Construction of a convergent scheme

1.4 Construction of a convergent scheme

In this section we construct a sequence (fn) which will eventually be shown to converge to a solution of (1.1). So suppose that f^◦ ∈C_c²(R⁶) is a given nonnegative function. Let T^∗ be as given in Proposition 1.2.1 and assume that 0< T < T^∗.

Step 1. Definition of the sequence.

Let C_σ denote the function given in the remark following Proposition 1.2.1 and let Φ : R³→R³ be a smooth function such that

Φ(x) =

(x if|x| ≤Cσ(T) + 1 (C_σ(T) + 2)_|x|^x if|x| ≥C_σ(T) + 2. Similarly let Ψ : R³ →R³ be a smooth function such that

Ψ(x) =

(x if|x| ≤C_ET(T) + 1 (C_E^T(T) + 2)_|x|^x if|x| ≥C_E^T(T) + 2, with C_E^T as in Proposition 1.3.3.

We define

f₀(t, x, v) :=f^◦(x, v).

If f_n is already defined, we set ρn(t, x) =

Z

|v|≤Q(T)

fn(t, x, v)dv (1.24)

jn(t, x) = Z

|v|≤Q(T)

fn(t, x, v)vdv, (1.25)

σn(t, x) = Z

|v|≤Q(T)

v⊗vfn(t, x, v)dv, (1.26) and

U_n(t, x) =

Z ρ_n(t, y)

|x−y|dy, (1.27)

A_n(t, x) =

Z j_n(t, y)

|x−y|dy. (1.28)

Define

E_n^L(t, x) =−∇U_n(t, x), B_n(t, x) =∇ ×A_n(t, x).

Finally, we define E_n^T as the solution of the equation

−∆E_n^T(t) + 4πρn(t)E_n^T(t) = 4π

Φ(div_(x)σn(t))−ρn(t)E_n^L(t)−jn(t)×Bn(t)

(1.29) with boundary condition E_n^T(t, x)→_|x|→∞0 as given by Corollary 1.3.2 and set

E_n(t) = Ψ(E_n^T(t)) +E_n^L(t).

Denote by (X_n, V_n)(s, t, x, v) the solution of the characteristic system

˙

x = v, (1.30)

˙

v = En(s, x) +v×Bn(s, x) (1.31) with initial condition (Xn, Vn)(t, t, x, v) = (x, v). Frequently we will also use the notation Zn(s, t, x, v) = (Xn, Vn)(s, t, x, v). The next iterate is then obtained by setting

fn+1(t, x, v) =f^◦(Xn(0, t, x, v), Vn(0, t, x, v)).

Remark. Note that our sequences are defined for 0 ≤ t < ∞ but in the sequel the convergence will only be proved for 0 ≤t≤ T. Moreover, note that in the construction of our sequence we have restricted the domain of integration when defining the quantities ρ_n,j_n, andσ_n. Furthermore, we have introduced bounds for div_(x)σ_n(t) in the equation defining E_n^T and also introduced a bound on E_n^T when definingE_n.

Concerning the regularity of the sequences constructed the following holds Lemma 1.4.1 Let (fn) be as defined above. Then

(a) fn∈C²([0,∞[×R⁶).

(b) ρ_n∈C²([0,∞[×R³), j_n∈C²([0,∞[×R³,R³),σ_n∈C²([0,∞[×R³,R^3×3).

(c) U_n, ∂_xjU_n∈C²([0,∞[×R³), A_n, ∂_xjA_n∈C²([0,∞[×R³,R³) for j= 1, . . . ,3.

(d) E_n^L∈C²([0,∞[×R³,R³), Bn∈C²([0,∞[×R³,R³).

(e) E_n^T, ∂xjE_n^T ∈C¹([0,∞[×R³,R³) for j= 1, . . . ,3.

(f ) Z_n∈C²([0,∞[×[0,∞[×R⁶,R⁶).

