construction, uniqueness, and careful continuity of the cell speed near p=0. The perturbative construction is coherent: the linearized operator has kernel spanned by a_p, the adjoint kernel yields the speed shift, and the fixed point argument produces small correctors. The large-time averaging result for a general class and its identification with the cell speed are natural and well integrated. The examples are honest and illustrate the assumptions.

The load-bearing point is Lemma 3.1, the uniform periodic-parabolic Schauder estimates and Fredholm alternative on skew tori T_p^{N+1}. The proof as written is a sketch: it cites Lieberman and LSU, then says translations patch the local estimates. A referee will want the dependence on p tracked explicitly—verifying that the Hölder norms of the coefficients b_p, d_p are uniform in the parabolic metric on T_p^{N+1}, and that the covering number and Schauder constants do not degenerate as p varies. I do not think the statement is false; the lattice generators are bounded on compact K and the fundamental domain has fixed volume, so the uniformity should be true. But as written, Section 3 rests on an under-proved lemma. This is a technical gap, not a conceptual one.

Minor caveats: the bootstrap regularity (Lemma 3.2 and Remark 3.1) also relies on the same lemma and is only sketched, and the smallness threshold for η is existential and depends on the initial datum's Lipschitz constant. These are stated honestly but limit the scope: this is a perturbative result, not a resolution of the general cell problem (1.2).

I would send this to peer review. The novel technique and the clean unperturbed part justify referee time. The referees should ask for a complete proof of Lemma 3.1 and a fuller write-up of the bootstrap. My own verdict would be accept after revision, provided those details are supplied. I would cite it if I work on HJ homogenization.

Referee Report

5 major / 4 minor

Summary. The paper studies periodic homogenization of the viscous Hamilton--Jacobi equation (1.1) with Hamiltonian H(y,s,p)=F(s,p)+\eta W(y,s,p), where the unknown enters through the fast scale u^\epsilon/\epsilon. In the unperturbed case the author constructs one-dimensional monotone correctors, proves uniqueness and continuity of the cell speed, and establishes local uniform convergence to u_t-c(Du)=0. In the perturbed case, for sufficiently small \eta depending on the Lipschitz constant of the initial datum, periodic-parabolic correctors on p-dependent skew tori are constructed by linearizing around the one-dimensional profile, applying a Fredholm alternative, and performing a contraction argument; this yields the homogenization theorem with effective Hamiltonian c^\eta(p). The paper also proves a large-time averaging result for a general class of cell evolutions and identifies the averaged constant with the corrector speed in the perturbative class, followed by three examples.

Significance. If fully substantiated, the paper is a genuine contribution to the homogenization of second-order contact-type Hamilton--Jacobi equations, since it combines fast x/\epsilon and u^\epsilon/\epsilon oscillations with a viscous term and gradient-dependent spatial perturbations. The unperturbed corrector construction is detailed and convincing, and the perturbative strategy based on periodic-parabolic Fredholm theory is natural and potentially reusable. The explicit examples in Section 5 are useful. However, the central perturbative claim, Theorem 5, currently rests on a uniform Schauder/Fredholm lemma and a bootstrap regularity step that are only sketched, and the continuity proof for the effective speed has a nontrivial passage-to-the-limit gap. These points are local and likely repairable, but they are load-bearing.

major comments (5)

