 # Discretization by euler's method for regular lagrangian flow

This paper is concerned with the numerical analysis of the explicit Euler scheme for ordinary differential equations with non-Lipschitz vector fields. We prove the convergence of the Euler scheme to regular lagrangian flow (Diperna-Lions flows) which is the right concept of the solution in this context. Moreover, we show that order of convergence is 1/2.

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## 1 Introduction

When is a bounded smooth vector field, the flow of is the smooth map such that

 {dXdt(t,x)=b(t,X(t,x)),t∈[0,T]X(0,x)=x, (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

 X(t,⋅)#m≤Lm.

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 for

vector-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.

## 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 .

###### 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 .

###### Lemma 2.1.

Let . The local maximal function of is finite for a.e. and we have

 ∫Bρ(0)Mλf(y)dy≤cd,p+cd∫Bρ+λ(0)|f(y)|log(2+|f(y)|)dy.

For and we have

 ∫Bρ(0)(Mλf(y))pdy≤cd,p∫Bρ+λ(0)|f(y)|pdy,

but this is false for .

###### Lemma 2.2.

If then there exists a negligible set such that

 |u(x)−u(y)|≤cd|x−y|(MλDu(x)+MλDu(y))

for with .

###### Definition 2.2 (One-sided Lipschitz).

The function are said to satisfy a one-sided Lipschitz condition on if

 ⟨b(t,x)−b(t,~x),x−~x⟩≤νK(t)|x−~x|2 (2.2)

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

 X(t,x)=x+∫t0b(X(s,x))ds% for 0≤t≤T. (2.3)

Let us define an equidistant time discretization of by

 h=tn+1−tn,

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

 Xn+1=Xn+hb(Xn),n=0,1,… (2.4)

with .

## 3 Main result

### 3.1 Result

###### Theorem 3.1.

Let be a bounded vector field belonging to for some , satisfies the one-sided Lipschitz condition (2.2) on any compact set and that . Let be a regular Lagrangian flow associated to , as in Definition 1.1. Then the numerical solution satisfies

 (3.5)
###### Proof.

During the proof we use the notation for the -norm in the ball . By definition of (1.1) we have

 X(tn+1)=X0+∫tn+10b(X(s))dseX(tn)=X0+∫tn0b(X(s))ds,

and then

 X(tn+1)=X(tn)+∫tn+1tnb(X(s))ds. (3.6)

By definition of the Euler scheme we have

 Xn+1=Xn+hb(Xn). (3.7)

We set

 Yn+1:=X(tn)+hb(X(tn)). (3.8)

From (3.7) and (3.8) we have

 Yn+1−Xn+1=X(tn)−Xn+h(b(X(tn))−b(Xn)),

and by simple calculation we obtain

 ⟨Yn+1−Xn+1,Yn+1−Xn+1⟩ =⟨X(tn)−Xn,Yn+1−Xn+1⟩ +h⟨b(X(tn))−b(Xn),Yn+1−Xn+1⟩.

Therefore we deduce

 |Yn+1−Xn+1|2=⟨X(tn)−Xn,Yn+1−Xn+1⟩+h⟨b(X(tn))−b(Xn),X(tn)−Xn⟩+h2|b(X(tn))−b(Xn)|2. (3.9)

By Cauchy-Schwarz and Young inequalities we have

 (3.10)

We observe that

 ∥X(tn)∥∞≤R+T∥b∥∞

and

 ∥Xn∥∞≤R+T∥b∥∞

Using that is one-sided Lipschitz condition on any compact set we obtain

 h⟨b(X(tn))−b(Xn),X(tn)−Xn⟩≤νh|X(tn)−Xn|2, (3.11)

where dependent on .

Now, we observe

 (3.12)

From (3.9), (3.10), (3.11), (3.12) we deduce

 |Yn+1−Xn+1|2 ≤|X(tn)−Xn|22+|Yn+1−Xn+1|22 +κh|X(tn)−Xn|2+4∥b∥2∞h2,

Thus we conclude

 |Yn+1−Xn+1|2≤|X(tn)−Xn|2(1+2κh)+8∥b∥2∞h2, (3.13)

Now , taking in (3.13) we obtain

 ∥∥|Yn+1−Xn+1|2∥∥p/2 =∥Yn+1−Xn+1∥2p ≤∥∥(1+2κh)|X(tn)−Xn|2+8∥b∥2∞h2∥∥p/2 ≤∥∥(1+2κh)|X(tn)−Xn|2∥∥p/2+∥∥8∥b∥2∞h2∥∥p/2 ≤(1+2κh)∥X(tn)−Xn∥2p+8∥b∥2∞cd,pRdh2,

its implies that

 ∥Yn+1−Xn+1∥2p≤(1+2κh)∥X(tn)−Xn∥2p+C1h2, (3.14)

with .

