| Title:
|
The microstructure of Lipschitz solutions for a one-dimensional logarithmic diffusion equation (English) |
| Author:
|
Schadewaldt, Nicole |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
52 |
| Issue:
|
2 |
| Year:
|
2011 |
| Pages:
|
227-255 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We consider the initial-boundary-value problem for the one-dimensional fast diffusion equation $u_t = [\operatorname{sign}(u_x) \log |u_x|]_x$ on $Q_T=[0,T]\times[0,l]$. For monotone initial data the existence of classical solutions is known. The case of non-monotone initial data is delicate since the equation is singular at $u_x=0$. We ‘explicitly’ construct infinitely many weak Lipschitz solutions to non-monotone initial data following an approach to the Perona-Malik equation. For this construction we rephrase the problem as a differential inclusion which enables us to use methods from the description of material microstructures. The Lipschitz solutions are constructed iteratively by adding ever finer oscillations to an approximate solution. These fine structures account for the fact that solutions are not continuously differentiable in any open subset of $Q_T$ and that the derivative $u_x$ is not of bounded variation in any such open set. We derive a characterization of the derivative, namely $u_x = d^+ \mathbbm{1}_A + d^- \mathbbm{1}_B$ with continuous functions $d^+>0$ and $d^-<0$ and dense sets $A$ and $B$, both of positive measure but with infinite perimeter. This characterization holds for any Lipschitz solution constructed with the same method, in particular for the ‘microstructured’ Lipschitz solutions to the one-dimensional Perona-Malik equation. (English) |
| Keyword:
|
logarithmic diffusion |
| Keyword:
|
one-dimensional |
| Keyword:
|
differential inclusion |
| Keyword:
|
microstructured Lipschitz solutions |
| MSC:
|
34A05 |
| MSC:
|
35B05 |
| MSC:
|
35B65 |
| idZBL:
|
Zbl 1240.35225 |
| idMR:
|
MR2849047 |
| . |
| Date available:
|
2011-05-17T08:37:07Z |
| Last updated:
|
2013-09-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141500 |
| . |
| Reference:
|
[1] Ambrosio L., Fusco N., Pallara D.: Functions of Bounded Variation and Free Discontinuity Problems.Oxford Mathematical Monographs, Clarendon Press, Oxford, 2000. Zbl 0957.49001, MR 1857292 |
| Reference:
|
[2] Ball J.M.: A version of the fundamental theorem of Young measures.in Partial Differential Equations and Continuum Models of Phase Transitions, M. Rascle, D. Serre and M. Slemrod, editors, Lecture Notes in Physics, 334, Springer, Berlin, 1989, pp. 207–215. MR 1036070 |
| Reference:
|
[3] Ball J.M., James R.D.: Proposed experimental test of a theory of fine microstructure and the two-well problem.Philosophical Transactions: Physical Sciences and Engineering, 338 (1992), no. 1650, 389–450. |
| Reference:
|
[4] Conti S., Dolzmann B., Kirchheim B.: Existence of Lipschitz minimizers for the three-well problem in solid-solid phase transitions.Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007), 953–962. Zbl 1131.74037, MR 2371114, 10.1016/j.anihpc.2006.10.002 |
| Reference:
|
[5] Dacorogna B., Marcellini P.: Implicit Partial Differential Equations.Progress in Nonlinear Differential Equations and their Applications, 37, Birkhäuser, Basel, 1999. Zbl 0939.49013, MR 1702252 |
| Reference:
|
[6] DiBenedetto E., Herrero M.A.: Non-negative solutions of the evolution $p$-Laplacian equation. Initial traces and Cauchy problem when $1<p<2$.Arch. Rational Mech. Anal. 111 (1990), 225–290. Zbl 0726.35066, MR 1066761, 10.1007/BF00400111 |
| Reference:
|
[7] DiBenedetto E., Urbano J.M., Vespri V.: Current issues on singular and degenerate evolution equations.in Handbook of Differential Equations 1. Evolutionary Equations, C. Dafermos and E. Feireisel, editors, Elsevier, Amsterdam, 2004, pp. 169–286. Zbl 1082.35002, MR 2103698 |
| Reference:
|
[8] Evans L.C.: Partial Differential Equations.Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, Rhode Island, 2002. Zbl 1194.35001, MR 1625845 |
| Reference:
|