Proof of Lemma 1.4.1 The proof is by induction. Assume that (a) holds and that there exists a monotone function θ_n: [0,∞[→ R⁺ such that suppf_n(t)⊂B_θ(t)(0) for all t ≥0. Then it is seen directly that (b) holds and that the support and the modulus of ρn(t), jn(t), σn(t) are bounded by a function of θ(t). Then we easily obtain (c) and (d) and we know thatkE_n^L(t)k_∞ andkB_n(t)k_∞ are also bounded by a function ofθ(t). From Proposition 1.3.1 and Eq. (1.29) we then obtainE^T_n ∈C([0,∞[×R³,R³). Differentiating Eq. (1.29) with respect totit follows that∂tE_n^T ∈C([0,∞[×R³,R³). Rewriting Eq. (1.29) as

∆E_n^T(t) = 4π

ρ_n(t)E_n^T(t)−Φ(div_(x)σ_n(t)) +ρ_n(t)E_n^L(t) +j_n(t)×B_n(t) ,

we may conclude that (e) holds. By well known theorems about ordinary differential equations we then get (f). Moreover, there exists a functionθn+1such that|Z_n(t,0, x, v)| ≤ θn+1(t) for all (x, v)∈supp f^◦,t≥0. But this closes the loop and the proof is complete.

2

1.4 Construction of a convergent scheme

Step 2. Bounds for the support.

Having defined this sequence our next major goal is to show its convergence. To do so we first have to establish a number of a priori bounds for this sequence. Note that we have

kf_n(t)k_p =kf^◦k_p, 0≤t≤T, 1≤p≤ ∞, since the flow Z_n is volume preserving. Consequently

kρ_n(t)k₁ ≤ kf^◦k₁, 0≤t≤T, 1≤p≤ ∞.

Since we have restricted the domain of integration in the definition of the quantitiesρn(t), jn(t), andσn(t), it follows that

kρ_n(t)k∞,kj_n(t)k₁,kj_n(t)k∞, and kσ_n(t)k∞

are bounded independently of nand t∈[0, T]. Consequently kE_n^L(t)k∞,kB_n(t)k∞

are also bounded by constant depending only onf^◦ andT. Define Pn(t) = sup{|v||∃0≤s≤t, x∈R³:fn(s, x, v)6= 0}, R_n(t) = sup{|x||∃0≤s≤t, v ∈R³:f_n(s, x, v)6= 0}.

These definitions imply

Pn+1(t) = sup{|V_n(s,0, x, v)||s∈[0, t],(x, v)∈supp f^◦}, R_n+1(t) = sup{|X_n(s,0, x, v)||s∈[0, t],(x, v)∈supp f^◦}, and from (1.31) we get that

|V_n(t)| ≤ |V_n(0)|+ Z t

0

kE_n(s)k∞+|V_n(s)|kB_n(s)k∞ds.

This allows us to conclude that

P_n(t)≤P ,˜ n≥1, t∈[0, T],

where ˜P > 0 is a properly chosen constant depending on f^◦ and T only. Using the characteristic equation for Xn it is clear that there also holds

R_n(t)≤R,˜ n≥1, t∈[0, T], and a certain constant ˜R >0.

Step 3. Bounds for derivatives.

We continue deriving bounds for our sequence. They are obtained by differentiating the integrated version of the characteristic system, Eqns. (1.30), (1.31), with respect to x.

So let 0≤s≤t≤T. Writing (X_n, V_n)(s) = (X_n, V_n)(s, t, x, v) we have V_n(s) = v+

Z s t

E_n(τ, X_n(τ)) +V_n(τ)×B_n(τ, X_n(τ))dτ, Xn(s) = x+

Z s t

Vn(τ)dτ.

Consequently

∂xjVn(s) = Z s

t

DEn(τ, Xn(τ))∂xjXn(τ) +∂xjVn(τ)×Bn(τ, Xn(τ)) +V_n(τ)×

DB_n(τ, X_n(τ))∂_xjX_n(τ)

dτ, (1.32)

∂xjXn(s) = ej+ Z s

t

∂xjVn(τ)dτ. (1.33)

For the second derivatives we obtain the estimates

|∂_x²V_n(s)| ≤ Z t

s

k∂_x²E_n(τ)k_∞|∂_xX_n(τ)|²+k∂_xE_n(τ)k_∞|∂_x²X_n(τ)|

+k∂_x²Bn(τ)k_∞|V_n(τ)||∂_xXn(τ)|²+ 2k∂_xBn(τ)k_∞|∂_xVn(τ)||∂_xXn(τ)|

+kB_n(τ)k∞|∂_x²Vn(τ)|+|V_n(τ)|k∂_xBn(τ)k∞|∂²_xXn(τ)| dτ (1.34)

|∂_x²X_n(s)| ≤ Z t

s

|∂_x²V_n(τ)|dτ. (1.35)

Supposing (x, v)∈suppf_n+1(t) we get (X_n(τ), V_n(τ))∈supp f_n+1(τ), i.e.,

|X_n(τ, t, x, v)| ≤R,˜ |V_n(τ, t, x, v)| ≤P˜ forτ, t∈[0, T], τ ≤t.

