A note on some microlocal estimates used to prove the convergence of splitting methods relying on pseudo-spectral discretizations

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A note on some microlocal estimates used to prove the convergence of splitting methods relying on

pseudo-spectral discretizations

Joackim Bernier, Fernando Casas, Nicolas Crouseilles

To cite this version:

Joackim Bernier, Fernando Casas, Nicolas Crouseilles. A note on some microlocal estimates used to prove the convergence of splitting methods relying on pseudo-spectral discretizations. 2020. �hal- 02929869�

THE CONVERGENCE OF SPLITTING METHODS RELYING ON PSEUDO-SPECTRAL DISCRETIZATIONS.

JOACKIM BERNIER, FERNANDO CASAS, AND NICOLAS CROUSEILLES

Abstract. In [BCC20], we used some classical microlocal estimates to prove the conver- gence of our splitting methods (for example page A671). In this note, through Corollary 2 and Remark 1, we provide a detailed proof of these estimates. All the proofs rely on results presented in [NR10].

We consider the classes of symbols S(hxi^s1 +hξi^s2;hξi,hxi), s₁, s₂ ∈ R. By definition (see Definition 1.1.1 page 19 in [NR10]), it contains the symbols a(x, ξ) such that

∀γ, δ∈N, |∂_x^γ∂_ξ^δa(x, ξ)|._γ,δ (hxi^s1 +hξi^s2)hxi^−γhξi^−δ. We are going to prove the following proposition.

Proposition 1. If s₁, s₂ ≥0 andb∈S(hxi^s1+hξi^s2;hξi,hxi) then

∀u∈S(R^d), |(u, b^wu)_L2|._b,s1,s2 (u,(hxi^s1 +hξi^s2)^wu)_L2. This proposition is useful to get the following corollaries.

Corollary 1. If s≥0 anda∈S(hxi^s+hξi^s;hξi,hxi) then

∀u∈S(R^d), ka^wuk_L2 .a,skuk_X^s where

kuk²_X^s :=

Z

R^d

hxi^2s|u(x)|²dx+ Z

R^d

hξi^2s|ˆu(ξ)|²dξ.

Proof of Corollary 1. Since we have

(hxi^s+hξi^s)²≤2(hxi^2s+hξi^2s),

applying the Proposition 1.2.9 page 29 and the Theorem 1.2.16 page 31 in [NR10], we get a symbol

c∈S((hxi^s+hξi^s)²;hξi,hxi)⊂S(hxi^2s+hξi^2s;hξi,hxi), such that

c^w = (a^w)^∗a^w.

Consequently, applying Proposition 1, for u∈S(R^d), we have

ka^wuk²_L2 = (u, c^wu)_L2 ≤c_a,s(u,(hxi^2s+hξi^2s)^wu)_L2 =c_a,skuk²_X^s

where c_a,s is a constant depending only ona ands.

J.B. thanks Paul Alphonse for his help and his advices to write this note.

1

2 J. BERNIER, F. CASAS, AND N. CROUSEILLES

Corollary 2. If s≥0 andα, β >0 are such that α+β ≤s then

∀u∈S(R^d), khxi^αh∇i^βuk_L2 ._a,skuk_X^s. Proof of Corollary 1. Since, by Young, we have

hxi^αhξi^β ≤ α

α+βhxi^α+β+ β

α+βhξi^α+β ≤ hxi^s+hξi^s, applying the Theorem 1.2.16 page 31 in [NR10], we get a symbol

a∈S(hxi^αhξi^β;hξi,hxi) ⊂S(hxi^s+hξi^s;hξi,hxi), such that

a^w =hxi^αh∇i^β.

Consequently, we conclude by applying the Corollary 1.

Remark 1. This corollary could be easily extended to control the terms we had in [BCC20]. For exemple, provided that α+β +γ ≤ s, we could also control things like kh∇i^γhxi^αh∇i^βuk_L2(R^d) or kh∂x₁i^γhx2i^αh∂x₂i^βuk_L2(R^d).