[Section 3.1, Lemma 3.1] Lemma 3.1 is the foundation of Propositions 3.2 and 3.3 and therefore of Theorem 5, but its proof is a sketch. The manuscript does not explicitly verify that the coefficients b_p(y,r)=D_pF(\chi^0_p(p\cdot y+r), p a_p(p\cdot y+r)) and d_p(y,r)=F_s(\chi^0_p(p\cdot y+r), p a_p(p\cdot y+r)) are uniformly H\"older continuous in the parabolic distance on T^{N+1}_p, nor does it track how the covering of the fundamental domain and the Schauder constant depend on p. The Fredholm alternative on the varying skew lattice is also asserted rather than proved. Please supply either a complete proof of the uniform estimates and Fredholm property or a precise statement of an existing theorem with all hypotheses verified on the skew tori.
[Section 3.1, Lemma 3.2 and Remark 3.1] The bootstrap regularity result is sketched but is load-bearing: Definition 3.1 requires w^\eta_p\in C^2(T^{N+1}), and the viscosity test in Theorem 5 uses (\chi^\eta_p)_{zz} and D^2_{yz}\chi^\eta_p. The differentiated equation for U=\Xi_r is written down, but the Schauder step is not justified in detail: one needs the coefficients F_s(\Xi,D_y\Xi)+\eta W_s(\Xi,D_y\Xi) and D_pF(\Xi,D_y\Xi)+\eta D_pW(\Xi,D_y\Xi) to be uniformly H\"older in the parabolic distance on T^{N+1}_p, uniformly in p\in K, and the argument that the resulting bounds pass back to C^2 in the original variables is only summarized.
[Section 3.1, Proposition 3.5] The proof of continuity of p\mapsto c^\eta(p) passes to a subsequence and then identifies the limit using the uniqueness of the small solution from Proposition 3.3. As written, it does not show that the limiting pair (\psi^*,\kappa^*) belongs to the small ball B^\eta_p in X_p, nor does it explain how the X_{p_n}-norms behave under the varying skew lattice when passing to the limit. Without such control the uniqueness step is not justified. Please add a compactness argument that keeps the limit inside the contraction class or otherwise identify the limit by a stability argument for the fixed-point equation.
[Section 3.2, Theorem 5] The smallness threshold \eta_K in Proposition 3.3 is compact-dependent, so for a fixed \eta\le\eta_{K_0} the constructed cell speed c^\eta(p) is a priori defined only on K_0=B_{Lip(u_0)}(0). Theorem 5 nevertheless states convergence to the viscosity solution of u_t-c^\eta(Du)=0 on R^N, which requires a Hamiltonian defined on all of R^N. Either prove that c^\eta admits a continuous extension to R^N, or formulate and justify a restricted viscosity solution whose test- function gradients lie in K_0.
[Section 4.2, Proposition 4.1] The maximum-principle proof of the oscillation estimate uses a penalized functional \Phi over the unbounded domain R^{2N}, but it does not establish the growth control needed for the maximum to be attained, nor does it justify the claim that the terms \alpha|\zeta_\alpha| and \alpha|\eta_\alpha| are O_\alpha(1) when passing to the limit \alpha\to 0. A standard barrier comparison using the linear growth estimate |H(\zeta,s,q)|\le\chi(|q|) and w(\zeta,0)=0 should close this gap, but it is not included. Since Theorem 6 depends on this proposition, the argument should be completed.

minor comments (4)

[Section 1.2 and Section 2] The symbol c_0 is used both for the positivity constant in assumption (F1) and for the unperturbed cell speed c_0(p) in Section 3; this is confusing and should be resolved by a change of notation.
[Appendix C, Lemma C.1] In the sentence before the displayed identity, 'there exists \epsilon and maximum points' should read 'there exist s_\epsilon and maximum points'; the parameter \epsilon is already fixed.
[Section 3.1, Proposition 3.2] In the contradiction argument it would help to state explicitly that the compact embedding is applied on a fixed fundamental domain after pulling back by (y,r)=(\theta,\sigma-p_n\cdot\theta), since the skew tori vary with p_n.
[Section 4.3, Theorem 7] The notation (4.3) is referenced in the proof but does not appear to be labeled in the text; please add the number or refer to the displayed comparison inequality directly.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the cell speed is computed from explicitly solved cell problems, and the homogenization theorems independently prove convergence to that speed.

full rationale

The derivation is self-contained in the relevant sense. The unperturbed cell speed c(p) is constructed by solving the periodic ODE (1.9) under assumptions (F1)-(F4); Theorem 4 then proves, by Evans' perturbed test function argument with the independently constructed corrector, that the solutions u^epsilon converge to u_t - c(Du)=0. Similarly, the perturbed effective Hamiltonian H^eta(p)=-c^eta(p) is defined by the fixed-point solution of the periodic-parabolic cell problem (3.2)-(3.3), and Theorem 5 proves convergence to that speed without using the limit equation as an input. The large-time averaging result (Theorem 6) is obtained from an independent almost-additivity argument, and Theorem 7 is a consistency check showing that the cell evolution averages to the same c^eta(p); no fitted parameter is renamed as a prediction. The proof of Lemma 3.1 is a brief sketch that invokes standard periodic-parabolic Schauder and Fredholm theory, which is a completeness or correctness concern, not a circularity concern, because the cited theory is external and independent of the paper's conclusions. No self-referential uniqueness theorem, ansatz smuggled in by citation, or renaming of a known empirical pattern is load-bearing in the argument.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The results depend on structural hypotheses on F and W and on the smallness of eta; no numerical constants are fitted to data, and no new entities are postulated. Proofs rely on standard tools (viscosity theory, periodic-parabolic Schauder estimates, Fredholm alternative, fixed point theorems) from cited literature.

