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On the optimal decay rate of global solutions to the convection-diffusion equation with zero-mass initial data

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Pith's one-line read Zero-mass data for the convection-diffusion equation force a faster optimal decay rate and a unique self-similar profile built from first moments.

desk verdict Solid, complete resolution of the zero-mass decay problem for scalar convection-diffusion, including the eta=1 case left open by Karch–Schonbek; the only real limitation is the smallness restriction for p ≤ 1+1/n. read the letter →

arxiv 2607.05158 v1 pith:5T32XFE7 submitted 2026-07-06 math.AP

Yi C. Huang , Ryunosuke Kusaba This is my paper · ORCID
classification math.AP MSC 35K5835B4035C20
keywords convection-diffusionequationoptimaldecayratezero-massinitialdataasymptoticprofileself-similarityhigher-orderexpansions
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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The reading

The paper studies the Cauchy problem for the scalar convection-diffusion equation when the initial mass vanishes. Under that zero-mass condition the usual heat-kernel decay is improved by an extra half power of time. The authors first prove the corresponding upper bound (with a smallness restriction when the nonlinearity is not supercritical) and then construct a self-similar asymptotic profile that is a linear combination of gradients of the heat kernel whose coefficients are first moments of the data and of the integrated nonlinearity. Because the profile is self-similar, the improved decay rate is attained if and only if at least one of those moments is nonzero, and is therefore optimal. The result also closes an earlier gap for the borderline exponent that appears in second-order expansions when mass is zero.

What carries the argument

The second-order free-solution expansion (Lemma 3.2) together with a Duhamel remainder that is shown to be $o\left(t^{-\frac{n}{2}(1-\frac{1}{q})-\frac{1}{2}}\right)$ once the zero-mass assumption and the a-priori bound are used; the resulting profile $A(t)$ inherits the exact scaling of a first-order heat-kernel derivative.

What would settle it

Exhibit a zero-mass initial datum (even a small one) for which the first-moment vector $M_1(u_0)-a M_0(\psi_0)$ vanishes while the solution still decays exactly like $t^{-\frac{n}{2}(1-\frac{1}{q})-\frac{1}{2}}$ in some $L^q$, or construct a large-data solution for $p\le 1+\frac{1}{n}$ that decays slower than the claimed rate.

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Extended reading notes

Core claim

Under the zero-mass condition and an a-priori decay bound of the same strength, every global solution satisfies $\lim t^{\frac{n}{2}(1-\frac{1}{q})+\frac{1}{2}} \|u(t)\|_q = \|A(1)\|_q$, where $A(t) = -(M_1(u_0)-a M_0(\psi_0))\cdot \nabla G_t$ is the unique self-similar profile of that order; the limit is positive precisely when one of the first-moment conditions $M_{e_j}(u_0)-a_j M_0(\psi_0)$ is nonzero.

Load-bearing premise

When the power $p$ is at most $1+\frac{1}{n}$ the initial datum must be sufficiently small in the weighted $L^1-L^\infty$ norm for the improved upper bound to be proved.

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Editorial analysis

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Referee Report

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Summary. The paper studies the Cauchy problem for the convection-diffusion equation ∂_t u - Δu = a · ∇f(u) with homogeneous nonlinearity of degree p > 1 and zero-mass initial data M_0(u_0) = 0. Under the weight assumption u_0 ∈ L^1_1 ∩ L^∞, Theorem 2.1 establishes the improved L^q decay ||u(t)||_q ≲ (1+t)^{-n/2(1-1/q)-1/2} for p > 1 + 1/n (no smallness) and for p ≤ 1 + 1/n under a smallness condition in L^1_1 ∩ L^∞. Theorem 2.3 then shows that any solution satisfying the corresponding a-priori bound (2.3) admits the self-similar asymptotic profile A(t) = -(M_1(u_0) - a M_0(ψ_0)) · ∇G_t, with ψ_0 = ∫_0^∞ f(u( au)) d au. Theorem 2.4 converts this into the sharp statement lim t^{n/2(1-1/q)+1/2} ||u(t)||_q = ||A(1)||_q, and identifies the necessary and sufficient first-moment condition for the constant to be positive. The proofs rely on the Duhamel formula, heat-kernel expansions (Lemmas 3.1–3.2), and previously established higher-order expansions (Lemmas 4.1–4.2).