[9] Evans L.C., Gariepy R.F.: Measure Theory and Fine Properties of Functions.Studies in Advanced Mathematics, CRC Press, Boca Raton, Florida, 1992. Zbl 0804.28001, MR 1158660 |
| Reference:
|
[10] Höllig K.: Existence of infinitely many solutions for a forward backward heat equation.Trans. Amer. Math. Soc. 278 (1983), no. 1, 299–316. MR 0697076, 10.2307/1999317 |
| Reference:
|
[11] Iagar R.G., Sánchez A., Vázquez J.L.: Radial equivalence for the two basic nonlinear degenerate diffusion equations.J. Math. Pures Appl. (9) 89 (2008), no. 1, 1–24. MR 2378087 |
| Reference:
|
[12] Kawohl B., Kutev N.: Maximum and comparison principle for one-dimensional anisotropic diffusion.Math. Ann. 311 (1998), no. 1, 107–123. Zbl 0909.35025, MR 1624275, 10.1007/s002080050179 |
| Reference:
|
[13] Kinderlehrer D., Pedregal P.: Characterizations of Young measures generated by gradients.Arch. Rational Mech. Anal. 115 (1991), 329–365. Zbl 0754.49020, MR 1120852, 10.1007/BF00375279 |
| Reference:
|
[14] Kirchheim B.: Rigidity and Geometry of Microstructures.Lecture Notes, MPI for Mathematics in the Sciences, Leipzig, 2003. |
| Reference:
|
[15] Ladyzenskaja O.A., Solonnikov V.A., Ural'ceva N.N.: Linear and Quasilinear Equations of Parabolic Type.Translations of Mathematical Monographs, 23, American Mathematical Society, Providence, R.I., 1968. MR 0241822 |
| Reference:
|
[16] Müller S., Šverák V.: Unexpected solutions of first and second order partial differential equations.Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), pp. 691–702. MR 1648117 |
| Reference:
|
[17] Müller S., Šverák V.: Convex integration for Lipschitz mappings and counterexamples to regularity.Ann. of Math. 157 (2003), no. 3, 715–742. MR 1983780, 10.4007/annals.2003.157.715 |
| Reference:
|
[18] Pedregal P.: Parametrized Measures and Variational Principles.Progress in Nonlinear Differential Equations and their Applications, 30, Birkhäuser, Basel, 1997. Zbl 0879.49017, MR 1452107 |
| Reference:
|
[19] Perona P., Malik J.: Scale-space and edge detection using anisotropic diffusion.Pattern Analysis and Machine Intelligence 12 (1990), no. 7, 629–639. 10.1109/34.56205 |
| Reference:
|
[20] A. Rodriguez A., Vázquez J.: Obstructions to existence in fast-diffusion equations.J. Differential Equations 184 (2002), no. 2, 348–385. MR 1929882, 10.1006/jdeq.2001.4144 |
| Reference:
|
[21] Saks S.: Theory of the Integral.second edition, Hafner, New York, 1937. Zbl 0017.30004 |
| Reference:
|
[22] Schadewaldt N.: Lipschitz solutions for a one-dimensional fast diffusion equation.Ph.D. Thesis, Dr. Hut Verlag, 2009. |
| Reference:
|
[23] Székelyhidi L., Jr.: The regularity of critical points of polyconvex functionals.Arch. Rational Mech. Anal. 172 (2004), no. 1, 133–152. MR 2048569, 10.1007/s00205-003-0300-7 |
| Reference:
|
[24] Tartar L.: Compensated compactness and applications to partial differential equations.Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, IV, Research Notes in Mathematics, 39, Pitman, Boston, Mass.-London, 1979, pp. 136–212. Zbl 0437.35004, MR 0584398 |
| Reference:
|
[25] Vázquez J.L.: Smoothing and Decay Estimates for Nonlinear Diffusion Equations – Equations of Porous Medium Type.Oxford Lecture Series in Mathematics and its Applications, 33, Oxford University Press, Oxford, 2006. MR 2282669 |
| Reference:
|
[26] Vázquez J.L.: The Porous Medium Equation – Mathematical Theory.Oxford Mathematical Monographs, Clarendon Press, Oxford, 2007. MR 2286292 |
| Reference:
|
[27] Zhang K.: Existence of infinitely many solutions for the one-dimensional Perona-Malik model.Calc. Var. Partial Differential Equations 26 (2006), no. 2, 171–199. Zbl 1096.35067, MR 2222243, 10.1007/s00526-005-0363-4 |
| Reference:
|
[28] Zhang K.: On existence of weak solutions for one-dimensional forward-backward diffusion equations.J. Differential Equations, 220 (2006), 322–353. Zbl 1092.35048, MR 2183375, 10.1016/j.jde.2005.01.011 |
| Reference:
|
[29] Zhang K.: On the principle of controlled $L^\infty $ convergence implies almost everywhere convergence for gradients.Commun. Contemp. Math. 9 (2007), no. 1, 21–30. MR 2293558, 10.1142/S0219199707002320 |
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