(李浩光)

School of Mathematics and Statistics, South-Central Minzu University, Wuhan 430074, China;Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles,MIIT, Nanjing 210016, China

E-mail: lihaoguang@scuec.edu.cn

Chaojiang XU (徐超江)

School of Mathematics and Key Laboratory of Mathematical MIIT,Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

E-mail: xuchaojiang@nuaa.edu.cn

Abstract In this work,we study the linearized Landau equation with soft potentials and show that the smooth solution to the Cauchy problem with initial datum in L2(R3) enjoys an analytic regularization effect,and that the evolution of the analytic radius is the same as the heat equations.

Key words linear Landau equation;analytic smoothing effect;soft potential

1 Introduction

In this work,we study the Cauchy problem of the spatially homogeneous Landau equation

whereF=F(t,v)≥0 is the density distribution function depending on the velocity variablesv ∈R3and the timet ≥0.The Landau bilinear collision operator is given by

One says that these are hard potentials ifγ>0,Maxwellian molecules ifγ=0,soft potentials ifγ ∈]-3,0[ and Coulombian potentials ifγ=-3.We shall study the linearization of the Landau equation (1.1) near the absolute Maxwellian distribution

Considering the fluctuation of the density distribution function

sinceQ(μ,μ)=0,the Cauchy problem (1.1) is reduced to the Cauchy problem

In this work,we study Cauchy problem of the linear Landau equation

Using reference [7],we show that the diffusion partLis written as

withA(v)=()1≤i,j≤3being a symmetric matrix where

For the hard potential case,the existence and uniqueness of the solution to the Cauchy problem for the spatially homogeneous Landau equation was already treated in [6,14]under a rather weak assumption on the initial datum.Moreover,the authors of these works proved the smoothness of the solution inC∞(]0,+∞[;S(R3)).In[2],Chen-Li-Xu improved this smoothing property and proved that the solution is in fact analytic for anyt>0 (see [3,4]for the Gevrey regularity).

For the Maxwellian molecules case,in[8],Lerner,Morimoto,Pravda-Starov and Xu studied the spatially homogeneous non-cutoffBoltzmann equation and the Landau equation in a closeto-equilibrium framework and showed that the solution enjoys the Gelfand-Shilov smoothing effect (see also [10,13]and [9]).This implies that the nonlinear spatial homogeneous Landau equation has the same smoothing effect properties as the classic heat equation or the harmonic oscillators heat equation.In addition,starting from anyL2initial datum att=0,the solution of the Cauchy problem is spatial analytic for anyt>0 and the analytic radius is.In the non-Maxwellian case,we cannot use the Fourier transformation and the spectral decomposition as in[8–10,13].Recently,Li and Xu in[11]proved the analytic smoothing effect of the solution to the nonlinear Landau equation with hard potentials and the Maxwellian molecules case;that is,γ ≥0.Concerning the soft potential and the Coulombic interaction,Guo [7]constructed global in time classical solutions near the Maxiwellians for any small initial datum belonging toHN,N ≥8.

In this work,we study the Cauchy problem of the linearized Landau equation with soft potentials-3<γ<0.We show that the smooth solution enjoys an analytic smoothing effect for a short time.The main theorem is stated below.

Theorem 1.1For soft potentials-3<γ<0 and for anyT>0,with the initial datumf0∈L2(R3),the Cauchy problem (1.3) admits a unique weak solution

Moreover,for anyα ∈N3and=min{t,1},there exists a positive constantCwhich depends only on‖f0‖L2(R3)andT,and we have that

Remark 1.2(i) Equivalently,for soft potentials and for anym ∈N,we have that

For the multi-indices,we use the notation from the binomial expansion:

(ii) We do not consider the Coulombic interaction in Theorem 1.1 because the inequality(2.4)only holds true forγ>-3.In fact,we think that the Coulombic case has similar estimate as Theorem 1.1,but the proofs are different.

The rest of the paper is arranged as follows: we prove the ultra-analyticity for the coefficient of the Landau operator in Section 2.In Section 3,we estimate the commutators and prove the coercivity property of the linear Landau operator.In Section 4,we study the Cauchy problem for the linear Landau equation,and show the existence and uniqueness properties of the weak solution.The analytic smoothing effect of the weak solution for the linear Landau equation with soft potentials will be proven in Section 5.In the Appendix,we introduce the Hermite operator and related results.

2 Ultra-analyticity for the Coefficient of the Landau Operator

Forγ ∈R,denote that

where we use the notations〈v〉=(1+|v|2)1/2.

