Asymptotics of Radially Symmetric Solutions for the Exterior Problem of Multidimensional Burgers Equation

Huijiang Zhao; Qingsong Zhao; Tong Yang

REVIEW 1 major objections 5 minor 15 references

Asymptotics of Radially Symmetric Solutions for the Exterior Problem of Multidimensional Burgers Equation

T0 review · 1 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The stationary wave of the radially symmetric multidimensional Burgers equation is nonlinearly stable over its entire existence range, with algebraic or exponential convergence rates for small perturbations.

desk verdict Extends the stability range and gives rates for the multidimensional Burgers stationary wave, but the exponential decay claim in Theorem 2.3 is overstated by a factor of two and needs fixing. read the letter →

arxiv 1908.03354 v1 pith:N4OFWDFO submitted 2019-08-09 math.AP

Tong Yang , Huijiang Zhao , Qingsong Zhao This is my paper · ORCID

classification math.AP MSC 35B3535B4035Q3535K55

keywords multidimensionalBurgersequationradiallysymmetricsolutionexteriordomainstationarywavenonlinearstabilityweightedenergymethodtemporaldecayrateanti-derivative

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

For radially symmetric solutions of the multidimensional Burgers equation outside a ball in dimensions $n\ge 4$, the paper proves that the stationary wave is nonlinearly stable for every boundary/far-field pair for which the wave exists: $v_+<0$ and $V_-\le |v_+|$. Earlier work had only established stability in the subcase $v_\pm<0$ with $V_-\le v_+$, so this paper closes the gap that includes non-monotone stationary waves. The proof passes to the anti-derivative of the perturbation, obtaining a scalar parabolic equation for $w$, and uses a custom positive weight function to control the non-monotonicity. With algebraic or exponential spatial weights on the initial anti-derivative, the paper obtains matching temporal rates, $(1+t)^{-\alpha/2}$ or $e^{-\gamma t}$, in the sup norm. If correct, this means the existence condition for the stationary wave is also a stability condition over the same parameter range.

What carries the argument

The central object is the anti-derivative $w(t,r)=-\int_r^{\infty}(v(t,y)-\varphi(y))\,dy$, which turns the perturbation equation into $w_t+\psi w_r-\mu w_{rr}=-\frac{1}{2}w_r^2$ with $\psi=\varphi-\mu(n-1)/r$. Because $\varphi$ is non-monotone, the usual energy method leaves an indefinite term, so the paper constructs the weight $\chi(r)=\exp\left(-\frac{1}{\mu}\int_{r_0}^r\psi(s)\,ds\right)\int_r^{\infty}\left(\frac{2}{r_0}-\frac{1}{s}\right)\exp\left(\frac{1}{\mu}\int_{r_0}^s\psi(\tau)\,d\tau\right)ds$. This $\chi$ is bounded above and below by positive constants and solves $\chi'+(\psi/\mu)\chi=1/r-2/r_0$, making the boundary term at $r=r_0$ strictly negative and the effective spatial coefficient positive. Multiplying the equation by $\chi w$ and integrating yields coercive zeroth-order estimates; higher derivatives and the space-time weighted method then convert the spatial weights into the algebraic and exponential temporal rates.