assumptions (6)

domain assumption (F1) F(s,p) >= c0 > 0
Used to construct positive periodic solutions and invert the ODE; stated in (1.4).
domain assumption (F2) rho D_p F(s,rho p) dot p - F(s,rho p) < 0
Ensures uniqueness of the one-dimensional corrector and strict monotonicity needed for the Fredholm kernel; stated in (1.5).
domain assumption (F3) F_infty(p) = lim_{rho to infinity} F(s,rho p)/rho exists
Used in the interval-mapping construction of the cell ODE and in the limits; stated in (1.6).
domain assumption (F4) F in Lip cap C^{3,alpha} and 1-periodic in s
Provides enough regularity for the viscosity test function and the parabolic estimates; stated in (1.7).
domain assumption W in C^{3,alpha}_loc(T^N x T x R^N) with global Lipschitz condition
The Lipschitz condition is used for the comparison principle (Appendix A) and for the cell problem; stated in Theorem 2.
domain assumption (H1)-(H2) Lipschitz and periodicity for the general cell evolution
Used in Section 4 for the large-time averaging theorem; stated in Section 4.

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Pith. "Pith review of Homogenization of Viscous Sublinear Hamilton--Jacobi Equations with $u/\epsilon$-Dependence." pith.science (2026). https://pith.science/paper/LMYMLQ2O

@misc{pith2026260800438,
  author       = {Pith},
  title        = {Pith review of: Homogenization of Viscous Sublinear Hamilton--Jacobi Equations with $u/\epsilon$-Dependence},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LMYMLQ2O}},
  note         = {Machine review of arXiv:2608.00438}
}

abstract

We study the periodic homogenization of a class of viscous Hamilton--Jacobi equations with fast dependence on the unknown. This problem combines features of first-order Hamilton--Jacobi equations with $u^\epsilon/\epsilon$-periodic Hamiltonians and semilinear heat equations with rapidly oscillating positive potentials. In this paper, we prove qualitative homogenization results for $H(y,s,p)=F(s,p)+\eta W(y,s,p)$ when $|\eta|$ is sufficiently small, with the smallness threshold depending on the Lipschitz constant of the initial datum, and establish a large-time averaging result for a general class of evolutionary cell problems.

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Works this paper leans on

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Homogenization of ﬁrst-o rder equations with u/ǫ-periodic Hamiltonian: Rate of convergence as ǫ → 0 and numerical methods

Achdou, Y ., Patrizi, S., 2011. Homogenization of ﬁrst-o rder equations with u/ǫ-periodic Hamiltonian: Rate of convergence as ǫ → 0 and numerical methods. Mathematical Models and Methods in A pplied Sciences 21, 1317–1353

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[2] [2]

Some homogenization results for non-c oercive Hamilton–Jacobi equations

Barles, G., 2007. Some homogenization results for non-c oercive Hamilton–Jacobi equations. Calculus of V ariations and Partial Differential Equations 30, 449–466

work page 2007

[3] [3]

Space-time periodi c solutions and long-time behavior of solutions to quasi-linear parabolic equations

Barles, G., Souganidis, P .E., 2001. Space-time periodi c solutions and long-time behavior of solutions to quasi-linear parabolic equations. SIAM Journal on Mathema tical Analysis 32, 1311–1323

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Berestycki, H., Hamel, F., 2007. Generalized travellin g waves for reaction-diffusion equations. Contemporary Mathematics 446, 101–124

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The s peed of propagation for kpp type problems

Berestycki, H., Hamel, F., Nadirashvili, N., 2005. The s peed of propagation for kpp type problems. i: Periodic framework. Journal of the European Mathematical Society 7, 173–213

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