Discretization by euler's method for regular lagrangian flow

1 Introduction

When $b : [0, T] \times R^{d} \to R^{d}$ is a bounded smooth vector field, the flow of $b$ is the smooth map $X : [0, T] \times R^{d} \to R^{d}$ such that

{\begin{matrix} \frac{d X}{d t} (t, x) = b (t, X (t, x)), t \in [0, T] X (0, x) = x, \end{matrix}

(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 $b \in L_{l o c}^{1} ([0, T] \times R^{d}; R^{d})$ . We say that a map $X : [0, T] \times R^{d} \to R^{d}$ is a regular Lagrangian flow for the vector field $b$ if

for $m$ -a.e. $x \in R^{d}$ the map $t \to X (t, x)$ is an absolutely continuous integral solution of $γ^{'} (t) = b (t, γ (t))$ for $t \in [0, T]$ , with $γ (0) = x$ ;
there exists a constant $L$ independent of $t$ such that

$X {(t, \cdot)}_{#} m \leq L m .$

The constant $L$ in $(i i)$ will be called the compressibility constant of $X$ .

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 [7] says that $X_{t}$ defines a regular Lagrangian flow when $b$ is a Sobolev vector field with bounded divergence. This theory was later extended to the case of $B V$ 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

W^{1, p}

vector-fields with

p > 1

but seemingly fails for

W^{1, 1}

vector-fields, 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 $1 / 2$ , 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 [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 $x \in R^{d}$ . However the usual assumptions required differentiability and/or Lipschitz(locally Lipschitz) regularity for the vector field $b$ see for instance [8]. In our result (theorem 3.1) we show the convergence in $L_{p}$ norm respect to the spatial variable which is coherent with respect to the definition of regular Lagrangian flow.

2 Preliminaries

When $A$ is measurable subset of $R^{d}$ we denote by $m (A)$ its Lebesgue measure. When $μ$ is a measure on $Ω$ and $f : Ω \to Ω^{'}$ a measurable map, $f_{#} μ$ will denote the push forward of $μ$ , i.e. the measure $ν$ such that $\int φ \circ f d μ = \int φ d ν$ for every $φ \in C_{c} (Ω^{'}) .$

We recall the following result, see for instance [4].

Theorem 2.1 (Existence and uniqueness of the flow).

Let $b$ be a bounded vector field belonging to $L^{1} ([0, T]; W^{1, p} (R^{d}; R^{d}))$ for some $p > 1$ . Assume that $[d i v b]^{-} \in L^{1} ([0, T]; L^{\infty} (R^{d}))$ . Then there exists a unique regular Lagrangian flow associated to $b$

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 $μ$ be a vector-field locally finite measure. For every $λ > 0$ , we define the maximal function of $μ$ as

When $μ = f m$ , where $f$ is a function in $L^{1} (R^{d}; R^{m})$ , we will often use the notation $M_{λ} f$ for $M_{λ} μ$ .

The proof of the following two lemmas can be found in [10].

Lemma 2.1.

Let $λ > 0$ . The local maximal function of $μ$ is finite for a.e. $x \in R^{d}$ and we have

\int_{B_{ρ} (0)} M_{λ} f (y) d y \leq c_{d, p} + c_{d} \int_{B_{ρ + λ} (0)} | f (y) | log (2 + | f (y) |) d y .

For $p > 1$ and $ρ > 0$ we have

\int_{B_{ρ} (0)} {(M_{λ} f (y))}_{λ}^{p} d y \leq c_{d, p} \int_{B_{ρ + λ} (0)} | f (y) |^{p} d y,

but this is false for $p = 1$ .

Lemma 2.2.

If $u \in B V (R^{d})$ then there exists a negligible set $N \subset R^{d}$ such that

| u (x) - u (y) | \leq c_{d} | x - y | (M_{λ} D u (x) + M_{λ} D u (y))

for $x, y \in R^{d} ∖ N$ with $| x - y | \leq λ$ .