What would settle it

Solve the exterior initial-boundary value problem numerically in $n=4$ with parameters in the non-monotone range, say $v_+=-1$, $V_-=0.5$, $\mu=r_0=1$, starting from a small $H^2$ anti-derivative $w_0$ with compact support and $w_0(r_0)=0$, and measure $\sup_{r\ge r_0}|v(t,r)-\varphi(r)|$ in time; observing no convergence, or slower decay than $(1+t)^{-\alpha/2}$, would contradict the theorems.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the stationary wave $\varphi(r)$ built from the boundary value $v_-$ and far-field state $v_+$ is time-asymptotically nonlinearly stable for the full range $v_+<0$, $V_-\le |v_+|$, where $V_-=v_- - \mu(n-1)/r_0$. Theorem 2.1 states that whenever the anti-derivative $w_0(r)=-\int_r^{\infty}(v_0(y)-\varphi(y))\,dy$ is small in $H^2$, the initial-boundary value problem has a unique global solution $v(t,r)$ and $\sup_{r\ge r_0}|v(t,r)-\varphi(r)|\to 0$ as $t\to\infty$. Theorems 2.2 and 2.3 sharpen this to explicit rates: initial data in $L^2_\alpha$ give the algebraic decay $(1+t)^{-\alpha/2}$, and initial data in $H^{2,\beta}_{\mathrm{exp}}$ give exponential decay $e^{-\gamma t}$. The previously open case is the non-monotone regime $V_->v_+$ with $V_-+v_+<0$; the paper includes it and thereby matches the full existence range with a stability result.

Load-bearing premise

The load-bearing premise is that the anti-derivative of the initial perturbation is small in $H^2$ and lies in the appropriate weighted space, which requires both small energy and zero integrated mass at infinity; the authors note in Remark 2.1 that large perturbations are left open.

Editorial extensions

If this is right

For every pair $(v_-,v_+)$ with $v_+<0$ and $V_-\le |v_+|$, the stationary wave attracts all sufficiently small $H^2$ perturbations of its anti-derivative.
Algebraic spatial weights on the initial perturbation produce the matching temporal decay $(1+t)^{-\alpha/2}$ uniformly in $r$.
Exponential spatial weights on the initial perturbation produce exponential decay $e^{-\gamma t}$, with the rate $\gamma$ fixed by the weight parameters in (2.20).
The previously open non-monotone regime $V_->v_+$ with $V_-+v_+<0$ is included in the stability statement.

Reading between the lines

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

In our reading, the weight construction in Lemma 2.2 is not tied to the Burgers equation specifically: any profile $\psi$ for which the chosen $f(r)=1/r-2/r_0$ satisfies $f'<0$, $f(r_0)<0$, and $f'\in L^1$ would yield the same coercivity, so the method should transfer to other exterior parabolic problems with non-monotone stationary states.
The algebraic exponent $\alpha/2$ is the natural parabolic diffusion rate; we would expect the same half-rate in other dissipative conservation laws after the same anti-derivative reduction, with the dimension entering only through the effective profile $\psi$.
The authors explicitly leave the large-perturbation regime open; a natural next test is whether smallness of $\|w_0\|_{H^2}$ can be relaxed while keeping the same weight function.

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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

Two things to know. First, the paper does what it claims: it extends the nonlinear stability of the stationary wave for the exterior multidimensional Burgers problem from the subrange V_- ≤ v+ to the full existence range V_- ≤ |v+| for n ≥ 4, and it proves algebraic convergence rates for that range. That is a real advance within the subfield. Second, the exponential decay stated in Theorem 2.3 is too strong: the lemmas as written give e^{-γt/2}, not e^{-γt}, for sup|v-φ|. The stress-test note holds up, and the paper needs a correction or a sharper estimate before the exponential claim can stand.

What is genuinely new: the weighted energy method around a carefully chosen χ(r). The identity (3.2) is clean, the sign conditions in Lemma 2.2 are the heart, and they make the zeroth-order estimate work despite the non-monotone stationary wave. The algebraic rate argument follows the standard Kawashima-Matsumura path and seems correct. The paper is honest about what is open: large perturbations and the other unsolved cases listed in the introduction.

Where it is soft. The exponential rate problem is the main one. Lemma 3.9 only yields e^{γt}|w_r|^2 and e^{γt}|w_rr|^2 bounded. The pointwise bound via Sobolev interpolation (3.24) then gives sup|w_r| ≤ C e^{-γt/2}, not e^{-γt}. The statement says 'follows easily', but no additional estimate is supplied. This is a quantitative overclaim rather than a fatal flaw: the stability and algebraic rates are unaffected, but Theorem 2.3 must either claim the half-rate or prove e^{2γt} bounds. A referee should demand this.

