EXISTENCE OF RADIAL GLOBAL SMOOTH SOLUTIONS TO THE PRESSURELESS EULER-POISSON EQUATIONS WITH QUADRATIC CONFINEMENT

2025-05-06 0 0 705.18KB 15 页 10玖币

侵权投诉

EXISTENCE OF RADIAL GLOBAL SMOOTH SOLUTIONS TO THE

PRESSURELESS EULER-POISSON EQUATIONS WITH QUADRATIC

CONFINEMENT

JOS´

E A. CARRILLO:, RUIWEN SHU;

Abstract. We consider the pressureless Euler-Poisson equations with quadratic conﬁnement. For spa-

tial dimension dě2, d ‰4, we give a necessary and suﬃcient condition for the existence of radial global

smooth solutions, which is formulated explicitly in terms of the initial data. This condition appears to

be much more restrictive than the critical-threshold conditions commonly seen in the study of Euler-

type equations. To obtain our results, the key observation is that every characteristic satisﬁes a periodic

ODE system, and the existence of global smooth solution requires the period of every characteristic to

be identical.

1. Introduction

In this work, we will deal with the pressureless Euler-Poisson equations with conﬁnement written as

Btρ`∇¨ pρuq “ 0

Btu`u¨∇u“ ´ż∇Npx´yqρpt, yqdy´x.(1.1)

Here xPRd, d ě2, ρpt, xqis the particle density function, and upt, xqis the velocity ﬁeld. Nis the

Newtonian repulsion potential, satisfying ´∆N“δ, given by

Npxq “ $

’

´1

2πln |x|, d “2

cd|x|2´d, d ě3, cd“1

|Sd´1|

The last term ´xin the velocity uequation represents the eﬀect of a quadratic conﬁning potential.

Notice that this is equivalent to say that the particles are subject to a potential force with the potential

being φ“ p´∆q´1pρ´dq, i.e., Newtonian repulsion with a positive charged background, see for instance

[23]. Our aim is to give a sharp result on the existence of global smooth solutions to (1.1) for radial

initial data.

The existence of global smooth solutions to Euler-Poisson systems has been thoroughly studied in

the literature. One popular approach for the study of Eulerian dynamics, which we will adopt in this

paper, is spectral dynamics [8, 13]. This method was originally designed to analyze the eigenvalues of the

deformation matrix ∇ualong the characteristics of the ﬂow. It was later generalized to analyze the time

evolution of certain quantities along characteristics, and derive the existence of global smooth solutions

of the PDE system as that of a family of ODE systems. For the pressureless Euler-Poisson system, some

criteria for the existence of global smooth solutions have been developed by [8, 13, 14, 15, 2, 23] in the

context of 1D or multi-D radial solutions. Similar approaches were also developed to study Eulerian

dynamics arising from models of collective behavior [20, 3, 4, 10, 19, 7, 12, 5, 18, 22, 23], which usually

involve other forcing terms like the Cucker-Smale alignment interaction [6] or linear damping.

The local-in-time existence and uniqueness of classical solutions to the Euler-Poisson system is known

for the initial data being a small perturbation of the stationary state, see [16, 17]. In these references,

the authors assume that the density is positive on the whole line with zero limit as xÑ ˘8. A local-

in-time well-posedness of the Cauchy problem for the pressureless Euler-Poisson system in the plane

without smallness assumptions in Sobolev spaces was given in [2, Section 5]. Besides the study of the

pressureless Euler-Poisson system, people have also studied the existence of global smooth solutions to

the Euler-poisson system with pressure [25, 9, 24, 21, 26, 11]. We can summarize by saying that ﬁnding

sharp criteria for the existence of global smooth solutions is a challenging problem for Euler-Poisson

(:)Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK. Email: carrillo@maths.ox.ac.uk

(;)Department of Mathematics, University of Georgia, Athens, GA 30602, USA. Email: ruiwen.shu@uga.edu

Date: October 26, 2022.

arXiv:2210.13657v1 [math.AP] 24 Oct 2022

2 PRESSURELESS EULER-POISSON EQUATIONS WITH QUADRATIC CONFINEMENT

type problems. One of the diﬃculties we need to face in this work is to deal with initial data that are

compactly supported in the density for (1.1), and thus we need to introduce a suitable notion of solution

consistent with free boundary conditions for the system (1.1).