Definition 2.2 (One-sided Lipschitz).

The function $b : [0, T] \times R^{d} \to R^{d}$ are said to satisfy a one-sided Lipschitz condition on $K \subset [0, T] \times R^{d} \to R^{d}$ if

⟨ b (t, x) - b (t, ~ x), x - ~ x ⟩ \leq ν_{K} (t) {| x - ~ x |}^{2}

(2.2)

holds for a.e in $K$ and with $ν \in L^{1} ([0, T])$ . The function $ν (t)$ is called a one-sided Lipschitz constant associated with $b$ .

By simplicity we consider the autonomous case. We shall consider the solution $X$ of (1.1) given by

X (t, x) = x + \int_{0}^{t} b (X (s, x)) d s % for 0 \leq t \leq T .

(2.3)

Let us define an equidistant time discretization of $[0, T]$ by

h = t_{n + 1} - t_{n},

where $h > 0$ is the time step, we denote by $X_{n}$ a numerical estimate of the exact solution $X (t_{n})$ , $n = 0, 1, \dots$ . Then we consider the Euler scheme wich has the form

X_{n + 1} = X_{n} + h b (X_{n}), n = 0, 1, \dots

(2.4)

with $X_{0} = x$ .

3 Main result

3.1 Result

Theorem 3.1.

Let $b$ be a bounded vector field belonging to $W^{1, p} (R^{d}; R^{d})$ for some $p > 1$ , $b$ satisfies the one-sided Lipschitz condition (2.2) on any compact set and that $[d i v b]^{-} \in L^{\infty} (R)$ . Let $X$ be a regular Lagrangian flow associated to $b$ , as in Definition 1.1. Then the numerical solution satisfies

(3.5)

Proof.

During the proof we use the notation $∥ . ∥_{L^{p}}$ for the $L^{p}$ -norm in the ball $B_{R} (0) \subset R^{d}$ . By definition of (1.1) we have

X (t_{n + 1}) = X_{0} + \int_{0}^{t_{n + 1}} b (X (s)) d s e X (t_{n}) = X_{0} + \int_{0}^{t_{n}} b (X (s)) d s,

and then

X (t_{n + 1}) = X (t_{n}) + \int_{t_{n}}^{t_{n + 1}} b (X (s)) d s .

(3.6)

By definition of the Euler scheme we have

X_{n + 1} = X_{n} + h b (X_{n}) .

(3.7)

We set

Y_{n + 1} := X (t_{n}) + h b (X (t_{n})) .

(3.8)

From (3.7) and (3.8) we have

Y_{n + 1} - X_{n + 1} = X (t_{n}) - X_{n} + h (b (X (t_{n})) - b (X_{n})),

and by simple calculation we obtain

	$⟨ Y_{n + 1} - X_{n + 1}, Y_{n + 1} - X_{n + 1} ⟩$	$= ⟨ X (t_{n}) - X_{n}, Y_{n + 1} - X_{n + 1} ⟩$
		$+ h ⟨ b (X (t_{n})) - b (X_{n}), Y_{n + 1} - X_{n + 1} ⟩ .$

Therefore we deduce

\begin{matrix} {| Y_{n + 1} - X_{n + 1} |}_{n + 1}^{2} & = ⟨ X (t_{n}) - X_{n}, Y_{n + 1} - X_{n + 1} ⟩ + h ⟨ b (X (t_{n})) - b (X_{n}), X (t_{n}) - X_{n} ⟩ + h^{2} {| b (X (t_{n})) - b (X_{n}) |}^{2} . \end{matrix}

(3.9)

By Cauchy-Schwarz and Young inequalities we have

(3.10)

We observe that

∥ X (t_{n}) ∥_{\infty} \leq R + T ∥ b ∥_{\infty}

and

∥ X_{n} ∥_{\infty} \leq R + T ∥ b ∥_{\infty}

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

\begin{matrix} h ⟨ b (X (t_{n})) - b (X_{n}), X (t_{n}) - X_{n} ⟩ \leq ν h {| X (t_{n}) - X_{n} |}_{n}^{2}, \end{matrix}

(3.11)

where $ν$ dependent on $T, ∥ b ∥_{\infty}, R$ .

