Ambrosio: The Flow Associated to Weakly Differentiable Vector Fields: Recent Results and Open Problems

Ambrosio, Luigi:
The Flow Associated to Weakly Differentiable Vector Fields: Recent Results and Open Problems
Bollettino dell'Unione Matematica Italiana Serie 8 10-B (2007), fasc. n.1, p. 25-41, Unione Matematica Italiana (English)
pdf (473 Kb), djvu (177 Kb). | MR 2310956 | Zbl 1178.34069

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In questa nota descriviamo alcuni recenti sviluppi della teoria dei flussi as- sociati a campi vettoriali poco regolari rispetto alle variabili spaziali, ad esempio con regolarità di tipo Sobolev o BV. Dopo aver illustrato alcune applicazioni a leggi di conservazione e equazioni della fluidodinamica, diamo una presentazione di tipo assiomatico del problema, usando un linguaggio di tipo probabilistico ispirato dalla teoria di L.C. Young. Nella parte finale discutiamo dei risultati ancora più recenti sulla regolarità del flusso stesso rispetto alle variabili spaziali.

Referenze Bibliografiche

[1] M. AIZENMAN, On vector fields as generators of flows: a counterexample to Nelson's conjecture, Ann. Math., 107 (1978), 287-296. | MR 482853 | Zbl 0394.28012

[2] G. ALBERTI, Rank-one properties for derivatives of functions with bounded variation, Proc. Roy. Soc. Edinburgh Sect. A, 123 (1993), 239-274. | fulltext (doi) | MR 1215412 | Zbl 0791.26008

[3] G. ALBERTI - L. AMBROSIO, A geometric approach to monotone functions in $\mathbb{R}^n$, Math. Z., 230 (1999), 259-316. | fulltext (doi) | MR 1676726 | Zbl 0934.49025

[4] G. ALBERTI - S.MÜLLER, A new approach to variational problems with multiple scales, Comm. Pure Appl. Math., 54 (2001), 761-825. | fulltext (doi) | MR 1823420 | Zbl 1021.49012

[5] F. J. ALMGREN, The theory of varifolds - A variational calculus in the large, Princeton University Press, 1972.

[6] L. AMBROSIO - N. FUSCO - D. PALLARA, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, 2000. | MR 1857292 | Zbl 0957.49001

[7] L. AMBROSIO, Transport equation and Cauchy problem for BV vector fields, Inventiones Mathematicae, 158 (2004), 227-260. | fulltext (doi) | MR 2096794 | Zbl 1075.35087

[8] L. AMBROSIO - C. DE LELLIS, Existence of solutions for a class of hyperbolic systems of conservation laws in several space dimensions, International Mathematical Research Notices, 41 (2003), 2205-2220. | fulltext (doi) | MR 2000967 | Zbl 1061.35048

[9] L. AMBROSIO - C. DE LELLIS, A note on admissible solutions of 1d scalar conservation laws and 2d Hamilton-Jacobi equations, Journal of Hyperbolic Differential Equations, 1 (4) (2004), 813-826. | fulltext (doi) | MR 2111584 | Zbl 1071.35032

[10] L. AMBROSIO, Lecture notes on transport equation and Cauchy problem for BV vector fields and applications, Preprint, 2004 (available at http://cvgmt.sns.it). | fulltext (doi) | MR 2096794

[11] L. AMBROSIO, Lecture notes on transport equation and Cauchy problem for non-smooth vector fields and applications, Preprint, 2005 (available at http://cvgmt.sns.it). | fulltext (doi) | MR 2408257

[12] L. AMBROSIO - G. CRIPPA, Existence, uniqueness, stability and differentiability properties of the flow associated to weakly differentiable vector fields, Preprint, 2006 (available at http://cvgmt.sns.it). | fulltext (doi) | MR 2409676

[13] L. AMBROSIO - F. BOUCHUT - C. DE LELLIS, Well-posedness for a class of hyperbolic systems of conservation laws in several space dimensions, Comm. PDE, 29 (2004), 1635-1651. | fulltext (doi) | MR 2103848 | Zbl 1072.35116

[14] L. AMBROSIO - G. CRIPPA - S. MANIGLIA, Traces and fine properties of a BD class of vector fields and applications, Ann. Sci. Toulouse, XIV (4) (2005), 527-561. | fulltext EuDML | MR 2188582 | Zbl 1091.35007

[15] L. AMBROSIO - N. GIGLI - G. SAVARÉ, Gradient flows in metric spaces and in the Wasserstein space of probability measures, Lectures in Mathematics, ETH Zurich, Birkhäuser, 2005. | MR 2129498

