Winter School on Transport Equations
and Control Theory for PDEs


An Introduction to the Theory
of Linear Transport Equations

Linear transport equations play a key role in hyperbolic problems, as the crossroads of the propagation phenomena. Still, optimal conditions for well-posedness, in terms of smoothness or of particular structures, are not completely clear.
This course is intended to give an overview on the subject for non specialists.

In particular, the following topics will be exposed:

(a) Quick review of the smooth theory

(b) Intuitive approach for a piecewise smooth coefficient, discussion on conservative form and advection form, absolute continuity or not of the divergence of the coefficient

(c) Particular 1-d, or 2-d autonomous equations

(d) Renormalized theory of DiPerna and Lions for bounded divergence coefficient, with its recent improvements

(e) Theory of reversible solutions for one-sided Lipschitz coefficient with measure divergence.

References

M. Aizenman,
On vector fields as generators of flows: a counterexample to Nelson's conjecture,
Ann. Math. (2) 107 (1978), no. 2, 287-296.

L. Ambrosio,
Transport equation and Cauchy problem for BV vector fields,
Preprint 2003.

L. Ambrosio, F. Bouchut and C. DeLellis,
Well-posedness for a class of hyperbolic systems of conservation laws in several space dimensions,
Preprint 2003.

F. Bouchut,
Renormalized solutions to the Vlasov equation with coefficients of bounded variation,
Arch. for Ration. Mech. and Anal. 157 (2001), 75-90.

F. Bouchut and L. Desvillettes,
On two-dimensional hamiltonian transport equations with continuous coefficients,
Diff. and Int. Eq. 14 (2001), 1015-1024.

F. Bouchut, F. Golse and M. Pulvirenti,
Kinetic equations and asymptotic theory,
Series in Applied mathematics 4, Gauthier Villars 2000.

F. Bouchut and F. James,
Differentiability with respect to initial data for a scalar conservation law,
International Series Num. Math. 129, Birkhauser, 113-118 (1999).

F. Bouchut and F. James,
Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness,
Comm. in Partial Diff. Eq. 24 (1999), 2173--2189.

F. Bouchut and F. James,
One-dimensional transport equations with discontinuous coefficients,
Nonlinear Analysis, Theory, Methods and Applications, Vol. 32 (1998), pp. 891-933.

F. Bouchut, F. James and S. Mancini,
Uniqueness and weak stability for multi-dimensional transport equations with one-sided Lipschitz coeffcient,
Preprint 2003.

A. Bressan,
An ill posed Cauchy problem for a hyperbolic system in two space dimensions,
Preprint 2003.

N. Depauw,
Non unicite des solutions bornees pour un champ de vecteurs BV en dehors d'un hyperplan, (French)
[Nonuniqueness of bounded solutions for a BV vector field outside of a hyperplane]
C. R. Math. Acad. Sci. Paris 337 (2003), no. 4, 249-252.

R.J. DiPerna and P.L. Lions,
Ordinary differential equations, transport theory and Sobolev spaces,
Invent. Math. 98 (1989), 511-547.

A.F. Filippov,
Differential Equations with Discontinuous Right-Hand Side,
A.M.S. Transl. (2) 42 (1964), 199-231.

P. Hartman,
Ordinary differential equations,
Corrected reprint of the second (1982) edition Classics in Applied Mathematics, 38 Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002.

M. Hauray,
On Liouville transport equation with a force field in BVloc,
to appear in Comm. Pure Applied Math.

M. Hauray,
On two dimmensionnal Hamiltonian transport equations with Lp coefficients,
Annales IHP Analyse Non Lin, 20 (2003), no. 4, 625--644.

P.L. Lions,
Mathematical topics in fluid mechanics. Vol. 2. Compressible models
Oxford Lecture Series in Mathematics and its Applications, 10, Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1998.

P.L. Lions,
Sur les equations differentielles ordinaires et les equations de transport, (French)
[On ordinary differential equations and transport equations]
C. R. Acad. Sci. Paris S. I Math. 326 (1998), no. 7, 833--838.

G. Petrova and B. Popov,
Linear transport equations with $?$-monotone coefficients,
J. Math. Anal. Appl. 260 (2001), no. 2, 307--324.

F. Poupaud and M. Rascle,
Measure solutions to the linear multi-dimensional transport equation with non-smooth coefficients,
Comm. Partial Differential Equations 22 (1997), no. 1-2, 337-358.