Now, we observe

(3.12)

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

	${\| Y_{n + 1} - X_{n + 1} \|}_{n + 1}^{2}$	$\leq \frac{{\| X (t_{n}) - X_{n} \|}_{n}^{2}}{2} + \frac{{\| Y_{n + 1} - X_{n + 1} \|}_{n + 1}^{2}}{2}$
		$+ κ h {\| X (t_{n}) - X_{n} \|}_{n}^{2} + 4 {∥ b ∥}_{\infty}^{2} h^{2},$

Thus we conclude

{| Y_{n + 1} - X_{n + 1} |}_{n + 1}^{2} \leq {| X (t_{n}) - X_{n} |}_{n}^{2} (1 + 2 κ h) + 8 {∥ b ∥}_{\infty}^{2} h^{2},

(3.13)

Now , taking $L^{p / 2}$ in (3.13) we obtain

	${∥ ∥ {\| Y_{n + 1} - X_{n + 1} \|}_{n + 1}^{2} ∥ ∥}_{p / 2}^{2}$	$= {∥ Y_{n + 1} - X_{n + 1} ∥}_{n + 1}^{2}$
		$\leq {∥ ∥ (1 + 2 κ h) {\| X (t_{n}) - X_{n} \|}_{n}^{2} + 8 {∥ b ∥}_{\infty}^{2} h^{2} ∥ ∥}_{p / 2}^{2}$
		$\leq {∥ ∥ (1 + 2 κ h) {\| X (t_{n}) - X_{n} \|}_{n}^{2} ∥ ∥}_{p / 2}^{2} + {∥ ∥ 8 {∥ b ∥}_{\infty}^{2} h^{2} ∥ ∥}_{p / 2}^{2}$
		$\leq (1 + 2 κ h) {∥ X (t_{n}) - X_{n} ∥}_{n}^{2} + 8 {∥ b ∥}_{\infty}^{2} c_{d, p} R^{d} h^{2},$

its implies that

{∥ Y_{n + 1} - X_{n + 1} ∥}_{n + 1}^{2} \leq (1 + 2 κ h) {∥ X (t_{n}) - X_{n} ∥}_{n}^{2} + C_{1} h^{2},

(3.14)

with $C_{1} = 8 {∥ b ∥}_{\infty}^{2} c_{d, p} R^{d}$ .

On other hand we have

	$\| X (t_{n + 1}) - Y_{n + 1} \|$	$= ∣ ∣ ∣ X (t_{n}) + \int_{t_{n}}^{t_{n + 1}} b (X (s)) d s - X (t_{n}) - h b (X (t_{n})) ∣ ∣ ∣$
		$= ∣ ∣ ∣ \int_{t_{n}}^{t_{n + 1}} (b (X (s)) - b (X (t_{n}))) d s ∣ ∣ ∣$
		$\leq \int_{t_{n}}^{t_{n + 1}} \| b (X (s)) - b (X (t_{n})) \| d s .$

Then

{| X (t_{n + 1}) - Y_{n + 1} |}_{n + 1}^{2} \leq {∣ ∣ ∣ \int_{t_{n}}^{t_{n + 1}} | b (X (s)) - b (X (t_{n})) | d s ∣ ∣ ∣}^{2} .