Also, several ingredients are deferred. Proposition 3.1 is standard iteration, fine. Lemmas 3.4-3.6 lean on 'repeating arguments in [8]' without page-level direction; a referee can verify but it slows things down. The smallness of the anti-derivative in H2 is an explicit hypothesis, not hidden. The citation pattern looks appropriate: [7] is the natural target to extend, [8] and [15] are the rate machinery.

Bottom line: the central stability result is credible and the paper belongs in the literature after the exponential rate is fixed or weakened. It deserves a serious referee. I would bring it to reading group because the weighted energy identity is instructive even if Theorem 2.3 ends up being a half-rate result. My recommendation is to send it to peer review with a clear request to address the exponential decay gap.

Referee Report

1 major / 5 minor

Summary. This paper studies the large-time behavior of radially symmetric solutions to the multidimensional Burgers equation in the exterior of a ball in R^n, n >= 3, with boundary data v_- and far-field state v_+ corresponding to a stationary wave. The main results are: (i) nonlinear stability of the stationary wave for the full parameter range v_+ < 0 and V_- <= |v_+|, extending the earlier result of Hashimoto-Matsumura which covered only the monotone case; (ii) algebraic temporal decay of order (1+t)^{-alpha/2} when the anti-derivative of the initial perturbation belongs to a weighted L^2 space; and (iii) exponential temporal decay of order e^{-gamma t} when the initial anti-derivative lies in an exponentially weighted Sobolev space. The analysis is based on an anti-derivative transformation that converts the problem into a scalar parabolic equation, combined with a space-weighted energy method using a specially constructed weight function chi(r), and a space-time weighted energy method for the decay rates.

Significance. If correct, the paper fills a genuine gap in the asymptotic classification of radially symmetric Burgers flow: it removes the monotonicity restriction on the stationary wave and covers the full existence range. The construction of the weight function in Lemma 2.2 and the resulting a priori estimates in Proposition 3.2 are nontrivial and carefully presented. The stability result for the whole range is a solid contribution to the literature. However, the exponential decay rate in Theorem 2.3 is not justified by the estimates actually proved; this is a concrete quantitative overclaim in a stated main result. The algebraic rate in Theorem 2.2 and the stability in Theorem 2.1 appear sound.

major comments (1)

[Section 3.3, Theorem 2.3] The exponential decay rate in (2.21) is not supported by the estimates in Lemmas 3.8 and 3.9. These lemmas yield e^{gamma t}|w(t)|^2_{beta,exp} <= C and e^{gamma t}|w_r(t)|^2_{beta,exp} + e^{gamma t}|w_rr(t)|^2_{beta,exp} <= C; hence each of ||w_r(t)|| and ||w_rr(t)|| is O(e^{-gamma t/2}). The Sobolev interpolation (3.24), sup|w_r| <= C||w_r||^{1/2}||w_rr||^{1/2}, then gives sup|w_r(t,r)| = O(e^{-gamma t/2}). The alternative weighted bound |w_r(r)| <= C e^{-beta r0/2}|w_rr|_{beta,exp} gives the same half-rate. To obtain O(e^{-gamma t}) one would need e^{2gamma t}-weighted estimates in Lemmas 3.8-3.9 or a separate pointwise decay argument. Thus inequality (2.21) as stated is an overclaim; the correct rate implied by the given estimates is e^{-gamma t/2}. This does not affect Theorems 2.1-2.2, but it is a quantitative error in a stated main result.

minor comments (5)

[Various] There are several typos throughout the manuscript: 'radiall y' in the abstract, 'is quite is different' in Section 1, and 'estiamtes' in Section 3.1.
[Section 2] The introduction states 'we will focus on the case n >= 4' after Lemma 2.1, but Theorems 2.1-2.3 do not explicitly state a dimension restriction; please clarify whether the results also apply to n=3 or are restricted to n>=4.
[Section 3.2] The proofs of Lemmas 3.4-3.6 are deferred to 'repeating the arguments used in [8]'. Since the problem here has a different weight and boundary term, a sketch or explicit statement of the adapted induction would make the proof more self-contained.
[Section 3.3] In the display following (3.29), the text contains LaTeX artifacts '/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright' that should be removed.
[Section 3.3] Lemma 3.9 is stated without proof ('Similarly...'). Since it is used for the exponential decay theorem, the details of its derivation should be provided or at least carefully outlined.