Taking the absolute value, estimating and adding we obtain from (1.32) and (1.33)

|∂_xXn(s)|+|∂_xVn(s)| ≤ C+C Z t

s

(1 +k∂_xEn(τ)k_∞+k∂_xBn(τ)k_∞)

·(|∂_xXn(τ)|+|∂_xVn(τ)|)dτ.

As a consequence of Eq. (1.29) and Proposition 1.3.1 we know that k∂_xE_n^T(τ)k_∞ is bounded independently of τ ∈ [0, T], so that in view of the boundedness of ∂_xΨ we get

k∂_xE_n(τ)k_∞≤ k∂_xΨk_∞k∂_xE^T_n(τ)k_∞+k∂_x²U_n(τ)k_∞≤C(1 +k∂_x²U_n(τ)k_∞).

Defining

H_n(s, t) := sup

(x,v)∈suppfn+1(t)

(|∂_xX_n(s, t, x, v)|+|∂_xX_n(s, t, x, v)|)

1.4 Construction of a convergent scheme We now proceed as in Section 1.2. According to [40], Lemma P1, we have

k∂_x²Un(τ)k∞ ≤ C(1 + log₊k∂_xρn(τ)k∞), and with Gronwall’s Lemma one obtains

Hn+1(s, t)≤Cexp

Choosing s = 0 in this last estimate and taking the logarithm on both sides of the inequality we have Since we may assume that the right hand side is nonnegative, it follows that

log₊Hn+1(0, t)≤C

i.e.,Hn(0, .) is bounded on [0, T] independently ofn. Consequently the following quantities are bounded on [0, T] by a constant depending only on f^◦ and T as well:

Hn(s, t),k∂_xfn(t)k∞,k∂_xρn(t)k∞,k∂_xjn(t)k∞,k∂_x²Un(t)k∞,k∂_x²An(t)k∞.

Inserting these bounds in our estimates for the second derivatives of the characteristic flow, Eqns. (1.34) and (1.35), we find

|∂_x²Vn(s)| ≤ C

By Gronwall’s Lemma it follows that

|∂_x²Xn(s)|+|∂_x²Vn(s)| ≤C

1 + Z t

s

k∂_x²E_n^T(τ)k_∞+k∂_x³Un(τ)k_∞+k∂_x³An(τ)k_∞dτ

. Next we differentiate Eqns. (1.27), (1.28) with respect toxand apply [40], Lemma P1, to find

k∂_x³Un+1(τ)k_∞ ≤ C(1 + log₊k∂²_xρn+1(τ)k_∞), k∂_x³An+1(τ)k_∞ ≤ C(1 + log₊k∂²_xjn+1(τ)k_∞).

In addition we also get

k∂_x²E_n+1^T (τ)k_∞≤C(1 + log₊k∂_x²σ_n+1(τ)k_∞), so that we obtain

k∂_x³U_n+1(t)k_∞+k∂_x³A_n+1(t)k_∞+k∂_x²E_n+1^T (t)k_∞

≤C(1 + log₊k∂²_xZn(0, t)k_{∞,suppfn+1(t)})

≤C

1 + log₊C

1 + Z t

0

k∂_x³U_n(τ)k_∞+k∂_x³A_n(τ)k_∞+k∂_x²E_n^T(τ)k_∞

. It follows from the above that

k∂³_xU_n(t)k_∞,k∂_x³A_n(t)k_∞,k∂_x²E_n^T(t)k_∞

are bounded by a constant depending only onf^◦andT and consequently the same is true for

k|∂_x²Z_n(s, t)k_{∞,suppfn+1(t)},k∂_x²ρ_n(t)k_∞,k∂_x²f_n(t)k_∞,k∂_x²σ_n(t)k_∞, and k∂_x²j_n(t)k_∞. Step 4. Proof of convergence.