(3.15)

Taking $L^{p / 2}$ in (3.15) we get

	${∥ ∥ {\| X (t_{n + 1}) - Y_{n + 1} \|}_{n + 1}^{2} ∥ ∥}_{p / 2}^{2}$	$= {∥ X (t_{n + 1}) - Y_{n + 1} ∥}_{n + 1}^{2}$
		$\leq {∥ ∥ ∥ \int_{t_{n}}^{t_{n + 1}} \| b (X (s)) - b (X (t_{n})) \| d s ∥ ∥ ∥}_{p}^{2} .$

We observe that

	${∥ ∥ ∥ \int_{t_{n}}^{t_{n + 1}} \| b (X (s)) - b (X (t_{n})) \| d s ∥ ∥ ∥}_{p}$
	$\leq {∥ ∥ ∥ \int_{t_{n}}^{t_{n + 1}} c_{d} \| X (s) - X (t_{n}) \| (M_{λ} D b (X (s)) + M_{λ} D b (X (t_{n}))) d s ∥ ∥ ∥}_{p}$
	$\leq c_{d} {∥ ∥ ∥ \int_{t_{n}}^{t_{n + 1}} (\int_{t_{n}}^{s} \| b (X (u)) \| d u) (M_{λ} D b (X (s)) + M_{λ} D b (X (t_{n}))) d s ∥ ∥ ∥}_{p}$
	$\leq c_{d} {∥ b ∥}_{\infty} {∥ ∥ ∥ \int_{t_{n}}^{t_{n + 1}} (\int_{t_{n}}^{s} d u) (M_{λ} D b (X (s)) + M_{λ} D b (X (t_{n}))) d s ∥ ∥ ∥}_{p}$

	$\leq c_{d} {∥ b ∥}_{\infty} \int_{t_{n}}^{t_{n + 1}} {∥ (s - t_{n}) (M_{λ} D b (X (s)) + M_{λ} D b (X (t_{n}))) ∥}_{p} d s$
	$\leq c_{d} {∥ b ∥}_{\infty} \int_{t_{n}}^{t_{n + 1}} (s - t_{n}) {∥ M_{λ} D b (X (s)) + M_{λ} D b (X (t_{n})) ∥}_{p} d s$
	$\leq c_{d} {∥ b ∥}_{\infty} \int_{t_{n}}^{t_{n + 1}} (s - t_{n}) ({∥ M_{λ} D b (X (s)) ∥}_{p} + {∥ M_{λ} D b (X (t_{n})) ∥}_{p}) d s$
	$\leq c_{d} {∥ b ∥}_{\infty} \int_{t_{n}}^{t_{n + 1}} (s - t_{n}) L^{1 / p} ({∥ M_{λ} D b (x) ∥}_{L^{p} (B_{R + T {∥ b ∥}_{\infty}} (0))} + {∥ M_{λ} D b (t_{n}, x) ∥}_{L^{p} (B_{R + T {∥ b ∥}_{\infty}} (0))}) d s$
	$\leq c_{d} {∥ b ∥}_{\infty} \int_{t_{n}}^{t_{n + 1}} (s - t_{n}) c_{d, p} L^{1 / p} {∥ D b (x) ∥}_{L^{p} (B_{R + λ + T {∥ b ∥}_{\infty}} (0))} d s$

	$= K h^{2},$

where $K = \frac{c_{d, p} M {∥ b ∥}_{\infty}}{2}$ and we use that

	$\int_{t_{n}}^{t_{n + 1}} (s - t_{n}) d s =$	${(\frac{s^{2}}{2} - s t_{n})}_{n}^{t_{n + 1}}$
	$=$	$(\frac{t_{n + 1}^{2}}{2} - t_{n + 1} t_{n}) - (\frac{t_{n}^{2}}{2} - t_{n}^{2})$
	$=$	$\frac{t_{n + 1}}{2} (t_{n + 1} - 2 t_{n}) - (- \frac{t_{n}^{2}}{2})$
	$=$	$\frac{(t_{n} + h)}{2} (h - t_{n}) + \frac{t_{n}^{2}}{2}$
	$=$	$\frac{h^{2} - t_{n}^{2}}{2} + \frac{t_{n}^{2}}{2}$
	$=$	$\frac{h^{2}}{2} .$

Then we arrive at

{∥ X (t_{n + 1}) - Y_{n + 1} ∥}_{n + 1}^{2} \leq C_{2} h^{4} .