Circularity Check

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Score 0: stability and decay rates are derived, not assumed; external stationary-wave existence and explicit weighted-energy estimates carry the argument.

full rationale

The paper's derivation is self-contained on top of an external stationary-wave existence result. Lemma 2.1 quotes [7] (Hashimoto-Matsumura, not the present authors) for existence, smoothness, and integrability of the stationary profile; this is an input hypothesis, not the target stability conclusion. The anti-derivative transform w(t,r)=-∫_r^∞(v(t,y)-φ(y))dy converts (1.2) into (2.6) algebraically, and the weight χ in Lemma 2.2 is explicitly constructed to solve an ODE; its positivity and bounds are proved, not assumed. The a priori estimates (3.3), (3.5), and (3.13) have the form: small H2 initial data imply uniform H2 bounds, with no fitted constants or target decay rates among the hypotheses. The algebraic and exponential decay claims are then deduced from weighted energy lemmas (3.18)-(3.20), (3.28), (3.33)-(3.34) via Sobolev interpolation such as (3.24); the convergence rates are outputs of those estimates, not inputs. The citations [8] and [15] are methodological; although [15] is a self-citation (Yin-Zhao), it is not load-bearing. Thus no prediction reduces by construction to an input and no load-bearing self-citation chain appears. A separate, non-circular quantitative concern: Theorem 2.3's e^{-γt} sup-bound may be overstated by a factor 2, since Lemmas 3.8-3.9 give e^{γt}|w_r|^2≤C and e^{γt}|w_rr|^2≤C, so the interpolation (3.24) yields sup|w_r|=O(e^{-γt/2}), not O(e^{-γt}); this is a correctness issue, not circularity. Proposition 3.1's proof is also omitted as standard, but that is likewise a completeness issue, not circularity.

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

No free parameters or invented physical entities. The proof relies on standard Sobolev and parabolic tools and on stationary wave properties established in [7] under the same condition (1.9). The cited result is external, with no author overlap, and is used only for existence of the stationary wave, not for stability.

assumptions (4)

standard math Local existence via standard iteration (Proposition 3.1).
Relied on to start the energy argument; the proof is omitted with the phrase 'standard iterative method'.
domain assumption Stationary wave existence and properties from [7] (Lemma 2.1).
Under (1.9), ψ-v+∈L1 and ψ has bounded derivatives; this is used to construct the weight χ and to control coefficients in the energy estimates.
standard math Sobolev embedding and interpolation inequalities in one spatial dimension.
Used throughout Section 3 to control nonlinear terms and the sup norm of w_r and its derivatives.
domain assumption Space-time weighted energy method of [8] and [15] is transferable to this setting.
Lemmas 3.4-3.6 and 3.9 are asserted by repeating arguments in [8] or marked as similar, without full proof.

