Large time behavior of exponential surface diffusion flows on $\mathbb{R}$

Michael G\"osswein; Sho Katayama; Yoshikazu Giga

REVIEW 1 major objections 5 minor 1 cited by

Large time behavior of exponential surface diffusion flows on $\mathbb{R}$

T0 review · 1 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For small initial slopes, the exponential surface diffusion flow has a unique global classical solution, and its rescaled profiles converge to the unique self-similar profile of the linearized flow.

desk verdict First global well-posedness and large-time self-similar convergence for exponential surface diffusion flows; the main proof is sound, with a fixable gap in the uniqueness-class verification. read the letter →

arxiv 2411.17175 v1 pith:723S645E submitted 2024-11-26 math.AP

Yoshikazu Giga , Michael Gösswein , Sho Katayama This is my paper · ORCID

classification math.AP MSC 35K5535K3535B40

keywords exponentialsurfacediffusionflowglobalwell-posednessself-similarsolutionslarge-timeasymptoticsbiharmonicheatequationparabolicHölderspacesthermalgrooving

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

The paper studies the exponential surface diffusion flow $V=\partial_s^2 f(-\kappa)$ for a curve given as a graph over the whole real line, with $f$ a strictly increasing smooth function, typically $f(r)=e^r$. It establishes that if the first and second derivatives of the initial profile are bounded and sufficiently small, a unique classical solution exists for all time. It then proves that the rescaling $u_\sigma(x,t)=\sigma^{-1}u(\sigma x,\sigma^4 t)$ converges uniformly on compact sets to the unique self-similar solution of the linearized surface diffusion equation $V=-f'(0)\kappa$, with derivatives converging on compact sets away from $t=0$. This matters because it justifies the thermal-grooving explanation derived from the Gibbs–Thomson law without first linearizing the curvature dependence of $f$ near $\kappa=0$.

What carries the argument

The proof differentiates the equation once and works with $v=u_x$, rewriting the flow as $v_t=((1-\alpha)v_{xx}+F)_{xx}$, where $\alpha=\alpha(v,v_x)$ and $F=F(v,v_x)$ are treated as small perturbations of the biharmonic heat equation $v_t=-v_{xxxx}$. The central technical objects are scaled parabolic Hölder norms $\|\cdot\|'_{BC^{k+\mu,(k+\mu)/4}}$ and weighted norms $\|\cdot\|_{Z^k_T}$, which respect the asymptotic scaling of the equation so that semigroup estimates for $e^{-t\partial_x^4}$ and parabolic Schauder estimates remain uniform. The limit mechanism is the convergence $f_\sigma(r)=\sigma f(\sigma^{-1}r)\to f'(0)r$ as $\sigma\to\infty$, which makes the explicit curvature nonlinearity disappear at the rescaled level and forces the limit $U$ to solve the linearized surface diffusion equation.

What would settle it

A numerical experiment would settle the central claim: solve (1.1) with $f(r)=e^r$ and initial data $u_0(x)=a x_+ + b x_-$ for small $a,b$, evolve to large times, form $u_\sigma(x,t)=\sigma^{-1}u(\sigma x,\sigma^4 t)$, and compare on a fixed compact set with the unique self-similar solution $U$ of (1.11) with initial data $a x_+ + b x_-$; two different subsequential limits, or a limit failing (1.11), would refute the convergence claim.

Watch

Extended reading notes

Core claim

The paper's central assertion is that the exponential surface diffusion flow is globally well-posed for small data and that its large-time behavior is exactly the self-similar profile of the linearized flow. More precisely, under the smallness condition $\|(u_0)_x\|_{W^{1,\infty}}<\varepsilon_*$ and the asymptotic slope conditions $u_0(x)=(a+o(1))x$ as $x\to+\infty$ and $u_0(x)=(b+o(1))x$ as $x\to-\infty$ with $|a|,|b|$ small, the rescaled solutions $u_\sigma$ converge uniformly on compact subsets of $\mathbb{R}\times[0,\infty)$ to the unique self-similar solution $U$ of the linearized equation (1.11), and derivatives up to order $k+1$ in the parabolic sense converge on compact subsets of $\mathbb{R}\times(0,\infty)$. The curvature nonlinearity is genuinely washed out by the scaling, since $f_\sigma(r)=\sigma f(\sigma^{-1}r)\to f'(0)r$ as $\sigma\to\infty$. The paper presents this as the first global existence and asymptotic-self-similarity theorem for this nonlinear flow, and as a direct justification of the thermal-grooving explanation based on the Gibbs–Thomson law without linearizing $f$ near $\kappa=0$.