(3.16)

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

	${∥ 2 \| X (t_{n + 1}) - Y_{n + 1} \| \| Y_{n + 1} - X_{n + 1} \| ∥}_{p / 2}$	$= 2 {({∥ ∥ {\| X (t_{n + 1}) - Y_{n + 1} \|}_{n + 1}^{p / 2} {\| Y_{n + 1} - X_{n + 1} \|}_{n + 1}^{p / 2} ∥ ∥}_{1}^{p / 2})}_{1}^{2 / p}$
		$\leq 2 {({∥ ∥ {\| X (t_{n + 1}) - Y_{n + 1} \|}_{n + 1}^{p / 2} ∥ ∥}_{2}^{p / 2} {∥ ∥ {\| Y_{n + 1} - X_{n + 1} \|}_{n + 1}^{p / 2} ∥ ∥}_{2}^{p / 2})}_{2}^{2 / p}$
		$\leq 2 {∥ X (t_{n + 1}) - Y_{n + 1} ∥}_{p} {∥ Y_{n + 1} - X_{n + 1} ∥}_{p}$
		$\leq 2 (K h^{2}) \sqrt{(1 + 2 κ h) {∥ X (t_{n}) - X_{n} ∥}_{n}^{2} + C_{1} h^{2}}$
		$\leq 2 (K h^{2}) (\sqrt{(1 + 2 κ h) {∥ X (t_{n}) - X_{n} ∥}_{n}^{2}} + \sqrt{C_{1} h^{2}})$
		$\leq 2 (K h^{2}) ((1 + 2 κ h) {∥ X (t_{n}) - X_{n} ∥}_{p} + \sqrt{C_{1}} h)$
		$\leq 2 (K h^{2}) ⎛ ⎝ (1 + 2 κ h) ⎛ ⎝ \frac{{∥ X (t_{n}) - X_{n} ∥}_{n}^{2}}{2} + \frac{1}{2} ⎞ ⎠ + \sqrt{C_{1}} h ⎞ ⎠$

and

\leq K h^{2} (1 + 2 κ h) {∥ X (t_{n}) - X_{n} ∥}_{n}^{2} + 2 K C_{3} h^{3} + K h^{2},

(3.17)

where $C_{3} = κ + \sqrt{C_{1}}$ .

We have

| X (t_{n + 1}) - X_{n + 1} | \leq | X (t_{n + 1}) - Y_{n + 1} | + | Y_{n + 1} - X_{n + 1} |,

and

	${\| X (t_{n + 1}) - X_{n + 1} \|}_{n + 1}^{2}$	$\leq {(\| X (t_{n + 1}) - Y_{n + 1} \| + \| Y_{n + 1} - X_{n + 1} \|)}^{2}$
		$= {\| X (t_{n + 1}) - Y_{n + 1} \|}_{n + 1}^{2} + 2 \| X (t_{n + 1}) - Y_{n + 1} \| \| Y_{n + 1} - X_{n + 1} \| + {\| Y_{n + 1} - X_{n + 1} \|}_{n + 1}^{2} .$