[16] L. AMBROSIO - M. LECUMBERRY - S. MANIGLIA, Lipschitz regularity and approx- imate differentiability of the DiPerna-Lions flow. Rendiconti del Seminario Fisico Matematico di Padova, 114 (2005), 29-50. | fulltext EuDML | MR 2207860 | Zbl 1370.26014

[17] L. AMBROSIO - J. MALÝ, Very weak notions of differentiability, Preprint, 2005 (available at http://cvgmt.sns.it). | fulltext (doi) | MR 2332676

[18] L. AMBROSIO - C. DE LELLIS - J. MALÝ, On the chain rule for the divergence of BV like vector fields: applications, partial results, open problems, Preprint, 2005 (available at http://cvgmt. sns.it). | fulltext (doi) | MR 2373724

[19] L. AMBROSIO - S. LISINI - G. SAVARÉ, Stability of flows associated to gradient vector fields and convergence of iterated transport maps, Preprint, 2005 (available at http://cvgmt.sns.it). | fulltext (doi) | MR 2258529

[20] E. J. BALDER, New fundamentals of Young measure convergence, CRC Res. Notes in Math. 411, 2001. | MR 1713855 | Zbl 0964.49011

[21] V. BANGERT, Minimal measures and minimizing closed normal one-currents, Geom. funct. anal., 9 (1999), 413-427. | fulltext (doi) | MR 1708452 | Zbl 0973.58004

[22] J. BALL - R. JAMES, Fine phase mixtures as minimizers of energy, Arch. Rat. Mech. Anal., 100 (1987), 13-52. | fulltext (doi) | MR 906132 | Zbl 0629.49020

[23] J.-D. BENAMOU - Y. BRENIER, Weak solutions for the semigeostrophic equation formulated as a couples Monge-Ampere transport problem, SIAM J. Appl. Math., 58 (1998), 1450-1461. | fulltext (doi) | MR 1627555 | Zbl 0915.35024

[24] P. BERNARD - B. BUFFONI, Optimal mass transportation and Mather theory, Preprint, 2004. | fulltext (doi) | MR 2283105

[25] M. BERNOT - V. CASELLES - J. M. MOREL, Traffic plans, Preprint, 2004. | fulltext (doi) | MR 2177636

[26] V. BOGACHEV - E. M. WOLF, Absolutely continuous flows generated by Sobolev class vector fields in finite and infinite dimensions, J. Funct. Anal., 167 (1999), 1-68. | fulltext (doi) | MR 1710649 | Zbl 0956.60079

[27] Y. BRENIER, The least action principle and the related concept of generalized flows for incompressible perfect fluids, J. Amer. Mat. Soc., 2 (1989), 225-255. | fulltext (doi) | MR 969419 | Zbl 0697.76030

[28] Y. BRENIER, The dual least action problem for an ideal, incompressible fluid, Arch. Rational Mech. Anal., 122 (1993), 323-351. | fulltext (doi) | MR 1217592 | Zbl 0797.76006

[29] Y. BRENIER, A homogenized model for vortex sheets, Arch. Rational Mech. Anal., 138 (1997), 319-353. | fulltext (doi) | MR 1467558 | Zbl 0962.35140

[30] Y. BRENIER, Minimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations, Comm. Pure Appl. Math., 52 (1999), 411-452. | fulltext (doi) | MR 1658919 | Zbl 0910.35098

[31] F. BOUCHUT - F. JAMES, One dimensional transport equation with discontinuous coefficients, Nonlinear Analysis, 32 (1998), 891-933. | fulltext (doi) | MR 1618393 | Zbl 0989.35130

[32] F. BOUCHUT - F. GOLSE - M. PULVIRENTI, Kinetic equations and asymptotic theory, Series in Appl. Math., Gauthiers-Villars, 2000. | MR 2065070 | Zbl 0979.82048

[33] F. BOUCHUT, Renormalized solutions to the Vlasov equation with coefficients of bounded variation, Arch. Rational Mech. Anal., 157 (2001), 75-90. | fulltext (doi) | MR 1822415 | Zbl 0979.35032

[34] F. BOUCHUT - F. JAMES - S. MANCINI, Uniqueness and weak stability for multi- dimensional transport equations with one-sided Lipschitz coefficients, Annali Scuola Normale Superiore, Ser. 5, 4 (2005), 1-25. | fulltext EuDML | MR 2165401 | Zbl 1170.35363