Now we want to prove convergence of our sequences. First

|f_n+1(t, z)−fn(t, z)| ≤C|Z_n(0, t, z)−Zn−1(0, t, z)|

and similarly

|∂_xfn+1(t, z)−∂xfn(t, z)| ≤ C(|Z_n(0, t, z)−Zn−1(0, t, z)|

+|∂_xZ_n(0, t, z)−∂_xZn−1(0, t, z)|).

Let 0 ≤ s≤ t ≤T and suppose z = (x, v) ∈ suppfn+1(t)∪suppfn(t) as otherwise we clearly have

|f_n+1(t, z)−f_n(t, z)|+|∂_xf_n+1(t, z)−∂_xf_n(t, z)|= 0.

1.4 Construction of a convergent scheme

We write Z_n(s) = (X_n, V_n)(s) = (X_n, V_n)(s, t, x, v). From the characteristic system we get for the differences on the right hand side of (1.4) the estimates

|X_n(s)−Xn−1(s)| ≤ Z t

s

|V_n(τ)−Vn−1(τ)|dτ,

|V_n(s)−Vn−1(s)| ≤ Z t

s

|E_n(τ, X_n(τ))−En−1(τ, Xn−1(τ))|

+|V_n(τ)×B_n(τ, X_n(τ))−Vn−1(τ)×Bn−1(τ, Xn−1(τ))|dτ.

For z∈suppfn(t) we have |Vn−1(τ)| ≤C, so we can estimate the term

|V_n(τ)×Bn(τ, Xn(τ))−Vn−1(τ)×Bn−1(τ, Xn−1(τ))|

by

|V_n(τ)×Bn(τ, Xn(τ))−Vn−1(τ)×Bn(τ, Xn(τ))|

+|Vn−1(τ)×Bn(τ, Xn(τ))−Vn−1(τ)×Bn−1(τ, Xn(τ))|

+|Vn−1(τ)×Bn−1(τ, X_n(τ))−Vn−1(τ)×Bn−1(τ, Xn−1(τ))|, which is clearly majorized by

C(kB_n(τ)−Bn−1(τ)k∞+|Z_n(τ)−Zn−1(τ)|).

Using a slightly different grouping of the terms we get the same result forz∈suppfn+1(t).

Combining with

|E_n(τ, X_n(τ))−En−1(τ, Xn−1(τ))| ≤C(kE_n(τ)−En−1(τ)k_∞+|Z_n(τ)−Zn−1(τ)|), we obtain

|Z_n(s)−Zn−1(s)| ≤ C Z t

0

(kE_n(τ)−En−1(τ)k_∞+kB_n(τ)−Bn−1(τ)k_∞)dτ +C

Z _t

0

|Z_n(τ)−Zn−1(τ)|dτ. (1.36)

The next step is to derive a similar estimate for|∂_xZ_n(s)−∂_xZn−1(s)|. Note that we can rewrite ∂xVn(s)−∂xVn−1(s) as

Z s t

{∂_xEn(τ, Xn(τ))∂xXn(τ)−∂xEn−1(τ, Xn−1(τ))∂xXn−1(τ) (1.37) +∂_xV_n(τ)×B_n(τ, X_n(τ))−∂_xVn−1(τ)×Bn−1(τ, Xn−1n(τ)) (1.38) +Vn(τ)×∂xBn(τ, Xn(τ))∂xXn(τ)−Vn−1(τ)×∂xBn−1(τ, Xn−1(τ))∂xXn−1(τ)}dτ.

(1.39)

We are now going to estimate the integrand in the last expression. In case that (x, v) ∈ suppfn(t) we have |∂_xXn−1(τ)| ≤C so that for the first difference

|∂_xEn(τ, Xn(τ))∂xXn(τ)−∂xEn−1(τ, Xn−1(τ))∂xXn−1(τ)|

we obtain the bound

|∂_xEn(τ, Xn(τ))∂xXn(τ)−∂xEn(τ, Xn(τ))∂xXn−1(τ)|

+|∂_xE_n(τ, X_n(τ))∂_xXn−1(τ)−∂_xE_n(τ, Xn−1(τ))∂_xXn−1(τ)|

+|∂_xE_n(τ, Xn−1(τ))∂_xXn−1(τ)−∂_xEn−1(τ, Xn−1(τ))∂_xXn−1(τ)|, which is estimated by

C(|∂_xXn(τ)−∂xXn−1(τ)|+|X_n(τ)−Xn−1(τ)|+k∂_xEn(τ)−∂xEn−1(τ)k_∞).