The proof of the Proposition 1 relies on the following technical lemma whose proof is given in the subsection 1.1 of the Appendix.

Lemma 1. If a∈ S(hxi^s1 +hξi^s2;hξi,hxi), there exists r₁ ∈ S(hxi^s1−1 +hξi^s2;hξi,hxi), r₂ ∈S(hxi^s1+hξi^s2−1;hξi,hxi) such that

A_a=a^w+r₁^w+r₂^w

whereA_a is the Anti-Wick operator with symbola(see Definition 1.7.3 page 53 in[NR10]).

Finally, we focus on the proof of the Proposition 1.

Proof of Proposition 1. Lets₁, s₂ ≥0andb∈S(hxi^s1+hξi^s2;hξi,hxi). We aim at proving that

(1) ∀u∈S(R^d), |(u, b^wu)_L2|.b,s₁,s₂ (u,(hxi^s1 +hξi^s2)^wu)_L2. Sinceb∈S(hxi^s1+hξi^s2;hξi,hxi), there exists a constant c_b >0such that

∀x, ξ∈R^d, a(x, ξ) :=c_b(hxi^s1 +hξi^s2)±b(x, ξ)≥0.

Since it is clear that hxi^s1 +hξi^s2 ∈ S(hxi^s1 +hξi^s2;hξi,hxi), we have a ∈ S(hxi^s1 + hξi^s2;hξi,hxi) and so, by applying the Lemma 1, we get r₁ ∈ S(hxi^s1−1 +hξi^s2;hξi,hxi) and r₂ ∈S(hxi^s1 +hξi^s2−1;hξi,hxi) such that

A_a=a^w+r₁^w+r₂^w.

Since the symbol a is nonnegative, by applying the Proposition 1.7.6 page 53 in [NR10], we know that A_a is a nonnegative operator and so

(2) (u, A_au)_L2 =c_b(u,(hxi^s1 +hξi^s2)^wu)_L2 ±(u, b^wu)_L2 + (u, r₁^wu)_L2 + (u, r₂^wu)_L2 ≥0.

Finally, we have to control (u, r^w₁u)_L2 and (u, r^w₂u)_L2. By symmetry, we only focus on (u, r₁^wu)_L2.

We proceed by induction.

• Case s₁>1. By the induction assumption, we know that

(u, r₁^wu)_L2 ._b,s1,s2 (u,(hxi^s1−1+hξi^s2)^wu)_L2, and so since hxi^s1−1≤ hxi^s1 we have

(u, r₁^wu)_L2 ._b,s1,s2 (u,(hxi^s1+hξi^s2)^wu)_L2.

• Case s₁ ≤ 1. Applying Theorem 1.2.16 page 31 in [NR10], we get a symbol f ∈ S(hξi^−s2hxi^s1−1+ 1;hξi,hxi)⊂S(1; 1,1) such that

f^w=h∇i^−s2/2r₁^wh∇i^−s2/2.

Consequently, applying the Calderón-Vaillancourt theorem (Theorem 1.7.14 page 58 in [NR10]), this operator is bounded in L² and so

∀u∈S(R^d), (r₁^wh∇i^−s2/2u,h∇i^−s2/2u)_L2 ._r^w

1,s₂ kuk²_L2. As a consequence, the change of function u← h∇i^−s2/2uprovides the estimate

∀u∈S(R^d), (r^w₁u, u)_L2 ._r1,s2 (u,h∇i^s2u)_L2 ._r1,s2 (u,(hxi^s1 +hξi^s2)^wu)_L2. In any case, we have

(r₁^wu, u)_L2 + (r^w₂u, u)_L2 ._b,s1,s2 (u,(hxi^s1 +hξi^s2)^wu)_L2.

Plugging this estimate in (2) provides naturally the estimate (1) we aimed at proving.