Load-bearing premise

The load-bearing premise is that estimate (6.5) forces every subsequential limit $U$ into the uniqueness class used in [15, Theorem 3.4]; the paper asserts the needed Hölder inequality without proof, so if its constant does not fall below the smallness threshold the argument would only establish convergence along subsequences.

Editorial extensions

If this is right

The thermal-grooving profile of the linearized surface diffusion equation is recovered as the actual large-time attractor of the fully nonlinear exponential flow.
For small-gradient data, the exponential flow and the conventional linear flow share the same self-similar limit, so the exponential nonlinearity does not change the leading-order shape of grooves at large times.
The a priori decay estimate (1.8) converts local-in-time existence into global existence and supplies quantitative derivative bounds; the argument is designed to extend to higher-dimensional graph-like surfaces as indicated in Remark 1.4.
Under the stated smallness condition, any two subsequential limits of the rescaled solutions coincide, so the large-time profile is unique rather than one of a family.

Reading between the lines

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

If the same scaled-Hölder machinery extends to $\mathbb{R}^n$ as Remark 1.4 suggests, the exponential flow should admit self-similar solutions for small Lipschitz data on the sphere at infinity, paralleling the linearized theory.
The paper proves convergence only on compact sets; extending the equi-decay argument that Remark 1.5 attributes to [3] would upgrade the result to uniform convergence on all of $\mathbb{R}$.
The Hölder inequality asserted in the proof of Theorem 1.2 could be written out with its constant; if the constant is not small enough, the theorem would still give subsequential convergence, and only the uniqueness of the limit would be open.
The theorem makes a concrete prediction for crystal-surface experiments: long-time groove profiles should match the linearized self-similar solution whenever the initial slope and its spatial derivative are small, even if curvature itself is not small initially.

Share X Bluesky LinkedIn Reddit HN

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

Bottom line: this is the first global well-posedness and large-time self-similar convergence result for the exponential surface diffusion flow (1.1), and the main proof line is sound. The authors write v=u_x as a perturbed biharmonic heat equation, introduce weighted Hölder norms tailored to break the scaling invariance caused by the nonlinear curvature term, prove global linear decay estimates, close the nonlinear smallness, then pass to the limit. For f(r)=e^r this is genuinely new: earlier surface diffusion results (Koch–Lamm, Du–Yip) treated the linear flow f(r)=r, and the second-order exponential flow needed viscosity methods. The weighted Hölder framework is the paper's main technical contribution and I expect it to be reusable.

Soft spots, in proportion. (1) The proof that the whole family u_σ converges, not just subsequences, depends on putting the limit U in Koch–Lamm's X∞ uniqueness class. The paper derives ∥U∥_{X∞} ≤ C sup_t(∥U_x∥_∞ + t^{-1/4}∥U_xx∥_∞) from (6.5) "by the Hölder inequality". That exponent is wrong as written: (6.5) gives t^{1/4}∥U_xx∥_∞ ≤ Cε, so t^{-1/4}∥U_xx∥_∞ generally blows up near t=0. The correct estimate is likely recoverable by integrating t^{1/4}∥U_xx∥_∞ over parabolic cylinders, but that computation plus constant tracking is absent. If Cε* were not below ρ*, the argument would only give subsequential convergence. This is a fixable gap, not a fatal one. (2) The display f_σ(r)=σ f(σ^{-1}r)→r in (6.6) is not literally true unless f(0)=0; for f(r)=e^r, f_σ diverges by a constant. Since only derivatives of f_σ matter, the limiting equation is still correct after subtracting f(0), but the statement should be repaired. (3) Minor: the honest comparison with Du–Yip in Remark 1.5 is good to see, and the convergence is only on compact sets, not uniformly in R.