On other hand we have

 |X(tn+1)−Yn+1| =∣∣∣X(tn)+∫tn+1tnb(X(s))ds−X(tn)−hb(X(tn))∣∣∣ =∣∣∣∫tn+1tn(b(X(s))−b(X(tn)))ds∣∣∣ ≤∫tn+1tn|b(X(s))−b(X(tn))|ds.

Then

 |X(tn+1)−Yn+1|2≤∣∣∣∫tn+1tn|b(X(s))−b(X(tn))|ds∣∣∣2. (3.15)

Taking in (3.15) we get

 ∥∥|X(tn+1)−Yn+1|2∥∥p/2 =∥X(tn+1)−Yn+1∥2p ≤∥∥∥∫tn+1tn|b(X(s))−b(X(tn))|ds∥∥∥2p.

We observe that

 ∥∥∥∫tn+1tn|b(X(s))−b(X(tn))|ds∥∥∥p ≤∥∥∥∫tn+1tncd|X(s)−X(tn)|(MλDb(X(s))+MλDb(X(tn)))ds∥∥∥p ≤cd∥∥∥∫tn+1tn(∫stn|b(X(u))|du)(MλDb(X(s))+MλDb(X(tn)))ds∥∥∥p ≤cd∥b∥∞∥∥∥∫tn+1tn(∫stndu)(MλDb(X(s))+MλDb(X(tn)))ds∥∥∥p ≤cd∥b∥∞∫tn+1tn∥(s−tn)(MλDb(X(s))+MλDb(X(tn)))∥pds ≤cd∥b∥∞∫tn+1tn(s−tn)∥MλDb(X(s))+MλDb(X(tn))∥pds ≤cd∥b∥∞∫tn+1tn(s−tn)(∥MλDb(X(s))∥p+∥MλDb(X(tn))∥p)ds ≤cd∥b∥∞∫tn+1tn(s−tn)L1/p(∥MλDb(x)∥Lp(BR+T∥b∥∞(0))+∥MλDb(tn,x)∥Lp(BR+T∥b∥∞(0)))ds ≤cd∥b∥∞∫tn+1tn(s−tn)cd,pL1/p∥Db(x)∥Lp(BR+λ+T∥b∥∞(0))ds =Kh2,

where and we use that

 ∫tn+1tn(s−tn)ds= (s22−stn)tn+1tn = (t2n+12−tn+1tn)−(t2n2−t2n) = tn+12(tn+1−2tn)−(−t2n2) = (tn+h)2(h−tn)+t2n2 = h2−t2n2+t2n2 = h22.

Then we arrive at

 ∥X(tn+1)−Yn+1∥2p≤C2h4. (3.16)

From Hölder and Young inequalities , (3.14) and (3.16) we deduce

 ∥2|X(tn+1)−Yn+1||Yn+1−Xn+1|∥p/2 =2(∥∥|X(tn+1)−Yn+1|p/2|Yn+1−Xn+1|p/2∥∥1)2/p ≤2(∥∥|X(tn+1)−Yn+1|p/2∥∥2∥∥|Yn+1−Xn+1|p/2∥∥2)2/p ≤2∥X(tn+1)−Yn+1∥p∥Yn+1−Xn+1∥p ≤2(Kh2)√(1+2κh)∥X(tn)−Xn∥2p+C1h2 ≤2(Kh2)(√(1+2κh)∥X(tn)−Xn∥2p+√C1h2) ≤2(Kh2)((1+2κh)∥X(tn)−Xn∥p+√C1h) ≤2(Kh2)⎛⎝(1+2κh)⎛⎝∥X(tn)−Xn∥2p2+12⎞⎠+√C1h⎞⎠

and

 ≤Kh2(1+2κh)∥X(tn)−Xn∥2p+2KC3h3+Kh2, (3.17)

where .

We have

 |X(tn+1)−Xn+1|≤|X(tn+1)−Yn+1|+|Yn+1−Xn+1|,

and

 |X(tn+1)−Xn+1|2 ≤(|X(tn+1)−Yn+1|+|Yn+1−Xn+1|)2 =|X(tn+1)−Yn+1|2+2|X(tn+1)−Yn+1||Yn+1−Xn+1|+|Yn+1−Xn+1|2.