In case (x, v) ∈supp f_n+1(t) we may argue analogously. Similarly we may estimate the integrand in the second difference (1.38) as

|∂_xVn(τ)×Bn(τ, Xn(τ))−∂xVn−1(τ)×Bn−1(τ, Xn−1(τ))|

≤ C(kB_n(τ)−Bn−1(τ)k+|∂_xZn(τ)−∂xZn−1(τ)|+|Z_n(τ)−Zn−1(τ)|).

Before estimating the third term, Eq. (1.39), note that we have (x, v)∈suppf_n(t) ⇒ |V_n(τ, t, x, v)| ≤C, (x, v)∈suppf_n+1(t) ⇒ |V_n−1(τ, t, x, v)| ≤C, which is easily deduced from the characteristic equations. It follows that

|V_n(τ)×∂xBn(τ, Xn(τ))∂xXn(τ)−Vn−1(τ)×∂xBn−1(τ, Xn−1(τ))∂xXn−1(τ)|

is less than or equal

|V_n(τ)×∂xBn(τ, Xn(τ))∂xXn(τ)−Vn(τ)×∂xBn(τ, Xn(τ))∂xXn−1(τ)|

+|V_n(τ)×∂_xB_n(τ, X_n(τ))∂_xXn−1(τ)−V_n(τ)×∂_xB_n(τ, Xn−1(τ))∂_xXn−1(τ)|

+|V_n(τ)×∂_xB_n(τ, Xn−1(τ))∂_xXn−1(τ)−V_n(τ)×∂_xBn−1(τ, Xn−1(τ))∂_xXn−1(τ)|

+|V_n(τ)×∂xBn−1(τ, Xn−1(τ))∂xXn−1(τ)−Vn−1(τ)×∂xBn−1(τ, Xn−1(τ))∂xXn−1(τ)|.

Again we may use that∂_xXn−1(τ) is bounded if (x, v)∈suppf_n(t) to obtain

|V_n(τ)×∂xBn(τ, Xn(τ))∂xXn(τ)−Vn−1(τ)×∂xBn−1(τ, Xn−1(τ))∂xXn−1(τ)|

≤C(|∂_xZ_n(τ)−∂_xZn−1(τ)|+|Z_n(τ)−Zn−1(τ)|+k∂_xB_n(τ)−∂_xBn−1(τ)k∞). Using another grouping of the terms we get the same result if (x, v) ∈supp fn+1(t), so that when defining

G_n(s, t) := sup|Z_n(s, t, x, v)−Zn−1(s, t, x, v)|+|∂_xZ_n(s, t, x, v)−∂_xZn−1(s, t, x, v)|

1.4 Construction of a convergent scheme

where the supremum is taken over the set

suppfn+1(t)∪suppfn(t), we arrive at

G_n(s, t) ≤ C Z t

s

G_n(τ, t)dτ +C

Z t s

kE_n(τ)−En−1(τ)k_∞+kB_n(τ)−Bn−1(τ)k_∞dτ +C

Z t s

k∂_xEn(τ)−∂xEn−1(τ)k∞+k∂_xBn(τ)−∂xBn−1(τ)k∞dτ.

It follows that

kf_n+1(s)−f_n(s)k_∞+k∂_xf_n+1(s)−∂_xf_n(s)k_∞

≤C Z t

s

kE_n(τ)−En−1(τ)k_∞+kB_n(τ)−Bn−1(τ)k_∞ +k∂_xE_n(τ)−∂_xEn−1(τ)k_∞+k∂_xB_n(τ)−∂_xBn−1(τ)k_∞dτ.