In addition,for the matrixAdefined in (1.4),we denote that

and the weighted norm,forθ ∈R,is

From formula (21) of Corollary 1 in [7],for anyθ ∈R,there existsC1>0 such that

where,for any vector-valued functionG(v)=(G1,G2,G3),we define the projection to the vectorv=(v1,v2,v3) as

Noticing that?f=Pv?f+(I-Pv)?f,we have that

we can also refer to [5]and the references thereins.We remark that the weights forfand?fare different in the definition of.

First,for anyγ>-3 andδ>0,we have that

This implies that

In the following,we prove that the coefficients of the linear Landau operator are ultraanalytic:

Lemma 2.1For anyβ ∈N3with|β|≥1 whereis as was defined in (1.4) with-3<γ<0,we have that,

Moreover,for anyβ ∈N3,

ProofForβ ∈N3with|β|≥1,without loss of generality,we set that

Direct calculation shows that,for any 1≤i ≤3,

For more details regarding the operatorsA±,i,we refer to the Appendix.By using the fact that

it follows from Cauchy-Schwartz’s inequality and (2.4) that

where we use the fact that

where{Ψα}α∈N3is the orthonormal basis inL2(R3).H?lder’s inequality and Poincaré’s inequality imply that

and along with the equalities (A.1) and (A.2),this shows that,

For the estimate of the remaining inequalities (2.7),an integration by parts inside the convolution and (2.8) show that

by using the fact that

From a calculation similar to that above,it follows that

For|β|≥1,the same estimate holds true for the last term,such that

This ends the proof of Lemma 2.1.

In order to prove the coercivity of the linear Landau operator,we need one more estimate to control the weighted.

Lemma 2.2Forf,g ∈S(R3),and for anyβ ∈N3andθ ∈R,we have that

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ProofIn fact,we have the inner product

We decompose the integration region [v,w]∈R3×R3into three parts:

For the first part,{|v|≤1},by Lemma 2.1 and (2.5),we have that

For the second part,{2|w|≥|v|,|v|≥1},we have that

Similarly to the proof of Lemma 2.1,one can verify that

We finally consider the third part,{2|w|≤|v|,|v|≥1}.Expandingaij(v-w) to get that

along with the fact that

we immediately have that

Since 2|w|≤|v|,|v|≥1,0

It follows from the inequality (2.9) and the norm equality (2.2) that

This is the inequality (2.10).This ends the proof of Lemma 2.2.

3 Estimations of Commutators

Proposition 3.1Letf ∈S(R3),and letLbe defined as in (1.3).For anyα ∈N3andθ ∈R,there exists a positive constantC0>0 which is independent ofαandθsuch that

Remark 3.2We have that

1.forα=0,θ=0,

2.for-3<γ<0,and anyN ∈N,0<δ<1,there existsCNsuch that

3.for|α|≥1,

whereαj0=max{α1,α2,α3}.

ProofRecalling the formulaLfin (1.3),for the smooth functionf,and integrating by parts,we have that

Then,using (2.1),we have that

Since,for any fixediorj,

where we use the result that

Similarly,one has that

Using the Leibniz formula,

and it follows that

Thus the proof of Proposition 3.1 is reduced to the estimations of terms R0(f) and R1(f),which we give by the following two lemmas:

Lemma 3.3We have,forα ∈N3that

ProofBy using (2.4),we get that

Then (2.3) implies that

For the term R01,we use that

It follows that

This gives the estimate of R0(f).

Lemma 3.4We have,forα ∈N3,|α|≥1 that

ProofNow we estimate R1.For the term R11(f),by using the inequality (2.10) in Lemma 2.2 directly,we obtain that

For the two terms,R13(f) and R14(f),we can deduce from the inequality (2.7) in Lemma 2.1 and (2.3) that

For the term R12(f),for|β|≥1,it follows from (2.6) in Lemma 2.1 that

Then we have that

We then conclude that

Substituting the estimates of R0(f),R1(f)into the decomposition(3.4)completes the proof of Proposition 3.1.

4 Existence and Uniqueness of Linear Landau Equation

Proposition 4.1For-3<γ<0,T>0,f0∈L2(R3),the Cauchy problem (1.3) admits a unique weak solution

satisfying that

whereC0is as was defined in Proposition 3.1.

