## Transport equation and Cauchy problem for BV vector fields.(English)Zbl 1075.35087

The author extends the result by DiPerna and Lions about weak solutions $$w$$ of the Cauchy problem for the transport equation $$B\cdot\nabla w=c$$, to the case of $$B$$ vector field of bounded variation only (DiPerna and Lions studied the case of $$B$$ in a Sobolev space). In particular, he proves that every locally bounded weak solution is renormalizable whenever $$B$$ has locally bounded variation, and its distributional divergence locally belongs to $$L^1$$. The main idea starts from one of the previous improvements of DiPerna’s and Lions’ result, given by Colombini and Lerner. They proved a uniqueness result in the case that the derivative of the vector field is a measure only along one direction, and there is absolute continuity along the other ones.
Using Alberti’s theorem about the rank-one structure of the singular part of the derivative, the present author is able to prove that, in some sense, any vector field with bounded variation asymptotically behaves on a small scale as the vector fields considered by Colombini and Lerner. The renormalization property permits then to prove uniqueness and comparison results for the Cauchy problem. Then, the author uses his results to investigate, in a measure-theoretic framework, the concept of characteristics for the transport equation, and also to study the Lagrangian flow for bounded vector fields of locally bounded variation.
Finally, it is worth noticing that the results of the present paper are applied in other works by the author and collaborators, to give an existence result for systems of conservation laws, in dimensions more than 2, and with only bounded variation initial datum. Such a result considerably improves previous ones.

### MSC:

 35Q72 Other PDE from mechanics (MSC2000) 35K15 Initial value problems for second-order parabolic equations 35L65 Hyperbolic conservation laws
Full Text:

