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Integrability-preserving regularizations of Laplacian Growth
Published 10 May 2024 in math-ph, math.DS, math.MP, math.ST, and stat.TH | (2405.06167v1)
Abstract: The Laplacian Growth (LG) model is known as a universality class of scale-free aggregation models in two dimensions, characterized by classical integrability and featuring finite-time boundary singularity formation. A discrete counterpart, Diffusion-Limited Aggregation (or DLA), has a similar local growth law, but significantly different global behavior. For both LG and DLA, a proper description for the scaling properties of long-time solutions is not available yet. In this note, we outline a possible approach towards finding the correct theory yielding a regularized LG and its relation to DLA.
- Equilibrium measure and the distribution of zeros of extremal polynomials. Mat. Sbornik, 125(2):117–127, 1984. translation from Mat. Sb., Nov. Ser. 134(176), No.3(11), 306-352 (1987).
- S. D. Howison, Fingering in Hele-Shaw cells, J. Fluid Mech., 167: 439–453, 1986.
- S. D. Howison, Complex variable methods in Hele-Shaw moving boundary problems, European J. Appl. Math., 3(3):209–224, 1992.
- P. P. Kufarev, A solution of the boundary problem of an oil well in a circle, Dokl. Acad. Nauk SSSR, 60 no. 8: 1333–1334, 1948.
- A proof of Freud’s conjecture for exponential weights. Constr. Approx., 4(1):65–83, 1988.
- E. Lundberg and D. Khavinson. Gravitational lensing by a collection of objects with radial densities. Analysis and Mathematical Physics, Volume 1, Numbers 2-3, 139-145, 2011.
- M. Y. Mo. The Riemann-Hilbert approach to double scaling limit of random matrix eigenvalues near the “birth of a cut” transition. Int. Math. Res. Not. IMRN, (13):Art. ID rnn042, 51, 2008.
- O. Marchal and M. Cafasso. Double-scaling limits of random matrices and minimal (2m,1)2𝑚1(2m,1)( 2 italic_m , 1 ) models: the merging of two cuts in a degenerate case. J. Stat. Mech. Theory Exp., P04013, 2011.
- Asymptotics of polynomial solutions of a class of generalized Lamé differential equations. Electron. Trans. Numer. Anal., 19:18–28, 2005.
- A. Martínez-Finkelshtein and E. A. Rakhmanov. Critical measures, quadratic differentials, and weak limits of zeros of Stieltjes polynomials. Comm. Math. Phys., 302(1):53–111, 2011.
- A. Martínez-Finkelshtein and E. B. Saff. Asymptotic properties of Heine-Stieltjes and Van Vleck polynomials. J. Approx. Theory, 118(1):131–151, 2002.
- Where does the sup norm of a weighted polynomial live? (A generalization of incomplete polynomials). Constr. Approx., 1:71–91, 1985.
- Integrable structure of interface dynamics, Physical Review Letters, 84:5106, 2000.
- On a class of equilibrium problems in the real axis. J. Comput. Appl. Math., 235(4):1065–1076, 2010.
- E. A. Rakhmanov. The convergence of diagonal Padé approximants. Mat. Sb. (N.S.), 104 (146)(2 (10)):271–291, 335, 1977. English translation: Math. USSR-Sb. 33 (1977), no. 2, 243–260.
- P. G. Saffman and G. Taylor, The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid, Proc. Roy. Soc. London. Ser. A, 245: 312–329, 1958.
- Dendritic and fractal patterns in electrolytic metal deposits, Phys. Rev. Lett., 56(12): 1260–1263, 1986.
- E. B. Saff and V. Totik. Logarithmic Potentials with External Fields, volume 316 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin, 1997.
- G. Taylor and P. G. Saffman, A note on the motion of bubbles in a Hele-Shaw cell and porous medium, Q J Mechanics Appl Math, 12(3): 265–279, 1959.
- R. Teodorescu, Generic critical points of normal matrix ensembles, J. Phys. A. Math. Teor., 39:8921, 2006.
- Why the boundary of a round drop becomes a curve of order four, vol. 3 of University Lecture Series, American Mathematical Society, Providence, RI, 1992.
- On some particular solutions of the problem of filtration, Doklady Akad. Nauk SSSR (N.S.), 57 no. 4: 335-338, 1947.
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