ProofThe existence of the weak solution is similar to that in [1,7,12].We consider the operator

For any? ∈C∞([0,T];S(R3)) with?(T)=0,it follows from (3.1) that

Sinceγ<0,we have that

This implies that

Then one can verify that

In what follows,we set the vector subspace as

Sincef0∈L2(R3),we define the linear functional as

where? ∈C∞([0,T],S(R3))with?(T)=0.From(4.1),the operatorP?is injective.Therefore the linear functionalGis well-defined,and moreover,we obtain that

This shows thatGis a continuous linear form on (U,‖·‖L1([0,T],L2(R3))).By using the Hahn-Banach theorem,Gcan be extended as a continuous linear form onL1([0,T];L2(R3)) with a norm smaller than 2e2C0T‖f0‖L2(R3).It follows from the Riesz representation theorem that there exists a uniquef ∈L∞([0,T];L2(R3)) satisfying that

Therefore,f ∈L∞([0,T];L2(R3)) is a weak solution of the Cauchy problem (1.3).Let∈L∞([0,T];L2(R3)) be another weak solution of the Cauchy problem (1.3) satisfying that

Settingw(t)=f(t)-(t),we have that,for all? ∈C∞0((0,T),S(R3)),

This shows thatw(t)=0 inL∞([0,T];L2(R3)).The proof of Proposition 4.1 is complete.

Remark 4.2If we use (3.2) in place of (3.1),we can prove that,for-3<γ<0,N ∈N,T>0,f0∈HN(R3),and Cauchy problem (1.3) admits a unique solution such that,for any 0≤δ<1,

5 Analytic Smoothing Effect for Linear Landau Equation

Suppose now thatf0∈L2(R3),and letf?be the solution of the linear Landau equation with the initial datumη??f0∈H∞(R3) whereη?is a mollifier function;that is,

We remark that

Using Remark 4.2,we have thatf?is a smooth solution for 0

Now we want to prove the estimate

withConly depending on‖f0‖L2(R3),so that by compactness of sequence{f?}and the uniqueness of the solution of the Cauchy problem (1.3),we get that

Letf?be the smooth solution of Cauchy problem (5.1) with 0

This implies that (sinceγ<0)

By using Gr?nwall’s inequality,for anyT>0 and 0

Letting|α|=1,it follows from Proposition 3.1 withθ=γthat

By using the inequality (2.3),one can verify that,for any 0

Substituting into the estimate (5.4),we have that

Integrating on [0,t]and using (5.3),one can verify that,for|α|=1,

We remark that,here,the constantCdepends only on‖f0‖L2(R3)andT.

Proposition 5.1For anym ∈N andα ∈N3,|α|=m,we have,for 0

whereConly depends on‖f0‖L2,and in particular,is independent onαand?.

This proposition implies Theorem 1.1.To simplify the notation,we omit the supreme index?off?.

ProofIn fact,we have proven that the assumption(5.6)holds true form=0,1,by(5.3)and (5.5).

Now assume that the assumption (5.6) holds true for|α|≤m-1.This means,for any|α|≤m-1,for 0

We intend to prove the validity of (5.6) form.First,

Lettingθ=in Proposition 3.1,it follows that

It follows from the inequality (2.3) again that

wherek0is chosen withαk0=max{α1,α2,α3},so that,by the induction assumption (5.7),for|α-ek0|=m-1,

We get then that

For the term B2(f),using the fact,for anyβ ≤α,γ<0,that

and by using the induction assumption (5.7) for|α-β|≤m-1,for 0

We then get,for 0

For the term B3(f),by using the induction assumption(5.7)for|α-β|≤m-2,|α-ek0|=m-1,

and we get then that

Finally,for the term B4(f),by using the induction assumption (5.7) for|α-ek0|=m-1,

We then get,for 0

Take the constantCsatisfying (5.5),and

where the constants are defined by (5.11),(5.13) and (5.15),so that it depends only on‖f0‖L2andT.Combining (5.8),(5.9),(5.10),(5.12) and (5.14) ends the proof of Proposition 5.1.

Conflict of InterestChaojiang Xu is an editorial board member for Acta Mathematica Scientia and was not involved in the editorial review or the decision to publish this article.All authors declare that there are no competing interests.

Appendix

The standard Hermite functions (?n)n∈Nare defined,forv ∈R,as

wherea+is the creation operator

The family (?n)n∈Nis an orthonormal basis ofL2(R) and we set,forn ≥0,α=(α1,α2,α3)∈N3,x ∈R,v ∈R3,that

with|α|=α1+α2+α3.The family (Ψα)α∈N3is an orthonormal basis ofL2(R3) composed by the eigenfunctions of the 3-dimensional harmonic oscillator

where Pkstands for the orthogonal projection

In particular,

whereμ(v) is the Maxwellian distribution.Setting

we have that

where (e1,e2,e3) stands for the canonical basis of R3.For more details regarding the Hermite functions,we refer to [13]and the references therein.