[35] F. BOUCHUT - G. CRIPPA, On the relations between uniqueness for the Cauchy problem and existence of smooth approximations for linear transport equations, Preprint, 2006 (available at http://cvgmt.sns.it). | fulltext (doi) | MR 2274485 | Zbl 1122.35104

[36] A. BRESSAN, An ill posed Cauchy problem for a hyperbolic system in two space dimensions, Rend. Sem. Mat. Univ. Padova, 110 (2003), 103-117. | fulltext EuDML | MR 2033003 | Zbl 1114.35123

[37] H. BREZIS, Analyse fonctionnelle. Théorie et applications, Masson, Paris, 1983. | MR 697382

[38] L. A. CAFFARELLI, Some regularity properties of solutions of Monge Ampére equation, Comm. Pure Appl. Math., 44 (1991), 965-969. | fulltext (doi) | MR 1127042 | Zbl 0761.35028

[39] L. A. CAFFARELLI, Boundary regularity of maps with convex potentials, Comm. Pure Appl. Math., 45 (1992), 1141-1151. | fulltext (doi) | MR 1177479 | Zbl 0778.35015

[40] L. A. CAFFARELLI, The regularity of mappings with a convex potential, J. Amer. Math. Soc., 5 (1992), 99-104. | fulltext (doi) | MR 1124980 | Zbl 0753.35031

[41] L. A. CAFFARELLI, Boundary regularity of maps with convex potentials, Ann. of Math., 144 (1996), 453-496. | fulltext (doi) | MR 1426885 | Zbl 0916.35016

[42] I. CAPUZZO DOLCETTA - B. PERTHAME, On some analogy between different ap- proaches to first order PDE's with nonsmooth coefficients, Adv. Math. Sci Appl., 6 (1996), 689-703. | MR 1411988 | Zbl 0865.35032

[43] A. CELLINA, On uniqueness almost everywhere for monotonic differential inclusions, Nonlinear Analysis, TMA, 25 (1995), 899-903. | fulltext (doi) | MR 1350714 | Zbl 0837.34023

[44] A. CELLINA - M. VORNICESCU, On gradient flows, Journal of Differential Equations, 145 (1998), 489-501. | fulltext (doi) | MR 1620979 | Zbl 0927.37007

[45] F. COLOMBINI - G. CRIPPA - J. RAUCH, A note on two dimensional transport with bounded divergence, Accepted by Comm. Partial Differential Equations, 2005. | fulltext (doi) | MR 2254607

[46] F. COLOMBINI - N. LERNER, Uniqueness of continuous solutions for BV vector fields, Duke Math. J., 111 (2002), 357-384. | fulltext (doi) | MR 1882138 | Zbl 1017.35029

[47] F. COLOMBINI - N. LERNER, Uniqueness of $L^\infty$ solutions for a class of conormal BV vector fields, Contemp. Math. 368 (2005), 133-156. | fulltext (doi) | MR 2126467 | Zbl 1064.35033

[48] F. COLOMBINI - T. LUO - J. RAUCH, Neraly Lipschitzean diverge free transport propagates neither continuity nor BV regularity, Commun. Math. Sci., 2 (2004), 207- 212. | MR 2119938 | Zbl 1088.35015

[49] G. CRIPPA - C. DE LELLIS, Oscillatory solutions to transport equations, Preprint, 2005 (available at http://cvgmt.sns.it). | fulltext (doi) | MR 2207545 | Zbl 1098.35101

[50] G. CRIPPA - C. DE LELLIS, Estimates for transport equations and regularity of the DiPerna-Lions flow, Preprint, 2006. | fulltext (doi) | MR 2369485 | Zbl 1144.35365

[51] A. B. CRUZEIRO, Équations différentielles ordinaires: non explosion et mesures quasi-invariantes, J. Funct. Anal., 54 (1983), 193-205. | fulltext (doi) | MR 724704 | Zbl 0523.28020

[52] A. B. CRUZEIRO, Équations différentielles sur l'espace de Wiener et formules de Cameron-Martin non linéaires, J. Funct. Anal., 54 (1983), 206-227. | fulltext (doi) | MR 724705 | Zbl 0524.47028

[53] A. B. CRUZEIRO, Unicité de solutions d'équations différentielles sur l'espace de Wiener, J. Funct. Anal., 58 (1984), 335-347. | fulltext (doi) | MR 759104 | Zbl 0551.47019

[54] M. CULLEN, On the accuracy of the semi-geostrophic approximation, Quart. J. Roy. Metereol. Soc., 126 (2000), 1099-1115.