1.1. Radial formulation & Notion of solution. As already mentioned, we are concerned with radial

solutions to (1.1), i.e., solutions with ρ“ρpt, rq,u“upt, rqx

r, where r“ |x|. To reformulate (1.1) into

radial variables, we introduce the quantities related to a density ρpxq “ ρprq:

Pprq “ |Sd´1|rd´1ρprq, mprq “ żr

Ppsqds“ż|y|ăr

ρpyqdy.

Similar notations will be used for time-dependent densities. We give a lemma on the Newtonian potential

generated by a radial density.

Lemma 1.1. Let ρpxq “ ρprqbe compactly supported and L8. Then

żNpx´yqρp|y|qdy“ż8

NpsqPpsqds`Nprqmprq.

Proof. Denote R“ |x|. We have

żNpx´yqρp|y|qdy“ż|y|ąR

Npx´yqρp|y|qdy`ż|y|ăR

Npx´yqρp|y|qdy

“ż|y|ąR

Npyqρp|y|qdy`Npxqż|y|ăR

ρp|y|qdy.

Here we treat the ﬁrst integral by the fact that ş|y|ąRNp¨´yqρp|y|qdyis a radial harmonic function on

Bp0; Rqand continuous on Rd, and thus constant on Ğ

Bp0; Rq. We use the mean-value property of the

harmonic function Non Bpx;|y|q in the second integral. Therefore the conclusion is obtained. 

Now we can write radial solutions to (1.1) as

BtP` BrpP uq “ 0

Btu`uBru“ ´BrNprqmpt, rq ´ r, mpt, rq “ żr

Ppt, sqds.(1.2)

We always assume that the radial initial data pρ0,u0qof (1.1) satisﬁes that ρ0is continuous and compactly

supported with ρ0ě0, ρ0p0q ą 0, and u0is C1on supp ρ0. As a consequence, the corresponding initial

data pP0, u0qof (1.2) satisﬁes that

‚P0is C1, compactly supported on r0, R0sfor some R0ą0, with limrÑ0`r1´dP0prq ą 0 and

Brpr1´dP0prqq|r“0“0.

‚u0is C1on r0, R0swith u0p0q “ 0.

The triple pP0prq, u0prq, R0qis said to be consistent if the above two conditions are satisﬁed.

Deﬁnition 1.2. A tuple pPpt, rq, upt, rq, Rptqq is called a classical bulk solution to (1.2) on r0, T s, T ą0

with the consistent initial data pP0, u0, R0qif

‚RPC1pr0,8qq.Pand uare supported on tpt, rq: 0 ďtďT, r P r0, Rptqsu and C1on this set.

P,u,Ragree with the initial data at t“0.

‚(1.2) is satisﬁed in tpt, rq: 0 ďtďT, r P p0, Rptqqu in the classical sense.

‚Rptqcorrectly describes the boundary motion, i.e., R1ptq “ upt, Rptqq.

It is called a global classical bulk solution if it is a classical bulk solution on r0, T sfor any Tą0.

We will provide a continuation criteria for bulk solutions to (1.2) in Section 2, see Lemma 2.1. This

allows us to analyse the global existence of bulk solutions by characteristic tracing.

1.2. Main results. With these preparations we can state the main contributions of this work. Because

of the speciﬁcity of the two dimensional Newtonian potential, we separate the general result from the

two dimensional case.

Theorem 1.3. Assume dě3,d‰4. Let pP0, u0, R0qbe a consistent initial data. Then there exists

a global classical bulk solution to (1.2) with this initial data if and only if the following conditions are

satisﬁed:

‚There exists a constant C0such that

m0prq´2{dˆ1

2u0prq2`m0prqNprq ` 1

2r2˙“C0,@rP p0, R0q.(1.3)

PRESSURELESS EULER-POISSON EQUATIONS WITH QUADRATIC CONFINEMENT 3

‚Either C0“min˜r Np˜rq ` 1

2˜r2(and pP0, u0, R0qis a stationary solution; or C0ąmin˜r Np˜rq ` 1

2˜r2(

and

min

p˜r,˜uqPKr"θprq˜u`P0prq

dm0prq˜r*ą0,@rP p0, R0q(1.4)

where the minimum is taken over the energy level set

Kr:“"p˜r, ˜uq P R`ˆR:m0prq´2{dˆ1

2˜u2`m0prqNp˜rq ` 1

2˜r2˙“C0*,

and θprqis deﬁned by

θprq:“$

’