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Pith. "Pith review of Asymptotics of Radially Symmetric Solutions for the Exterior Problem of Multidimensional Burgers Equation." pith.science (2026). https://pith.science/paper/N4OFWDFO

@misc{pith2026190803354,
  author       = {Pith},
  title        = {Pith review of: Asymptotics of Radially Symmetric Solutions for the Exterior Problem of Multidimensional Burgers Equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/N4OFWDFO}},
  note         = {Machine review of arXiv:1908.03354}
}

abstract

We are concerned with the large-time behavior of the radially symmetric solution for multidimensional Burgers equation on the exterior of a ball $\mathbb{B}_{r_0}(0)\subset \mathbb{R}^n$ for $n\geq 3$ and some positive constant $r_0>0$, where the boundary data $v_-$ and the far field state $v_+$ of the initial data are prescribed and correspond to a stationary wave. It is shown in \cite{Hashimoto-Matsumura-JDE-2019} that a sufficient condition to guarantee the existence of such a stationary wave is $v_+<0, v_-\leq |v_+|+\mu(n-1)/r_0$. Since the stationary wave is no longer monotonic, its nonlinear stability is justified only recently in \cite{Hashimoto-Matsumura-JDE-2019} for the case when $v_\pm<0, v_-\leq v_++\mu(n-1)/r_0$. The main purpose of this paper is to verify the time asymptotically nonlinear stability of such a stationary wave for the whole range of $v_\pm$ satisfying $v_+<0, v_-\leq |v_+|+\mu(n-1)/r_0$. Furthermore, we also derive the temporal convergence rate, both algebraically and exponentially. Our stability analysis is based on a space weighted energy method with a suitable chosen weight function, while for the temporal decay rates, in addition to such a space weighted energy method, we also use the space-time weighted energy method employed in \cite{Kawashima-Matsumura-CMP-1985} and \cite{Yin-Zhao-KRM-2009}.

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Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

[8]

Kawashima and A

S. Kawashima and A. Matsumura, Asymptotic stability of t raveling wave solutions of systems for one-dimensional gas motion. Commun. Math. Phys. 101 (1985), 97-127

work page 1985
[7]

Hashimoto, A

I. Hashimoto, A. Matsumura, Asymptotic behavior toward nonlinear waves for radially symmetric solutions of the multi-dimensional Burgers equation. J. Diﬀerential Equations 266 (2019), 2805-2829

work page 2019
[15]

Yin and H.-J

H. Yin and H.-J. Zhao, Nonlinear stability of boundary l ayer solutions for generalized Benjamin-Bona-Mahony- Burgers equation in the half space. Kinetic and Ralated Models 2 (2009), 521-550

work page 2009
[1]

L.-L Fan, H.-X. Liu, T. Wang, and H.-J. Zhao, Inﬂow proble m for the one-dimensional compressible Navier-Stokes equations under large initial perturbation. J. Diﬀerential Equations 257 (2014), no. 10, 3521-3553

work page 2014
[2]

Fan, H.-X

L.-L. Fan, H.-X. Liu, and H.-J. Zhao, One-dimensional da mped wave equation with large initial perturbation. Anal. Appl. (Singap.) 11 (2013), no. 4, 1350013, 40 pp

work page 2013
[3]

Fan, H.-X

L.-L. Fan, H.-X. Liu, and H.-J. Zhao, Nonlinear stabilit y of planar boundary layer solutions for damped wave equation. J. Hyperbolic Diﬀer. Equ. 8 (2011), no. 3, 545-590

work page 2011
[4]

Fan, H.-X

L.-L. Fan, H.-X. Liu, H.-J. Zhao, and Q.-Y. Zou, Global st ability of stationary waves for damped wave equations. Kinet. Relat. Models 6 (2013), no. 4, 729-760

work page 2013
[5]

Hashimoto, Asymptotic behavior of radially symmetri c solutions for Burgers equation in several space dimension s

I. Hashimoto, Asymptotic behavior of radially symmetri c solutions for Burgers equation in several space dimension s. Nonlinear Anal. 100 (2014), 43-58

work page 2014

Show all 15 references

[6]

Hashimoto, Behavior of solutions for radially symmet ric solutions for Burgers equation with a boundary corre- sponding to the rarefaction wave

I. Hashimoto, Behavior of solutions for radially symmet ric solutions for Burgers equation with a boundary corre- sponding to the rarefaction wave. Osaka J. Math. 53 (2016), 799-811

2016
[9]