Who is this for: geometric analysts and anyone working on fourth-order quasilinear parabolic equations. The core theorem is new and the main estimates appear solid. I would send it to peer review; the referee should ask for a written proof of the uniqueness-class verification and a correction of the f_σ display, but not for a rewrite.

Referee Report

1 major / 5 minor

Summary. The paper studies the exponential surface diffusion flow V = ∂_s^2 f(-κ) for graphs u(x,t) on R, where f is strictly increasing and f'(0)=1. It establishes global-in-time existence in Hölder spaces for initial data whose first two derivatives are small, with weighted decay estimates for all derivatives. It then studies the rescaling u_σ(x,t)=σ^{-1}u(σx,σ^4t) and claims convergence, uniformly on compact subsets of R×[0,∞), to the unique self-similar solution of the linearized surface diffusion equation (1.11), with derivatives converging on compact subsets of R×(0,∞). The proof rewrites the slope equation as a perturbed biharmonic heat equation, proves global linear decay estimates in weighted scaled Hölder norms, propagates smallness nonlinearly, and passes to the limit by Ascoli–Arzelà.

Significance. If correct, this is the first global well-posedness and large-time asymptotic profile theorem for the exponential surface diffusion flow on the whole line, going beyond the conventional linear case studied by Koch–Lamm and Du–Yip. The physical motivation, justifying Mullins grooving directly from the Gibbs–Thomson law without linearizing f near zero, is clearly explained. A particular strength is that the central decay estimate is a genuine a priori estimate against the biharmonic heat equation, with no fitting parameters and no circular use of the conclusion. The proof is detailed and the weighted Hölder framework is well adapted to the fourth-order quasilinear problem. The main caveat is a gap in the final uniqueness-class verification needed for full, rather than subsequential, convergence in Theorem 1.2.

major comments (1)

[Section 6, proof of Theorem 1.2, after Eq. (6.5)] The argument that every subsequential limit U lies in the Koch–Lamm uniqueness class is incomplete. The manuscript invokes the display ||U||_{X∞} ≤ C sup_{t>0}(||U_x||_{L∞} + t^{-1/4}||U_{xx}||_{L∞})(t) 'by the Hölder inequality', but this inequality is not correct as written: estimate (6.5) gives t^{1/4}||U_{xx}||_{L∞(R×(t/2,t))} ≤ C||(u0)_x||_{W^{1,∞}}, not t^{-1/4}, and for a self-similar profile the quantity t^{-1/4}||U_{xx}||_{L∞} is typically unbounded near t=0. The intended bound on ||U||_{X∞} can be recovered by using t^{1/4}||U_{xx}|| and integrating the pointwise bound over parabolic cylinders of radius R^4, but this computation is not written down. In addition, the constant C is not tracked through the passage u_σ→U, and the choice of ε* satisfying Cε*<ρ* is only asserted as 'sufficiently small'. Since this step upgrades subsequential convergence to convergence of the full family u_σ in Theorem 1.2, it is load-bearing and needs to be supplied.

minor comments (5)

[Section 6, proof of Theorem 1.2, last displayed line] The sentence 'Hence lim_{σ→∞} ||u_σ−U||_{L∞(K)} = 0 for every compact set σ → ∞' is garbled; it should read 'for every compact set K ⊂ R×[0,∞)'.
[Section 6, Eq. (6.2)] The displayed chain is missing a factor σ^3 on the middle right-hand side: from (5.10) one obtains σ^3||u_t||'... ≤ C||(u0)_x|| σ^3(σ^4t)^{-1/2}(1+σ^4t)^{-1/4}, which simplifies to the stated final bound ≤ C||(u0)_x|| t^{-3/4}; the intermediate inequality as written is not literally correct.
[Lemma 5.4, end of induction proof] The sentence 'By induction, we obtain v ∈ Z^k_T for all k ∈ Z^k_T for all k ∈ Z≥4' contains a typo and should read 'for all k ∈ Z with k ≥ 4'.
[Theorem 1.2 statement and proof] Under the stated fixed-k assumption, the proof yields U ∈ C^{k+1+μ,(k+1+μ)/4}_{time loc}(R×(0,∞)) rather than U ∈ C^∞(R×(0,∞)); the C^∞ conclusion requires either an additional smoothness argument allowing arbitrary k, or a weakened regularity statement.
[Lemma 4.2 proof] The symbol III is used both for the Duhamel integral ∫_{t̃}^{t} ∂_x^2 e^{-(t-s)∂_x^4}F(s)ds and, implicitly, for the function to which Proposition 2.6 is applied; since the two coincide only after identifying ∂_x^2 e^{-(t-s)∂_x^4}F with e^{-(t-s)∂_x^4}F_{xx}, a clarifying sentence would remove ambiguity.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: the large-time self-similar convergence is a genuine estimate against the biharmonic heat equation, with uniqueness imported from independent Koch–Lamm theory.

