1 Introduction
When is a bounded smooth vector field, the flow of is the smooth map such that
(1.1) 
Out of the smooth context (1.1) has been studied by several authors. In particular, the following is a common definition of generalized flow for vector fields which are merely integrable.
Definition 1.1 (Regular Lagrangian flow).
Let . We say that a map is a regular Lagrangian flow for the vector field if

for a.e. the map is an absolutely continuous integral solution of for , with ;

there exists a constant independent of such that
The constant in will be called the compressibility constant of .
This paper is concerned with the numerical analysis of the euler scheme for solving ordinary differential equation (1.1). We are interested in situations in which the coefficients in the equation are rough, but still within the range in which the associated Cauchy problem is wellposed. We now give a brief stateoftheart survey on the theory of ordinary differential equations with vector fields of low regularity. In a celebrated theory established by DiPerna and Lions [7] says that defines a regular Lagrangian flow when is a Sobolev vector field with bounded divergence. This theory was later extended to the case of vector fields by Ambrosio [1]. The central of DiPerna and Lions’ theory is based on the connection between ODE and the Cauchy problem for the linear transport equation. C. De Lellis and G. Crippa have recently given in [4] a new proof of the existence and uniqueness of the flow solution of (1.1
), not using the the associated transport equation. Their very interesting approach provides regularity estimates for
vectorfields with but seemingly fails for vectorfields, unfortunately. We refer the readers to the two excellent summaries in [6] and more recently [2] and [9].In our result, we show that the rate of convergence of the approximate solution given by the explicit Euler scheme towards the unique solution of the problem is at least of order , uniformly in time. The proof is based on the De LellisCrippa estimations for regular Lagrangian flow. We mentioned that future work we are interested in applying this scheme in PDEs with Lagrangian formulation like transportcontinuity equation [4], Euler equation [Crippa2] and Vlasov–Poisson system [3].
Finally we point that the classical convergence result for Euler approximation get the convergence for any initial data in . However the usual assumptions required differentiability and/or Lipschitz(locally Lipschitz) regularity for the vector field see for instance [8]. In our result (theorem 3.1) we show the convergence in norm respect to the spatial variable which is coherent with respect to the definition of regular Lagrangian flow.
2 Preliminaries
When is measurable subset of we denote by its Lebesgue measure. When is a measure on and a measurable map, will denote the push forward of , i.e. the measure such that for every
We recall the following result, see for instance [4].
Theorem 2.1 (Existence and uniqueness of the flow).
Let be a bounded vector field belonging to for some . Assume that . Then there exists a unique regular Lagrangian flow associated to
We recall here the definition of the maximal function of a locally finite measure and of a locally summable function and we recollect some wellknown properties which are used throughout all this paper.
Definition 2.1 ( Maximal function).
Let be a vectorfield locally finite measure. For every , we define the maximal function of as
When , where is a function in , we will often use the notation for .
The proof of the following two lemmas can be found in [10].
Lemma 2.1.
Let . The local maximal function of is finite for a.e. and we have
For and we have
but this is false for .
Lemma 2.2.
If then there exists a negligible set such that
for with .
Definition 2.2 (Onesided Lipschitz).
The function are said to satisfy a onesided Lipschitz condition on if
(2.2) 
holds for a.e in and with . The function is called a onesided Lipschitz constant associated with .
By simplicity we consider the autonomous case. We shall consider the solution of (1.1) given by
(2.3) 
Let us define an equidistant time discretization of by
where is the time step, we denote by a numerical estimate of the exact solution , . Then we consider the Euler scheme wich has the form
(2.4) 
with .
3 Main result
3.1 Result
Theorem 3.1.
Proof.
During the proof we use the notation for the norm in the ball . By definition of (1.1) we have
and then
(3.6) 
By definition of the Euler scheme we have
(3.7) 
We set
(3.8) 
and by simple calculation we obtain
Therefore we deduce
(3.9) 
By CauchySchwarz and Young inequalities we have
(3.10) 
We observe that
and
Using that is onesided Lipschitz condition on any compact set we obtain
(3.11) 
where dependent on .
Now, we observe
(3.12) 
Thus we conclude
(3.13) 
Now , taking in (3.13) we obtain
its implies that
(3.14) 
with .
On other hand we have
Then
(3.15) 
Taking in (3.15) we get
We observe that
where and we use that
Then we arrive at
(3.16) 
(3.17) 
where .
We have
and
Then we have
(3.18) 
where , and .
Applying the formula 3.18 recursively we deduce
By induction we easily have that
(3.19) 
We noted that
and
Therefore we have
(3.20) 
and
(3.21) 
Thus we have
(3.22) 
and
(3.23) 
If we take then
(3.24) 
and this proof the theorem ∎
3.2 Example
We present one example of vector fields which satisfies the hypothesis of the theorem (3.1). We consider with and . Also we consider . Now, we define . Then by young inequality we have that and . We shall prove that the function verifies onesided Lipschitz condition on compact sets, we assume that
taking big enough.
References
 [1] L. Ambrosio, Transport equation and Cauchy problem for vector fields, Invent. Math., 158, 227260, 2004.
 [2] L. Ambrosio G. Crippa Continuity equations and ODE fows with nonsmooth velocity, Lecture Notes of a course given at HeriottWatt University, Edinburgh. Proceeding of the Royal Society of Edinburgh, Section A: Mathematics. In press.
 [3] A. Bohun, F. Bouchut, G Crippa Lagrangian solutions to the Vlasov–Poisson system with density, Journal of Differential Equations, 260, 35763597, 2016.
 [4] G. Crippa and C. De Lellis, Estimates and regularity results for the DiPernaLions flow, J. Reine Angew. Math., 616:15–46, 2008.
 [5] G. Crippa, C. Nobili, C. Seis, and S. Spirito, Eulerian and Lagrangian Solutions to the Continuity and Euler Equations with Vorticity, SIAM J. Math. Anal., 49, 39733998, 2017.
 [6] C. De Lellis, Ordinary differential equations with rough coefficients and the renormalization theorem of Ambrosio, Bourbaki Seminar, Preprint, 126, 2007.
 [7] R. DiPerna and P.L. Lions, Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math., 98, 1989.
 [8] L. Grune, P. E. KloedenPathwise Approximation of Random Ordinary Differential Equations, BIT Numerical Mathematics, 41, 711721, 2001.
 [9] P.E. Jabin, Differential equations with singular fields, J. Math. Pures Appl., 94, 2010.
 [10] Elias M. Stein Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N.J., 1970.
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