T.-P. Liu, A. Matsumura, and K. Nishihara, Behaviors of s olutions for the Burgers equation with boundary corre- sponding to rarefaction waves. SIAM J. Math. Anal. 29 (1998), 293-308

1998
[10]

Liu and K

T.-P. Liu and K. Nishihara, Asymptotic behavior for sca lar viscous conservation laws with boundary eﬀect. J. Diﬀerential Equations 133 (1997), 296-320

1997
[11]

Liu and S.-H

T.-P. Liu and S.-H. Yu, Propagation of a stationary shoc k layer in the presence of a boundary. Arch. Rational Mech. Anal. 139 (1997), no. 1, 57-82

1997
[12]

Matsumura, Inﬂow and outﬂow problems in the half spac e for a one-dimensional isentropic model system of compressible viscous gas

A. Matsumura, Inﬂow and outﬂow problems in the half spac e for a one-dimensional isentropic model system of compressible viscous gas. IMS Conference on Diﬀerential Eq uations from Mechanics (Hong Kong, 1999). Methods Appl. Anal. 8 (2001), no. 4, 645-666

2001
[13]

Nishihara, Boundary eﬀect on a stationary viscous sh ock wave for scalar viscous conservation laws

K. Nishihara, Boundary eﬀect on a stationary viscous sh ock wave for scalar viscous conservation laws. J. Math. Anal. Appl. 255 (2001), no. 2, 535-550

2001
[14]

Nishihara, Asymptotic behaviors of solutions to vis cous conservation laws via L2− energy method

K. Nishihara, Asymptotic behaviors of solutions to vis cous conservation laws via L2− energy method. Adv. Math. (China) 30 (2001), no. 4, 293-321

2001

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [8]

Kawashima and A

S. Kawashima and A. Matsumura, Asymptotic stability of t raveling wave solutions of systems for one-dimensional gas motion. Commun. Math. Phys. 101 (1985), 97-127

work page 1985

[2] [7]

Hashimoto, A

I. Hashimoto, A. Matsumura, Asymptotic behavior toward nonlinear waves for radially symmetric solutions of the multi-dimensional Burgers equation. J. Diﬀerential Equations 266 (2019), 2805-2829

work page 2019

[3] [15]

Yin and H.-J

H. Yin and H.-J. Zhao, Nonlinear stability of boundary l ayer solutions for generalized Benjamin-Bona-Mahony- Burgers equation in the half space. Kinetic and Ralated Models 2 (2009), 521-550

work page 2009

[4] [1]

L.-L Fan, H.-X. Liu, T. Wang, and H.-J. Zhao, Inﬂow proble m for the one-dimensional compressible Navier-Stokes equations under large initial perturbation. J. Diﬀerential Equations 257 (2014), no. 10, 3521-3553

work page 2014

[5] [2]

Fan, H.-X

L.-L. Fan, H.-X. Liu, and H.-J. Zhao, One-dimensional da mped wave equation with large initial perturbation. Anal. Appl. (Singap.) 11 (2013), no. 4, 1350013, 40 pp

work page 2013

[6] [3]

Fan, H.-X

L.-L. Fan, H.-X. Liu, and H.-J. Zhao, Nonlinear stabilit y of planar boundary layer solutions for damped wave equation. J. Hyperbolic Diﬀer. Equ. 8 (2011), no. 3, 545-590

work page 2011

[7] [4]

Fan, H.-X

L.-L. Fan, H.-X. Liu, H.-J. Zhao, and Q.-Y. Zou, Global st ability of stationary waves for damped wave equations. Kinet. Relat. Models 6 (2013), no. 4, 729-760

work page 2013

[8] [5]

Hashimoto, Asymptotic behavior of radially symmetri c solutions for Burgers equation in several space dimension s

I. Hashimoto, Asymptotic behavior of radially symmetri c solutions for Burgers equation in several space dimension s. Nonlinear Anal. 100 (2014), 43-58

work page 2014