full rationale

The central claim (Theorem 1.2) is a convergence result for the rescaled flow u_sigma to the unique self-similar solution of the linearized problem V = -f'(0)∂_s^2 κ. The proof rests on (i) the global decay estimate (1.8)/(5.9) obtained from a perturbative treatment of v = u_x against the biharmonic heat semigroup, with all nonlinear remainders estimated by the weighted Hölder norm Z_T^k; (ii) a compactness argument via Ascoli–Arzelà; and (iii) identification of any subsequential limit U as a solution of (1.11) with piecewise-linear initial data, followed by uniqueness imported from Koch–Lamm [15, Theorem 3.4]. No parameter is fitted, and no 'prediction' is equivalent to an input by construction. The only intra-paper citations are to standard Hölder-norm estimates ([12, Lemmas 2.16 and 2.17]) from the second author's thesis; these are technical tools and do not carry the central claim, so they do not amount to load-bearing circularity. The passage from (6.5) to the X_∞-bound on U is asserted via a Hölder inequality with a t^{-1/4} factor that appears to be a typo (the natural factor is t^{1/4}); even if this is a genuine gap affecting the proof of full convergence versus subsequential convergence, it is a correctness issue, not a circularity, because it does not reduce the theorem to its assumptions or to a self-citation. The paper is otherwise self-contained against external benchmarks such as Schauder theory and Koch–Lamm's uniqueness theorem.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters or invented entities. The proof relies on standard parabolic regularity theory, the biharmonic heat kernel, and an external uniqueness theorem for the limiting self-similar profile. The theorem hypotheses, small initial slope and f'(0)=1 with f'>0, are explicit in the statement, not hidden assumptions.

assumptions (5)

standard math Parabolic Schauder estimates for linear parabolic systems (Solonnikov [23, Theorems 4.10, 4.11])
Used in Sections 3 and 4 to solve the linearized problem and to control the perturbed biharmonic equation.
standard math Pointwise estimates for the biharmonic heat kernel and its derivatives [23, Section 4]
Provides the smoothing estimates in Proposition 2.3.
standard math Contractivity property of lower order parabolic Hölder norms (Gösswein [12, Lemma 2.17])
Used in Lemma 5.5 and elsewhere to control lower-order terms by higher-order norms on short time intervals.
standard math Koch-Lamm uniqueness and existence of small self-similar solutions for the linearized flow [15, Theorem 3.4]
Used at the end of the proof of Theorem 1.2 to identify the unique limit U of rescaled solutions.
standard math Ascoli-Arzela compactness theorem
Used in Section 6 to extract convergent subsequences from the equicontinuous family u_σ.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Large time behavior of exponential surface diffusion flows on $\mathbb{R}$." pith.science (2026). https://pith.science/paper/723S645E

@misc{pith2026241117175,
  author       = {Pith},
  title        = {Pith review of: Large time behavior of exponential surface diffusion flows on $\mathbbR$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/723S645E}},
  note         = {Machine review of arXiv:2411.17175}
}

abstract

We consider a surface diffusion flow of the form $V=\partial_s^2f(-\kappa)$ with a strictly increasing smooth function $f$ typically, $f(r)=e^r$, for a curve with arc-length parameter $s$, where $\kappa$ denotes the curvature and $V$ denotes the normal velocity. The conventional surface diffusion flow corresponds to the case when $f(r)=r$. We consider this equation for the graph of a function defined on the whole real line $\mathbb{R}$. We prove that there exists a unique global-in-time classical solution provided that the first and the second derivatives are bounded and small. We further prove that the solution behaves like a solution to a self-similar solution to the equation $V=-f'(0)\kappa$. Our result justifies the explanation for grooving modeled by Mullins (1957) directly obtained by Gibbs--Thomson law without linearization of $f$ near $\kappa=0$.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Remarks on graph-like forward self-similar solutions to the surface diffusion flow equations
math.AP 2025-06 conditional novelty 6.0 of 10

For the planar surface diffusion flow, small V-shaped data produce genuine nonlinear forward self-similar graph solutions, while a growth condition forces any profile to be a line.

Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages · cited by 1 Pith paper

[1]

Asai, Quasilinear parabolic equation and its applications to fou rth order equations with rough initial data , J

T. Asai, Quasilinear parabolic equation and its applications to fou rth order equations with rough initial data , J. Math. Sci. Univ. Tokyo 19 (2012), 507–532 (2013)

work page 2012
[2]

Asai and Y

T. Asai and Y. Giga, On self-similar solutions to the surface diﬀusion ﬂow equati ons with contact angle boundary conditions, Interfaces Free Bound. 16 (2014), 539–573

work page 2014
[3]

Du and N

H. Du and N. K. Yip, Stability of self-similar solutions to geometric ﬂows , Interfaces Free Bound. 25 (2023), 155–191

work page 2023
[4]

Escher, U

J. Escher, U. F. Mayer, and G. Simonett, The surface diﬀusion ﬂow for immersed hypersurfaces , SIAM J. Math. Anal. 29 (1998), 1419–1433

work page 1998
[5]

Escher and P

J. Escher and P. B. Mucha, The surface diﬀusion ﬂow on rough phase spaces , Discrete Contin. Dyn. Syst. 26 (2010), 431–453

work page 2010
[6]

L. C. Evans, Partial diﬀerential equations , Second, Grad. Stud. Math., vol. 19, American Mathematical Society, Providence, RI, 2010

work page 2010
[7]

Garcke and M

H. Garcke and M. G¨ oßwein, On the surface diﬀusion ﬂow with triple junctions in higher sp ace dimensions , Geom. Flows 5 (2020), 1–39

work page 2020
[8]

Diﬀerential Equations 302 (2021), 617–661

, Non-linear stability of double bubbles under surface diﬀusi on, J. Diﬀerential Equations 302 (2021), 617–661

work page 2021

Show all 26 references

[9]

Garcke, K

H. Garcke, K. Ito, and Y. Kohsaka, Linearized stability analysis of stationary solutions for surface diﬀusion with boundary conditions , SIAM J. Math. Anal. 36 (2005), 1031–1056

2005
[10]

M.-H. Giga, Y. Giga, and J. Saal, Nonlinear Partial Diﬀerential Equations. Asymptotic Behav ior of Solutions and Self-Similar Solutions , Progress in Nonlinear Diﬀerential Equations and their App lications, vol. 79, Birkh¨ auser, Boston-Basel-Berlin, 2010

2010
[11]

Giga and T

Y. Giga and T. Miyakawa, Navier-Stokes ﬂow in R3 with measures as initial vorticity and Morrey spaces , Comm. Partial Diﬀerential Equations 14 (1989), 577–618

1989
[12]

G¨ oßwein, Surface diﬀusion ﬂow of triple junction clusters in higher sp ace dimensions , Ph.D

M. G¨ oßwein, Surface diﬀusion ﬂow of triple junction clusters in higher sp ace dimensions , Ph.D. Thesis, posted on 2019, DOI 10.5283/epub.38376

2019 doi
[13]

N. Hamamuki, Asymptotically self-similar solutions to curvature ﬂow eq uations with prescribed contact angle and their applications to groove proﬁles due to evaporation -condensation, Adv. Diﬀerential Equations 19 (2014), 317–358

2014
[14]

Ito and Y

K. Ito and Y. Kohsaka, Three-phase boundary motion by surface diﬀusion: stability of a mirror symmetric stationary solution , Interfaces Free Bound. 3 (2001), 45–80

2001
[15]

Koch and T

H. Koch and T. Lamm, Geometric ﬂows with rough initial data , Asian J. Math. 16 (2012), no. 2, 209–235

2012
[16]