### References:

 [1] Aizenman, M.: On vector fields as generators of flows: a counterexample to Nelson?s conjecture. Ann. Math. 107, 287-296 (1978) · Zbl 0394.28012 [2] Alberti, G.: Rank-one properties for derivatives of functions with bounded variation. Proc. R. Soc. Edinb., Sect. A, Math. 123, 239-274 (1993) · Zbl 0791.26008 [3] Alberti, G., Ambrosio, L.: A geometric approach to monotone functions in ?n. Math. Z. 230, 259-316 (1999) · Zbl 0934.49025 [4] Alberti, G., Müller, S.: A new approach to variational problems with multiple scales. Commun. Pure Appl. Math. 54, 761-825 (2001) · Zbl 1021.49012 [5] Alberti, G.: Personal communication [6] Ambrosio, L., Fusco, N., Pallara, D.: Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs 2000 · Zbl 0957.49001 [7] Ambrosio, L., Kirchheim, B., Lecumberry, M., Riviere, T.: On the rectifiability of defect measures arising in a micromagnetics model. Nonlinear Problems in Mathematical Physics and related topics II, in honor of O.A. Ladyzhenskaya, ed. by M.S. Birman, S. Hildebrandt, V.A. Solonnikov and N. Uraltseva pp. 29-60. International Mathematical Series, Kluwer/Plenum 2002 · Zbl 1055.49008 [8] Ambrosio, L., De Lellis, C.: Existence of solutions for a class of hyperbolic systems of conservation laws in several space dimensions. Int. Math. Res. Not. 41, 2205-2220 (2003) · Zbl 1061.35048 [9] Ambrosio, L., Bouchut, F., De Lellis, C.: Well-posedness for a class of hyperbolic systems of conservation laws in several space dimensions. Preprint 2003 (submitted to Comm. PDE and available at http://cvgmt.sns.it) · Zbl 1072.35116 [10] Benamou, J.-D., Brenier, Y.: Weak solutions for the semigeostrophic equation formulated as a couples Monge-Ampere transport problem. SIAM J. Appl. Math. 58, 1450-1461 (1998) · Zbl 0915.35024 [11] Bouchut, F., James, F.: One dimensional transport equation with discontinuous coefficients. Nonlinear Anal. 32, 891-933 (1998) · Zbl 0989.35130 [12] Bouchut, F.: Renormalized solutions to the Vlasov equation with coefficients of bounded variation. Arch. Ration. Mech. Anal. 157, 75-90 (2001) · Zbl 0979.35032 [13] Bressan, A.: An ill posed Cauchy problem for a hyperbolic system in two space dimensions. Preprint 2003 · Zbl 1114.35123 [14] Capuzzo Dolcetta, I., Perthame, B.: On some analogy between different approaches to first order PDE?s with nonsmooth coefficients. Adv. Math. Sci. Appl. 6, 689-703 (1996) · Zbl 0865.35032 [15] Castaing, C., Valadier, M.: Convex analysis and measurable multifunctions. Lect. Notes Math. 580. Berlin: Springer 1977 · Zbl 0346.46038 [16] Cellina, A.: On uniqueness almost everywhere for monotonic differential inclusions. Nonlinear Anal., Theory Methods Appl. 25, 899-903 (1995) · Zbl 0837.34023 [17] Cellina, A., Vornicescu, M.: On gradient flows. J. Differ. Equations 145, 489-501 (1998) · Zbl 0927.37007 [18] Colombini, F., Lerner, N.: Uniqueness of continuous solutions for BV vector fields. Duke Math. J. 111, 357-384 (2002) · Zbl 1017.35029 [19] Colombini, F., Lerner, N.: Uniqueness of L? solutions for a class of conormal BV vector fields. Preprint 2003 · Zbl 1064.35033 [20] Colombini, F., Rauch, J.: Unicity and nonunicity for nonsmooth divergence free transport. Preprint 2003 · Zbl 1065.35089 [21] Cullen, M., Gangbo, W.: A variational approach for the 2-dimensional semi-geostrophic shallow water equations. Arch. Ration. Mech. Anal. 156, 241-273 (2001) · Zbl 0985.76008 [22] Cullen, M., Feldman, M.: Lagrangian solutions of semigeostrophic equations in physical space. To appear · Zbl 1097.35004 [23] De Lellis, C., Otto, F.: Structure of entropy solutions to the eikonal equation. J. Eur. Math. Soc. (JEMS) 5, 107-145 (2003) · Zbl 1053.49028 [24] De Lellis, C., Otto, F., Westdickenberg, M.: Structure of entropy solutions for multi-dimensional conservation laws. Arch. Ration. Mech. Anal. 170, 137-184 (2003) · Zbl 1036.35127 [25] De Pauw, N.: Non unicité des solutions bornées pour un champ de vecteurs BV en dehors d?un hyperplan. C.R. Math. Sci. Acad. Paris 337, 249-252 (2003) [26] Di Perna, R.J., Lions, P.L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, 511-547 (1989) · Zbl 0696.34049 [27] Evans, L.C., Gariepy, R.F.: Lecture notes on measure theory and fine properties of functions. CRC Press 1992 · Zbl 0804.28001 [28] Federer, H.: Geometric measure theory. Springer 1969 · Zbl 0176.00801 [29] Hauray, M.: On Liouville transport equation with potential in BVloc. To appear on Commun. Partial Differ. Equations (2003) · Zbl 1028.35148 [30] Hauray, M.: On two-dimensional Hamiltonian transport equations with Lploc coefficients. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 20, 625-644 (2003) · Zbl 1028.35148 [31] Lions, P.L.: Sur les équations différentielles ordinaires et les équations de transport. C. R. Acad. Sci., Paris, Sér. I, Math. 326, 833-838 (1998) · Zbl 0919.34028 [32] Petrova, G., Popov, B.: Linear transport equation with discontinuous coefficients. Commun. Partial Differ. Equations 24, 1849-1873 (1999) · Zbl 0992.35104 [33] Poupaud, F., Rascle, M.: Measure solutions to the liner multidimensional transport equation with non-smooth coefficients. Commun. Partial Differ. Equations 22, 337-358 (1997) · Zbl 0882.35026 [34] Young, L.C.: Lectures on the calculus of variations and optimal control theory. Saunders 1969 · Zbl 0177.37801
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.