The field equations imply

kE_n^L(τ)−E_n−1^L (τ)k∞ ≤ Ckρ_n(τ)−ρn−1(τ)k∞, kB_n(τ)−Bn−1(τ)k_∞ ≤ Ckj_n(τ)−jn−1(τ)k_∞, k∂_xE_n^L(τ)−∂_xE_n−1^L (τ)k_∞ ≤ Ck∂_xρ_n(τ)−∂_xρn−1(τ)k_∞, k∂_xBn(τ)−∂xBn−1(τ)k_∞ ≤ Ck∂_xjn(τ)−∂xjn−1(τ)k_∞, and, furthermore, we have

kρ_n(τ)−ρn−1(τ)k_∞ ≤ Ckf_n(τ)−fn−1(τ)k_∞, kj_n(τ)−jn−1(τ)k_∞ ≤ Ckf_n(τ)−fn−1(τ)k_∞, k∂_xρn(τ)−∂xρn−1(τ)k_∞ ≤ Ck∂_xfn(τ)−∂xfn−1(τ)k_∞,

k∂_xjn(τ)−∂xjn−1(τ)k∞ ≤ Ck∂_xfn(τ)−∂xfn−1(τ)k∞. To estimate the terms involving E_n^T note that

− 1

4π∆ E_n^T(τ)−E_n−1^T (τ)

+ρn(τ)(E_n^T(τ)−E_n−1^T (τ)) =Gn(τ) (1.40) where

Gn(τ) = −(ρ_n(τ)−ρn−1(τ))E_n−1^T (τ) + (Φ(div_(x)σn(t))−Φ(div_(x)σn−1(t)))

−(ρ_n(τ)−ρn−1(τ))E_n^L(τ) +ρn−1(τ)(E_n−1^L (τ)−E_n^L(τ))

−(j_n(τ)−jn−1(τ))×Bn(τ)−jn−1(τ)×(Bn(τ)−Bn−1(τ)).

Remember supp ρ_n(τ)⊂B_R˜(0) for alln∈N, 0≤τ ≤T. So we get kE_n^T(τ)−E_n−1^T (τ)k_∞,BR(0) ≤ C(kG_n(τ)k₂+kG_n(τ)k_6/5)

from Proposition 1.3.1. It then follows from (1.40) that

k∂_xE_n^T(τ)−∂_xE_n−1^T (τ)k∞ ≤ C(kG_n(τ)k₁+kρ_n(τ)(E_n^T(τ)−E_n−1^T (τ))k₁)^1/3

·(kG_n(τ)k∞+kρ_n(τ)(E_n^T(τ)−E_n−1^T (τ))k∞)^2/3

≤ C(kG_n(τ)k₁+kG_n(τ)k₂+kG_n(τ)k_6/5)^1/3

·(kG_n(τ)k∞+kG_n(τ)k₂+kG_n(τ)k_6/5)^2/3. Using suppGn(τ)⊂BR˜(0) it is seen that

kG_n(τ)k_p ≤CpkG_n(τ)k_∞, and consequently

kG_n(τ)k_p ≤ C(kρ_n(τ)−ρn−1(τ)k_∞+kρ_n(τ)−ρn−1(τ)k_∞+kj_n(τ)−jn−1(τ)k_∞ +kE_n^L(τ)−E_n−1^L (τ)k_∞+kB_n(τ)−Bn−1(τ)k_∞

+kdiv_(x)σn(τ)−div_(x)σn−1(τ)k_∞)

≤ C(kf_n(τ)−fn−1(τ)k_∞+k∂_xfn(τ)−∂xfn−1(τ)k_∞).

Therefore we obtain

k∂_xE_n^T(τ)−∂_xE_n^T(τ)k_∞ ≤ C(kf_n(τ)−fn−1(τ)k_∞+k∂_xf_n(τ)−∂_xfn−1(τ)k_∞) and by a similar reasoning

kE_n^T(τ)−E_n^T(τ)k∞≤C(kf_n(τ)−fn−1(τ)k∞+k∂_xfn(τ)−∂xfn−1(τ)k∞).

So we finally arrive at

kf_n+1(t)−f_n(t)k_∞+k∂_xf_n+1(t)−∂_xf_n(t)k_∞

≤C Z _t

0

kf_n(τ)−fn−1(τ)k_∞+k∂_xf_n(τ)−∂_xfn−1(τ)k_∞dτ, which allows us to conclude that

kf_n+1(τ)−f_n(τ)k_∞+k∂_xf_n+1(τ)−∂_xf_n(τ)k_∞≤CCⁿ n!. It follows directly that the sequences (f_n) and (∂_xf_n) are uniformly Cauchy.

Im Dokument Existence Results for Plasma Physics Models Containing a Fully Coupled Magnetic Field (Seite 31-40)