When is a bounded smooth vector field, the flow of is the smooth map such that
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 well-posed. We now give a brief state-of-the-art survey on the theory of ordinary differential equations with vector fields of low regularity. In a celebrated theory established by DiPerna and Lions  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 . 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  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 forvector-fields with but seemingly fails for vector-fields, unfortunately. We refer the readers to the two excellent summaries in  and more recently  and .
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 Lellis-Crippa estimations for regular Lagrangian flow. We mentioned that future work we are interested in applying this scheme in PDEs with Lagrangian formulation like transport-continuity equation , Euler equation [Crippa2] and Vlasov–Poisson system .
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 . 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.
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 .
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 well-known properties which are used throughout all this paper.
Definition 2.1 ( Maximal function).
Let be a vector-field 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 .
Let . The local maximal function of is finite for a.e. and we have
For and we have
but this is false for .
If then there exists a negligible set such that
for with .
Definition 2.2 (One-sided Lipschitz).
The function are said to satisfy a one-sided Lipschitz condition on if
holds for a.e in and with . The function is called a one-sided Lipschitz constant associated with .
By simplicity we consider the autonomous case. We shall consider the solution of (1.1) given by
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
3 Main result
During the proof we use the notation for the -norm in the ball . By definition of (1.1) we have
By definition of the Euler scheme we have
and by simple calculation we obtain
Therefore we deduce
By Cauchy-Schwarz and Young inequalities we have
We observe that
Using that is one-sided Lipschitz condition on any compact set we obtain
where dependent on .
Now, we observe
Thus we conclude
Now , taking in (3.13) we obtain
its implies that
On other hand we have
Taking in (3.15) we get
We observe that
where and we use that
Then we arrive at
Then we have
where , and .
Applying the formula 3.18 recursively we deduce
By induction we easily have that
We noted that
Therefore we have
Thus we have
If we take then
and this proof the theorem ∎
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 one-sided Lipschitz condition on compact sets, we assume that
taking big enough.
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