[55] M. CULLEN - W. GANGBO, A variational approach for the 2-dimensional semi- geostrophic shallow water equations, Arch. Rational Mech. Anal., 156 (2001), 241-273. | fulltext (doi) | MR 1816477 | Zbl 0985.76008

[56] M. CULLEN - M. FELDMAN, Lagrangian solutions of semigeostrophic equations in physical space, Preprint, 2003. | fulltext (doi) | MR 2215268 | Zbl 1097.35004

[57] C. DAFERMOS, Hyperbolic conservation laws in continuum physics, Springer Verlag, 2000. | fulltext (doi) | MR 1763936 | Zbl 0940.35002

[58] C. DE LELLIS, Blow-up of the BV norm in the multidimensional Keyfitz and Kranzer system, Duke Math. J., 127 (2004), 313-339. | fulltext (doi) | MR 2130415 | Zbl 1074.35073

[59] L. DE PASCALE - M. S. GELLI - L. GRANIERI, Minimal measures, one-dimensional currents and the Monge-Kantorovich problem, Preprint, 2004 (available at http://cvgmt.sns.it). | fulltext (doi) | MR 2241304 | Zbl 1096.37033

[60] N. DEPAUW, Non unicité des solutions bornées pour un champ de vecteurs BV en dehors d'un hyperplan, C. R. Math. Sci. Acad. Paris, 337 (2003), 249-252. | fulltext (doi) | MR 2009116 | Zbl 1024.35029

[61] N. G. DE BRUIJN, On almost additive functions, Colloq. Math. 15 (1966), 59-63. | fulltext EuDML | fulltext (doi) | MR 196026 | Zbl 0147.12602

[62] R. J. DIPERNA, Measure-valued solutions to conservation laws. Arch. Rational Mech. Anal., 88 (1985), 223-270. | fulltext (doi) | MR 775191 | Zbl 0616.35055

[63] R. J. DI PERNA - P. L. LIONS, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547. | fulltext EuDML | fulltext (doi) | MR 1022305 | Zbl 0696.34049

[64] R. J. DI PERNA - P. L. LIONS, On the Cauchy problem for the Boltzmann equation: global existence and weak stability, Ann. of Math., 130 (1989), 312-366. | fulltext (doi) | MR 1014927

[65] L. C. EVANS - R. F. GARIEPY, Lecture notes on measure theory and fine properties of functions, CRC Press, 1992. | MR 1158660 | Zbl 0804.28001

[66] L. C. EVANS, Partial Differential Equations, Graduate studies in Mathematics, 19 (1998), American Mathematical Society. | MR 1625845

[67] L. C. EVANS, Partial Differential Equations and Monge-Kantorovich Mass Transfer, Current Developments in Mathematics, 1997, 65-126. | MR 1698853 | Zbl 0954.35011

[68] L. C. EVANS - W. GANGBO, Differential equations methods for the Monge-Kantorovich mass transfer problem, Memoirs AMS, 653, 1999. | fulltext (doi) | MR 1464149 | Zbl 0920.49004

[69] L. C. EVANS - W. GANGBO - O. SAVIN, Nonlinear heat flows and diffeomorphisms, Preprint, 2004. | fulltext (doi) | MR 2191774 | Zbl 1096.35061

[70] H. FEDERER, Geometric measure theory, Springer, 1969. | MR 257325 | Zbl 0176.00801

[71] A. FIGALLI, Existence and uniqueness of martingale solutions for SDE with rough or degenerate coefficients, Preprint, 2006, available at http://cvgmt.sns.it. | fulltext (doi) | MR 2375067

[72] M. HAURAY, On Liouville transport equation with potential in $\operatorname{BV}_{\text{loc}}$, Comm. in PDE, 29 (2004), 207-217. | fulltext (doi) | MR 2038150 | Zbl 1103.35003

[73] M. HAURAY, On two-dimensional Hamiltonian transport equations with $L^p_{\text{loc}}$ coefficients, Ann. IHP Nonlinear Anal. Non Linéaire, 20 (2003), 625-644. | fulltext EuDML | fulltext (doi) | MR 1981402 | Zbl 1028.35148

[74] W. B. JURKAT, On Cauchy's functional equation, Proc. Amer. Math. Soc., 16 (1965), 683-686. | fulltext (doi) | MR 179496 | Zbl 0133.37702

[75] B. L. KEYFITZ - H. C. KRANZER, A system of nonstrictly hyperbolic conservation laws arising in elasticity theory, Arch. Rational Mech. Anal. 1980, 72, 219-241. | fulltext (doi) | MR 549642 | Zbl 0434.73019