R. V. Kohn and D. Margetis, Continuum Relaxation of Interacting Steps on Crystal Surfa ces in 2 + 1 Dimensions, Multiscale Modeling & Simulation 5 (2006), 729-758

2006
[17]

LeCrone, Y

J. LeCrone, Y. Shao, and G. Simonett, The surface diﬀusion and the Willmore ﬂow for uniformly regul ar hypersurfaces, Discrete Contin. Dyn. Syst. Ser. S 13 (2020), 3503–3524

2020
[18]

LeCrone and G

J. LeCrone and G. Simonett, On well-posedness, stability, and bifurcation for the axis ymmetric surface dif- fusion ﬂow , SIAM J. Math. Anal. 45 (2013), 2834–2869

2013
[19]

, On quasilinear parabolic equations and continuous maximal regularity, Evol. Equ. Control Theory 9 (2020), 61–86

2020
[20]

J.-G. Liu, J. Lu, D. Margetis, and J. L. Marzuola, Asymmetry in crystal facet dynamics of homoepitaxy by a continuum model , Physica D: Nonlinear Phenomena 393 (2019), 54-67

2019
[21]

W. W. Mullins, Two-dimensional motion of idealized grain boundaries , J. Appl. Phys. 27 (1956), 900–904

1956
[22]

, Theory of thermal grooving , J. Appl. Phys. 28 (1957), 333–339

1957
[23]

V. A. Solonnikov, On boundary value problems for linear parabolic systems of d iﬀerential equations of general form, Trudy Mat. Inst. Steklov. 83 (1965), 3–163 (Russian). 23

1965
[24]

Wheeler, Surface diﬀusion ﬂow near spheres , Calc

G. Wheeler, Surface diﬀusion ﬂow near spheres , Calc. Var. Partial Diﬀerential Equations 44 (2012), 131–151

2012
[25]

, On the curve diﬀusion ﬂow of closed plane curves , Ann. Mat. Pura Appl. (4) 192 (2013), 931–950

2013
[26]

, Convergence for global curve diﬀusion ﬂows , Math. Eng. 4 (2022), Paper No. 001, 13. (Y. Giga) Graduate School of Mathematical Sciences, The University o f Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan. Email address : labgiga@ms.u-tokyo.ac.jp (M. G¨ oßwein)Departmen...

2022

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Asai, Quasilinear parabolic equation and its applications to fou rth order equations with rough initial data , J

T. Asai, Quasilinear parabolic equation and its applications to fou rth order equations with rough initial data , J. Math. Sci. Univ. Tokyo 19 (2012), 507–532 (2013)

work page 2012

[2] [2]

Asai and Y

T. Asai and Y. Giga, On self-similar solutions to the surface diﬀusion ﬂow equati ons with contact angle boundary conditions, Interfaces Free Bound. 16 (2014), 539–573

work page 2014

[3] [3]

Du and N

H. Du and N. K. Yip, Stability of self-similar solutions to geometric ﬂows , Interfaces Free Bound. 25 (2023), 155–191

work page 2023

[4] [4]

Escher, U

J. Escher, U. F. Mayer, and G. Simonett, The surface diﬀusion ﬂow for immersed hypersurfaces , SIAM J. Math. Anal. 29 (1998), 1419–1433

work page 1998

[5] [5]

Escher and P

J. Escher and P. B. Mucha, The surface diﬀusion ﬂow on rough phase spaces , Discrete Contin. Dyn. Syst. 26 (2010), 431–453

work page 2010

[6] [6]

L. C. Evans, Partial diﬀerential equations , Second, Grad. Stud. Math., vol. 19, American Mathematical Society, Providence, RI, 2010

work page 2010

[7] [7]

Garcke and M

H. Garcke and M. G¨ oßwein, On the surface diﬀusion ﬂow with triple junctions in higher sp ace dimensions , Geom. Flows 5 (2020), 1–39

work page 2020

[8] [8]

Diﬀerential Equations 302 (2021), 617–661

, Non-linear stability of double bubbles under surface diﬀusi on, J. Diﬀerential Equations 302 (2021), 617–661

work page 2021