Applying $L^{p / 2}$ and by (3.14), (3.16) e (3.17) we obtain

	${∥ ∥ {\| X (t_{n + 1}) - X_{n + 1} \|}_{n + 1}^{2} ∥ ∥}_{p / 2}^{2}$	$= {∥ X (t_{n + 1}) - X_{n + 1} ∥}_{n + 1}^{2}$
		$\leq {∥ X (t_{n + 1}) - Y_{n + 1} ∥}_{n + 1}^{2} + 2 {∥ X (t_{n + 1}) - Y_{n + 1} ∥}_{p} {∥ Y_{n + 1} - X_{n + 1} ∥}_{p}$
		$+ {∥ Y_{n + 1} - X_{n + 1} ∥}_{n + 1}^{2}$
		$\leq C_{2} h^{4} + K h^{2} (1 + 2 κ h) {∥ X (t_{n}) - X_{n} ∥}_{n}^{2} + 2 K C_{3} h^{3} + K h^{2}$
		$+ (1 + 2 κ h) {∥ X (t_{n}) - X_{n} ∥}_{n}^{2} + C_{1} h^{2}$
		$\leq C_{2} h^{2} + K h^{2} (1 + 2 κ h) {∥ X (t_{n}) - X_{n} ∥}_{n}^{2} + 2 K C_{3} h^{2} + K h^{2}$
		$+ (1 + 2 κ h) {∥ X (t_{n}) - X_{n} ∥}_{n}^{2} + C_{1} h^{2}$
		$= C h^{2} + (1 + 2 κ h) (1 + K h^{2}) {∥ X (t_{n}) - X_{n} ∥}_{n}^{2},$

where $C = C_{2} + 2 K C_{3} + K + C_{1}$ .

Then we have

E_{n + 1} \leq α β E_{n} + C h^{2} .

(3.18)

where $α = (1 + 2 κ h)$ , $β = (1 + K h^{2})$ and $E_{n} = {∥ X (t_{n}) - X_{n} ∥}_{n}^{2}$ .

Applying the formula 3.18 recursively we deduce

	$E_{1} \leq α β E_{0} + C h^{2}$
	$E_{2} \leq α β E_{1} + C h^{2} \leq α β (α β E_{0} + C h^{2}) + C h^{2}$
	$\leq {(α β)}^{2} E_{0} + C h^{2} (1 + α β)$
	$E_{3} \leq α β E_{2} + C h^{2} \leq α β ({(α β)}^{2} E_{0} + C h^{2} (1 + α β)) + C h^{2}$
	$\leq {(α β)}^{3} E_{0} + C h^{2} (1 + α β + {(α β)}^{2})$
	$E_{4} \leq \dots .$

By induction we easily have that

E_{n} \leq {(α β)}^{n} E_{0} + C h^{2} n - 1 \sum m = 0 {(α β)}^{m} .

(3.19)

We noted that

n - 1 \sum m = 0 r^{m} = \frac{r^{n} - 1}{r - 1}, para r \neq 1; e que | 1 + z h | \leq exp (| z | h) % p a r a z \in R,

	$α^{n}$	$= {(1 + 2 κ h)}^{n} \leq {(1 + 2 κ h)}^{N}$
		$\leq exp (2 κ N h)$
		$= exp (2 κ (T - t_{0})),$

and

	$β^{n}$	$\leq {(1 + K h^{2})}^{N} \leq {(1 + K h^{2})}^{N^{2}}$
		$\leq exp (K N^{2} h^{2})$
		$= exp (K {(T - t_{0})}_{0}^{2}) .$

Therefore we have

{(α β)}^{n} \leq exp {(T - t_{0}) (2 κ + K (T - t_{0}))} = C_{exp}

(3.20)

and

α β - 1 = h (2 κ + K h + 2 K κ h^{2}) .

(3.21)

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

	$E_{n}$	$\leq C h^{2} (\frac{{(α β)}^{n} - 1}{α β - 1}) + {(α β)}^{n} E_{0}$
		$\leq C h^{2} (\frac{C_{exp} - 1}{h (2 κ + K h + 2 K κ h^{2})}) + C_{exp} E_{0}$
		$\leq C h (\frac{C_{exp} - 1}{2 κ + K h + 2 K κ h^{2}}) + C_{exp} E_{0} .$

Thus we have

E_{n} \leq C h \frac{C_{exp} - 1}{2 κ} + C_{exp} E_{0},

(3.22)

and

(3.23)

If we take $X (t_{0}) = X_{0} = x$ 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 $f \in W^{1, p} (R^{d})$ with $p > 1$ and $d i v f \in L^{\infty} (R^{d})$ . Also we consider $g \in L^{1} (R^{d})$ . Now, we define $b (x) = (f * g) (x)$ . Then by young inequality we have that $b \in W^{1, p} (R^{d})$ and $d i v b \in L^{\infty} (R^{d})$ . We shall prove that the function $b$ verifies one-sided Lipschitz condition on compact sets, we assume that $(x, y) \in K \subset R^{d} \times R^{d}$