[76] C. LE BRIS - P. L. LIONS, Renormalized solutions of some transport equations with partially W 1Y1 velocities and applications, Annali di Matematica, 183 (2004), 97-130. | fulltext (doi) | MR 2044334 | Zbl 1170.35364

[77] N. LERNER, Transport equations with partially BV velocities, Preprint, 2004. | MR 2124585 | Zbl 1170.35362

[78] P. L. LIONS, Sur les équations différentielles ordinaires et les équations de transport, C. R. Acad. Sci. Paris Sér. I, 326 (1998), 833-838. | fulltext (doi) | MR 1648524

[79] P. L. LIONS, Mathematical topics in fluid mechanics, Vol. I: incompressible models, Oxford Lecture Series in Mathematics and its applications, 3 (1996), Oxford University Press. | MR 1422251 | Zbl 0866.76002

[80] P. L. LIONS, Mathematical topics in fluid mechanics, Vol. II: compressible models, Oxford Lecture Series in Mathematics and its applications, 10 (1998), Oxford University Press. | MR 1422251 | Zbl 0908.76004

[81] J. LOTT - C. VILLANI, Weak curvature conditions and Poincaré inequalities, Preprint, 2005. | fulltext (doi) | MR 2311627

[82] S. MANIGLIA, Probabilistic representation and uniqueness results for measure-valued solutions of transport equations, Preprint, 2005. | fulltext (doi) | MR 2335089 | Zbl 1123.60048

[83] J. N. MATHER, Minimal measures, Comment. Math. Helv., 64 (1989), 375-394. | fulltext EuDML | fulltext (doi) | MR 998855

[84] J. N. MATHER, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z., 207 (1991), 169-207. | fulltext EuDML | fulltext (doi) | MR 1109661 | Zbl 0696.58027

[85] E. Y. PANOV, On strong precompactness of bounded sets of measure-valued solutions of a first order quasilinear equation, Math. Sb., 186 (1995), 729-740. | fulltext (doi) | MR 1341087 | Zbl 0839.35139

[86] G. PETROVA - B. POPOV, Linear transport equation with discontinuous coefficients, Comm. PDE, 24 (1999), 1849-1873. | fulltext (doi) | MR 1708110 | Zbl 0992.35104

[87] F. POUPAUD - M. RASCLE, Measure solutions to the liner multidimensional transport equation with non-smooth coefficients, Comm. PDE, 22 (1997), 337-358. | fulltext (doi) | MR 1434148 | Zbl 0882.35026

[88] A. PRATELLI, Equivalence between some definitions for the optimal transport problem and for the transport density on manifolds, preprint, 2003, to appear on Ann. Mat. Pura Appl (available at http://cvgmt.sns.it). | fulltext (doi) | MR 2149093

[89] S. K. SMIRNOV, Decomposition of solenoidal vector charges into elementary solenoids and the structure of normal one-dimensional currents, St. Petersburg Math. J., 5 (1994), 841-867. | MR 1246427

[90] E. M. STEIN, Singular integrals and differentiability properties of functions, Princeton University Press, 1970. | MR 290095 | Zbl 0207.13501

[91] L. TARTAR, Compensated compactness and applications to partial differential equations, Research Notes in Mathematics, Nonlinear Analysis and Mechanics, ed. R. J. Knops, vol. 4, Pitman Press, New York, 1979, 136-211. | MR 584398 | Zbl 0437.35004

[92] R. TEMAM, Problémes mathématiques en plasticité, Gauthier-Villars, Paris, 1983. | MR 711964 | Zbl 0547.73026

[93] J. I. E. URBAS, Global Hölder estimates for equations of Monge-Ampère type, Invent. Math., 91 (1988), 1-29. | fulltext EuDML | fulltext (doi) | MR 918234

[94] J. I. E. URBAS, Regularity of generalized solutions of Monge-Ampère equations, Math. Z., 197 (1988), 365-393. | fulltext EuDML | fulltext (doi) | MR 926846 | Zbl 0617.35017

[95] A. VASSEUR, Strong traces for solutions of multidimensional scalar conservation laws, Arch. Ration. Mech. Anal., 160 (2001), 181-193. | fulltext (doi) | MR 1869441 | Zbl 0999.35018

[96] C. VILLANI, Topics in mass transportation, Graduate Studies in Mathematics, 58 (2004), American Mathematical Society.

[97] C. VILLANI, Optimal transport: old and new, Lecture Notes of the 2005 Saint-Flour Summer school.

[98] L. C. YOUNG, Lectures on the calculus of variations and optimal control theory, Saunders, 1969. | MR 259704 | Zbl 0177.37801

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