Applying and by (3.14), (3.16) e (3.17) we obtain

 ∥∥|X(tn+1)−Xn+1|2∥∥p/2 =∥X(tn+1)−Xn+1∥2p ≤∥X(tn+1)−Yn+1∥2p+2∥X(tn+1)−Yn+1∥p∥Yn+1−Xn+1∥p +∥Yn+1−Xn+1∥2p ≤C2h4+Kh2(1+2κh)∥X(tn)−Xn∥2p+2KC3h3+Kh2 +(1+2κh)∥X(tn)−Xn∥2p+C1h2 ≤C2h2+Kh2(1+2κh)∥X(tn)−Xn∥2p+2KC3h2+Kh2 +(1+2κh)∥X(tn)−Xn∥2p+C1h2 =Ch2+(1+2κh)(1+Kh2)∥X(tn)−Xn∥2p,

where .

Then we have

 En+1≤αβEn+Ch2. (3.18)

where , and .

Applying the formula 3.18 recursively we deduce

 E1≤αβE0+Ch2 E2≤αβE1+Ch2≤αβ(αβE0+Ch2)+Ch2 ≤(αβ)2E0+Ch2(1+αβ) E3≤αβE2+Ch2≤αβ((αβ)2E0+Ch2(1+αβ))+Ch2 ≤(αβ)3E0+Ch2(1+αβ+(αβ)2) E4≤⋯.

By induction we easily have that

 En≤(αβ)nE0+Ch2n−1∑m=0(αβ)m. (3.19)

We noted that

 n−1∑m=0rm=rn−1r−1,parar≠1;e que |1+zh|≤exp(|z|h)%paraz∈R,
 αn =(1+2κh)n≤(1+2κh)N ≤exp(2κNh) =exp(2κ(T−t0)),

and

 βn ≤(1+Kh2)N≤(1+Kh2)N2 ≤exp(KN2h2) =exp(K(T−t0)2).

Therefore we have

 (αβ)n≤exp{(T−t0)(2κ+K(T−t0))}=Cexp (3.20)

and

 αβ−1=h(2κ+Kh+2Kκh2). (3.21)

From (3.19), (3.20) and (3.21) we conclude

 En ≤Ch2((αβ)n−1αβ−1)+(αβ)nE0 ≤Ch2(Cexp−1h(2κ+Kh+2Kκh2))+CexpE0 ≤Ch(Cexp−12κ+Kh+2Kκh2)+CexpE0.

Thus we have

 En≤ChCexp−12κ+CexpE0, (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 one-sided Lipschitz condition on compact sets, we assume that

 (b(x)−b(y)),x−y)=∫Rdg(z)(f(x−z)−f(y−z)),x−y)dz
 =∫Rdg(z)(f(x−z)−f(y−z)),(x−z)−(y−z))dz
 ≤C|x−y|2∫Rd|g(z)|(MλDf(x−z)+MλDf(y−z))dz
 ≤C|x−y|2∥g∥1∥MλDf∥p
 ≤C|x−y|2∥g∥1∥DF∥p≤C|x−y|2.

taking big enough.

## References

•  L. Ambrosio, Transport equation and Cauchy problem for vector fields, Invent. Math., 158, 227-260, 2004.
•  L. Ambrosio G. Crippa Continuity equations and ODE fows with non-smooth velocity, Lecture Notes of a course given at HeriottWatt University, Edinburgh. Proceeding of the Royal Society of Edinburgh, Section A: Mathematics. In press.
•  A. Bohun, F. Bouchut, G Crippa Lagrangian solutions to the Vlasov–Poisson system with density, Journal of Differential Equations, 260, 3576-3597, 2016.
•  G. Crippa and C. De Lellis, Estimates and regularity results for the DiPernaLions flow, J. Reine Angew. Math., 616:15–46, 2008.
•  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, 3973-3998, 2017.
•  C. De Lellis, Ordinary differential equations with rough coefficients and the renormalization theorem of Ambrosio, Bourbaki Seminar, Preprint, 1-26, 2007.
•  R. DiPerna and P.L. Lions, Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math., 98, 1989.
•  L. Grune, P. E. KloedenPathwise Approximation of Random Ordinary Differential Equations, BIT Numerical Mathematics, 41, 711-721, 2001.
•  P.-E. Jabin, Differential equations with singular fields, J. Math. Pures Appl., 94, 2010.
•  Elias M. Stein Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N.J., 1970.