(b (x) - b (y)), x - y) = \int_{R^{d}} g (z) (f (x - z) - f (y - z)), x - y) d z

= \int_{R^{d}} g (z) (f (x - z) - f (y - z)), (x - z) - (y - z)) d z

\leq C | x - y |^{2} \int_{R^{d}} | g (z) | (M_{λ} D f (x - z) + M_{λ} D f (y - z)) d z

\leq C | x - y |^{2} ∥ g ∥_{1} ∥ M_{λ} D f ∥_{p}

\leq C | x - y |^{2} ∥ g ∥_{1} ∥ D F ∥_{p} \leq C | x - y |^{2} .

taking $λ$ big enough.

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	${∥ ∥ {\| Y_{n + 1} - X_{n + 1} \|}_{n + 1}^{2} ∥ ∥}_{p / 2}^{2}$	$= {∥ Y_{n + 1} - X_{n + 1} ∥}_{n + 1}^{2}$
		$\leq {∥ ∥ (1 + 2 κ h) {\| X (t_{n}) - X_{n} \|}_{n}^{2} + 8 {∥ b ∥}_{\infty}^{2} h^{2} ∥ ∥}_{p / 2}^{2}$
		$\leq {∥ ∥ (1 + 2 κ h) {\| X (t_{n}) - X_{n} \|}_{n}^{2} ∥ ∥}_{p / 2}^{2} + {∥ ∥ 8 {∥ b ∥}_{\infty}^{2} h^{2} ∥ ∥}_{p / 2}^{2}$
		$\leq (1 + 2 κ h) {∥ X (t_{n}) - X_{n} ∥}_{n}^{2} + 8 {∥ b ∥}_{\infty}^{2} c_{d, p} R^{d} h^{2},$

Discretization by euler's method for regular lagrangian flow

Authors

Convergence analysis of a Lagrangian numerical scheme in computing effective diffusivity of 3D time-dependent flows

The weak convergence order of two Euler-type discretization schemes for the log-Heston model

Lagrangian Constraints and Differential Thomas Decomposition

Invariant Variational Schemes for Ordinary Differential Equations

A Characteristic Mapping Method for the three-dimensional incompressible Euler equations

Modelling of a spherical deflagration at constant speed

Approximation of solutions of DDEs under nonstandard assumptions via Euler scheme

1 Introduction

Definition 1.1 (Regular Lagrangian flow).

2 Preliminaries

Theorem 2.1 (Existence and uniqueness of the flow).

Definition 2.1 ( Maximal function).

Lemma 2.1.

Lemma 2.2.

Definition 2.2 (One-sided Lipschitz).

3 Main result

3.1 Result

Theorem 3.1.

Proof.

3.2 Example

References

Discretization by euler's method for regular lagrangian flow

Authors

Convergence analysis of a Lagrangian numerical scheme in computing effective diffusivity of 3D time-dependent flows

The weak convergence order of two Euler-type discretization schemes for the log-Heston model

Lagrangian Constraints and Differential Thomas Decomposition

Invariant Variational Schemes for Ordinary Differential Equations

A Characteristic Mapping Method for the three-dimensional incompressible Euler equations

Modelling of a spherical deflagration at constant speed

Approximation of solutions of DDEs under nonstandard assumptions via Euler scheme

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1 Introduction

Definition 1.1 (Regular Lagrangian flow).

2 Preliminaries

Theorem 2.1 (Existence and uniqueness of the flow).

Definition 2.1 ( Maximal function).

Lemma 2.1.

Lemma 2.2.

Definition 2.2 (One-sided Lipschitz).

3 Main result

3.1 Result

Theorem 3.1.

Proof.

3.2 Example

References