Significance. The work closes a natural gap left by the classical Escobedo–Zuazua theory and the subsequent higher-order expansions of Zuazua, Kusaba and others: when the mass vanishes, the first-order profiles disappear and the decay improves by exactly one half-derivative. The identification of the critical threshold p > 1 + 1/(n+1) for the Duhamel term to be integrable, the self-similar profile A(t), and the necessary-and-sufficient moment condition for optimality are clean and definitive. The results also supply a positive answer to the open optimality question for the second-order remainder when p = 1 + 3/(2n) under zero mass, and they complete the eta = 1 case left open by Karch–Schonbek. The parallel with the zero-mass Navier–Stokes setting is made explicit and is of independent interest. All estimates are explicit and the logical structure is free of circularity once the cited expansions are granted.

minor comments (4)
  1. In the statement of Theorem 2.1(2) the smallness threshold ε_0 is asserted to be independent of ||u_0||_{L^1_1 ∩ L^∞}, yet the bootstrap condition (4.1) is written in terms of that norm; a one-line clarification that ε_0 depends only on n, p and a would remove any ambiguity.
  2. Lemma 4.3 and the subsequent bootstrap are valid for all p > 1 + 1/(n+1), as noted in Remark 4.4; it would be helpful to the reader if this fact were already mentioned in the statement of Theorem 2.1 rather than only in the remark.
  3. The representation of A(1) used at the end of the proof of Theorem 2.4 (A(1) = (1/2) ∑ (M_{e_j}(u_0) - a_j M_0(ψ_0)) x_j G_1) is elementary but not written earlier; inserting it once when A(t) is introduced would make the positivity argument self-contained.
  4. A few typographical inconsistencies appear (e.g., the spacing around the multi-index notation in Section 3 and the occasional missing space after commas in the bibliography). They do not affect readability but should be cleaned in the final version.

Circularity Check

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No significant circularity: asymptotic profile and optimality follow from heat-kernel expansions and the integral equation under an independent a-priori bound.

full rationale

The paper derives the improved decay (Theorem 2.1) and the self-similar profile A(t) (Theorem 2.3) from the Duhamel formula (1.1), the free-solution expansions of Lemmas 3.1–3.2, and the integrability of f(u) that follows from the a-priori bound (2.3) once p>1+1/(n+1). The limit identity (2.5) and the algebraic criterion (2.6) for ||A(1)||_{L^q}>0 are then immediate consequences of the representation A(t)=t^{-1/2}δ_t A(1). Higher-order expansions from prior work (including the second author’s own papers) are used only as black-box upper bounds that vanish under M_0(u_0)=0; they are not rewritten into the present claim. The smallness hypothesis needed for (2.3) when p≤1+1/n is an independent restriction, not a definitional loop. No step equates a fitted quantity to a prediction, imports uniqueness by self-citation, or renames a known pattern as a new derivation.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper works entirely within classical parabolic theory. The only non-standard inputs are the range p>1+1/(n+1) forced by integrability of the Duhamel term and the smallness assumption needed for the bootstrap when p is subcritical. No free parameters are fitted; no new physical entities are postulated.

assumptions (4)
  • standard math Heat-kernel L^p–L^q estimates and moment expansions (Lemmas 3.1–3.2)
    Standard Gaussian bounds used throughout Sections 3–5.
  • domain assumption Global well-posedness and basic decay for (P) in L^1\cap L^\infty (Proposition 1.1)
    Taken from Escobedo–Zuazua and subsequent improvements; used as the starting point for all asymptotic analysis.
  • domain assumption Higher-order asymptotic expansions for non-zero mass (Lemmas 4.1–4.2)
    Quoted from Kusaba (2025) and used to treat the zero-mass case by vanishing of the leading profiles.
  • ad hoc to paper p>1+1/(n+1) and u_0\in L^1_1\cap L^\infty with M_0(u_0)=0
    The critical lower bound on p is forced by the time integrability of ||u||_p^p; the weighted space supplies the first-moment control needed for the improved rate.

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Pith. "Pith review of On the optimal decay rate of global solutions to the convection-diffusion equation with zero-mass initial data." pith.science (2026). https://pith.science/paper/5T32XFE7

@misc{pith2026260705158,
  author       = {Pith},
  title        = {Pith review of: On the optimal decay rate of global solutions to the convection-diffusion equation with zero-mass initial data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5T32XFE7}},
  note         = {Machine review of arXiv:2607.05158}
}
read the original abstract

We consider the optimal decay rate of global solutions to the convection-diffusion equation with zero-mass initial data. We establish an improved decay estimate of the global solutions under the zero-mass condition. Moreover, we derive a self-similar asymptotic profile of the global solutions. This result provides necessary and sufficient conditions for attaining the improved decay rate and shows that the decay rate is optimal.

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Works this paper leans on

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