1. Appendix

1.1. Proof of Lemma 1. We aim at proving that if a ∈ S(hxi^s1 +hξi^s2;hξi,hxi) then there existsr₁ ∈S(hxi^s1−1+hξi^s2;hξi,hxi), r₂ ∈S(hxi^s1 +hξi^s2−1;hξi,hxi) such that

A_a=a^w+r₁^w+r₂^w. Applying the Proposition 1.7.9 page 55, we know that

A_a=b^w where

(3) b(x, ξ) := 2^d

Z

R^d

a(y, η)e^{−|x−y|2−|ξ−η|2} dy (2π)^d/2

dη (2π)^d/2. Consequently, we just have to decomposeb in the good classes.

Naturally, the Taylor expansion at the order1 of ain(x, ξ) is a(y, η) =a(x, ξ) +

Z 1 0

∂_xa(x+t(y−x), ξ+t(η−ξ))dt(y−x) +

Z 1 0

∂_ξa(x+t(y−x), ξ+t(η−ξ))dt(η−ξ).

Plugging this expansion in (3) and realizing the change of coordinate(y, η)←(y−x, η− ξ), we are naturally led to set

r₁(x, ξ) = 2^d Z

R^d

Z 1 0

∂_xa(x+ty, ξ+tη) y e^{−|y|2−|η|2}dt dy (2π)^d/2

dη (2π)^d/2

4 J. BERNIER, F. CASAS, AND N. CROUSEILLES

and

r₂(x, ξ) = 2^d Z

R^d

Z 1 0

∂_ξa(x+ty, ξ+tη) η e^{−|y|2−|η|2}dt dy (2π)^d/2

dη (2π)^d/2. By symmetry, we just check that r₁ ∈S(hxi^s1−1+hξi^s2;hξi,hxi).

By definition, we just have to prove that

(4) |∂_x^α∂_ξ^βr₁(x, ξ)|.α,β (hxi^s1−1+hξi^s2)hxi^−αhξi^−β. Since, by assumption, we know that

|∂_x^γ∂_ξ^δa(x, ξ)|._γ,δ (hxi^s1+hξi^s2)hxi^−γhξi^−δ, we deduce that

|∂_x^α+1∂_ξ^βa(x+ty, ξ+tη)|.α,β(hx+tyi^s1 +hξ+tηi^s2)hx+tyi^−α−1hξ+tηi^−β. Recalling the Peetre’s inequality (0.1.2) page 19 :

hx+yi^s .shxi^shyi^|s|, ∀x, y∈R^d, s∈R, we get

|∂_x^α+1∂_ξ^βa(x+ty, ξ+tη)|._α,β,s1,s2 (hxi^s1hyi^s1+hξi^s2hηi^s2)hxi^−α−1hξi^−βhyi^α+1hηi^β ._α,β,s1,s2 (hxi^s1+hξi^s2)hxi^−α−1hξi^−βhyi^α+1+s1hηi^β+s2. (5)

Finally, observing that

(hxi^s1+hξi^s2)hxi⁻¹ ≤ hxi^s1−1+hξi^s2,

plugging (5) in the definition ofr₁ yields to (4), i.e. r₁ ∈S(hxi^s1−1+hξi^s2;hξi,hxi).

References

[BCC20] J. Bernier, F. Casas, N. Crouseilles,Splitting Methods for Rotations: Application to Vlasov Equations, SIAM Journal on Scientific Computing, 42(2) 2020.

[NR10] F. Nicola, L. G. Rodino, Global Pseudo-differential Calculus on Euclidean Spaces, Springer Birkhäuser Basel, 2010.

Institut de Mathématiques de Toulouse ; UMR5219, Université de Toulouse ; CNRS, Université Paul Sabatier, F-31062 Toulouse Cedex 9, France

E-mail address: [email protected] Universitat Jaume I, 12071 Castellon, Spain.

E-mail address: [email protected]

Univ Rennes, INRIA, MINGuS team, IRMAR - UMR 6625, F-35042 Rennes, France E-mail address: [email protected]