1´P0prqr

dm0prq

u0prqif u0prq ‰ 0

dP0prqu0prq ´ m0prqBru0prq

m0prqp´cdm0prqpd´2qr1´d`rqif ´cdm0prqpd´2qr1´d`r‰0

.(1.5)

Here item 1 and C0ąmin˜r Np˜rq ` 1

2˜r2(guarantee that at least one of the above fractions have

nonzero denominator, and they are equal when both having nonzero denominators.

It is easy to see that item 1 implies the equivalence of the two fractions in (1.5). In fact, diﬀerentiating

the energy level equation (1.3) with respect to rgives

´2

dP0prq´1

2u0prq2`m0prqNprq ` 1

2r2¯`m0prq´u0prqBru0prq ` P0prqNprq ` m0prqBrNprq ` r¯“0.

Using Nprq “ cdr2´d, one can rewrite it as

u0prq´´1

dP0prqu0prq ` m0prqBru0prq¯`m0prq´1´P0prqr

dm0prq¯´´cdm0prqpd´2qr1´d`r¯“0 (1.6)

which shows that the two fractions in (1.5) are equal whenever they have nonzero denominators. Also,

if both denominators are zero, then one has u0prq“´cdm0prqpd´2qr1´d`r“0, in which case we will

show that the same is true for any 0 ărăR0and we necessarily have a stationary solution to (1.2).

Theorem 1.4. Assume d“2. Let pP0, u0, R0qbe a consistent initial data. Then there exists a global

classical bulk solution to (1.2) with this initial data if and only if the following conditions are satisﬁed:

‚There exists a constant C0such that

m0prq´1´1

2u0prq2`m0prqNprq ` 1

2r2¯“C0´1

4πln m0prq,@rP p0, R0q.(1.7)

‚Either C0“min˜rtNp˜rq ` 1

2˜r2uand pP0, u0, R0qis a stationary solution; or C0ąmin˜rtNp˜rq `

2˜r2uand

min

p˜r,˜uqPKr"θprq˜u`P0prq

2m0prq˜r*ą0,@rP p0, R0q,(1.8)

where the minimum is taken over the energy level set

Kr:“"p˜r, ˜uq P R`ˆR:m0prq´1´1

2˜u2`m0prqNp˜rq ` 1

2˜r2¯“C0´1

4πln m0prq*,

and θprqis deﬁned by

θprq:“$

’

1´P0prqr

2m0prq

u0prqif u0prq ‰ 0

2P0prqu0prq ´ m0prqBru0prq

m0prqp´ 1

2πm0prqr´1`rqif ´1

2πm0prqr´1`r‰0

.(1.9)

Here item 1 and C0ąmin˜rtNp˜rq ` 1

2˜r2uguarantee that at least one of the above fractions have

nonzero denominator, and they are equal when both having nonzero denominators.

Similarly, diﬀerentiating the energy level equation (1.7) with respect to rgives

´P0prq´1

2u0prq2`m0prqNprq ` 1

2r2¯

`m0prq´u0prqBru0prq ` P0prqNprq ` m0prqBrNprq ` r¯“ ´ 1

4πm0prqP0prq.

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

EXISTENCEOFRADIALGLOBALSMOOTHSOLUTIONSTOTHEPRESSURELESSEULER-POISSONEQUATIONSWITHQUADRATICCONFINEMENTJOSEA.CARRILLO:,RUIWENSHU;Abstract.WeconsiderthepressurelessEuler-Poissonequationswithquadraticconnement.Forspa-tialdimensiond¥2;d4,wegiveanecessaryandsucientconditionfortheexistenceofradialgloba...

展开>> 收起<<

EXISTENCE OF RADIAL GLOBAL SMOOTH SOLUTIONS TO THE PRESSURELESS EULER-POISSON EQUATIONS WITH QUADRATIC CONFINEMENT.pdf

共15页,预览3页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

EXISTENCE OF RADIAL GLOBAL SMOOTH SOLUTIONS TO THE PRESSURELESS EULER-POISSON EQUATIONS WITH QUADRATIC CONFINEMENT

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: