TY - GEN

T1 - Contractive metrics for nonsmooth evolutions

AU - Bressan, Alberto

PY - 2012

Y1 - 2012

N2 - Given an evolution equation, a standard way to prove the well posedness of the Cauchy problem is to establish a Gronwall type estimate, bounding the distance between any two trajectories. There are important cases, however, where such estimates cannot hold, in the usual distance determined by the Euclidean norm or by a Banach space norm. In alternative, one can construct different distance functions, related to a Riemannian structure or to an optimal transportation problem. This paper reviews various cases where this approach can be implemented, in connection with discontinuous ODEs on ℝn, nonlinear wave equations, and systems of conservation laws. For all the evolution equations considered here, a metric can be constructed such that the distance between any two solutions satisfies a Gronwall type estimate. This yields the uniqueness of solutions, and estimates on their continuous dependence on the initial data.

AB - Given an evolution equation, a standard way to prove the well posedness of the Cauchy problem is to establish a Gronwall type estimate, bounding the distance between any two trajectories. There are important cases, however, where such estimates cannot hold, in the usual distance determined by the Euclidean norm or by a Banach space norm. In alternative, one can construct different distance functions, related to a Riemannian structure or to an optimal transportation problem. This paper reviews various cases where this approach can be implemented, in connection with discontinuous ODEs on ℝn, nonlinear wave equations, and systems of conservation laws. For all the evolution equations considered here, a metric can be constructed such that the distance between any two solutions satisfies a Gronwall type estimate. This yields the uniqueness of solutions, and estimates on their continuous dependence on the initial data.

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U2 - 10.1007/978-3-642-25361-4_2

DO - 10.1007/978-3-642-25361-4_2

M3 - Conference contribution

AN - SCOPUS:84883607238

SN - 9783642253607

T3 - Nonlinear Partial Differential Equations: The Abel Symposium 2010

SP - 13

EP - 35

BT - Nonlinear Partial Differential Equations

T2 - Abel Symposium 2010: Nonlinear Partial Differential Equations

Y2 - 28 September 2010 through 2 October 2010

ER -