Formation of shifted shock for the 3D compressible Euler equations with damping Chen Zhendong 1

2025-05-06 0 0 684.05KB 97 页 10玖币

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Formation of shifted shock for the 3D compressible Euler equations with

damping

Chen Zhendong*1

1Institute of Mathematical Science, The Chinese University of Hong Kong, Shatin, NT, Hong Kong.

Abstract

In this paper, we show the shock formation of the solutions to the 3-dimensional (3D)

compressible isentropic and irrotational Euler equations with damping for the initial short

pulse data which was ﬁrst introduced by D.Christodoulou [10]. Due to the damping eﬀect,

the largeness of the initial data is necessary for the shock formation and we will work on

the class of large data (in energy sense). Similar to the undamped case, the formation of

shock is characterized by the collapse of the characteristic hypersurfaces and the vanishing

of the inverse foliation density function µ, at which the ﬁrst derivatives of the velocity and

the density blow up. However, the damping eﬀect changes the asymptotic behavior of the

inverse foliation density function µand then shifts the time of shock formation compared

with the undamped case. The methods in the paper can also be extended to a class of 3D

quasilinear wave equations for the short pulse initial data.

Keywords: Compressible Euler equations with damping, delayed shock formation,

inverse foliation density function, characteristic null-hypersurfaces

Contents

1 Introduction 3

1.1 Brief review of known results .......................... 4

1.1.1 Christodoulou’ s theory of shock formation .............. 5

1.2 Notations ..................................... 9

*Acknowledgement: This is part of the Ph. D thesis of the author written under the supervision of Professor

Zhouping Xin at the Institute of Mathematical Sciences of The Chinese University of Hong Kong. The research

is supported in part by the Zheng Ge Ru Foundation and by the Hong Kong RGC Earmarked Research Grants:

CUHK-14305315, CUHK-14300917, and CUHK-14302819.

arXiv:2210.13796v1 [math.AP] 25 Oct 2022

1.3 Geometric blow-up for the Burgers’ equation with damping ........ 10

1.4 The inhomogeneous nonlinear wave equation for the isentropic and irrota-

tional Euler equations with damping ...................... 12

1.5 Initial data .................................... 14

2 The Geometric formulation, basic structure equations and the main results 16

2.1 The maximal development ........................... 16

2.2 Frames and coordinates ............................. 18

2.3 Connection coeﬃcients and 2nd fundamental forms k, χ and θ........ 22

2.4 Decomposition of wave equation in the null frame .............. 23

2.5 Curvature tensor ................................. 24

2.6 The rotation vector ﬁelds ............................ 25

2.7 Transport equations for µand χ........................ 25

2.8 The main results ................................. 26

3 Bootstrap assumptions and preliminary estimates 28

3.1 Preliminary estimates for the metric, µand the second fundamental forms

under the bootstrap assumptions ........................ 29

3.2 Accurate estimates for µand its derivatives .................. 33

3.2.1 Two key properties of µnear shock .................. 35

4 Estimates for deformation tensors 39

5 Multiplier and commuting vector ﬁelds methods, estimates on lower order terms 43

5.1 Multiplier method, fundamental energy estimates .............. 45

5.1.1 Estimates for the error integrals Qi.................. 48

5.2 Some top order acoustical terms ........................ 52

5.3 Estimates for lower order terms ........................ 56

6 Regularization of the transport equations for µand trχand estimates for the top

order acoustical terms 64

6.1 Estimates for the top order angular derivatives of trχ............ 64

6.1.1 Elliptic estimates on St,uand estimates for I2and I3......... 67

6.1.2 Estimate for I1.............................. 69

6.2 Estimates for the top order spatial derivatives of µ.............. 73

6.2.1 Elliptic estimates on St,uand estimates for I1,I2,I3.......... 75

7 Top order energy estimates 77

7.1 Contributions of the top order acoustical terms to the error integrals . . . 77

7.1.1 Contributions associated with K0................... 77

7.1.2 Contributions associated with K1................... 79

7.2 The top order energy estimates ......................... 84

7.2.1 Estimates associated with K1...................... 85

7.2.2 Estimates associated with K0...................... 87

8 Decent scheme 88

8.1 The next-to-top order energy estimates .................... 89

8.2 The decent scheme ................................ 92

9 Recovery of the bootstrap assumptions and completion of the proof 93

1 Introduction

In this paper, we will consider the following compressible Euler equations with damping in

R3:









∂

∂tρ+∇ · (ρv)=0,

∂

∂t(ρv)+∇(ρv⊗v+pI3)+aρv=0,

∂

∂tS+v· ∇S=0,

(1.1)

where ρ,v=(v1,v2,v3), p,Srepresent the density, the velocity, the pressure and the entropy

of the ﬂow, respectively. I3is a 3 ×3 identical matrix and ais the damping constant. These

equations describe the motion of a perfect ﬂuid which follow from the conservation of mass,

momentum and energy respectively. The enthalpy hof a thermodynamic system is deﬁned as

the sum of its internal energy and the product of its pressure and volume:

h=e+pV.(1.2)

Here eand V=1

ρrepresent the internal energy and the volume of the system, respectively. It

follows from the relation of thermodynamics that

dh =T dS +Vdp,(1.3)

where Tis the absolute temperature. The sound speed ηis deﬁned as η=r∂p

∂ρ . The equation

of state is given by p=p(ρ, S) and we assume that:

pis not linear in 1

ρ.(1.4)

This assumption will be clariﬁed later.

1.1 Brief review of known results

If the damping term vanishes, then the system (1.1) is returned to the classical compressible

Euler system, which is a prototype of hyperbolic systems of conservation laws:

∂tU+div(F(U)) =∂tU+A(U)∇U=0,

U(x,0) =U0(x),(1.5)

where U:Rm→Rn. Understanding the shock formation and development to (1.5) is funda-

mental since the system will develop singularities in general in ﬁnite time for initial smooth

small data. B.Riemann was the ﬁrst one to study the nonlinear eﬀects to the 1D isentropic Euler

equations. In [28], Riemann proved that the wave compression leads to the shock waves and the

wave expansion leads to the rarefaction waves. Later, P.Lax generalized Riemann’s result into

the 2 ×2 system (i.e. n=2) in one space dimension in [17]. Lax used the Riemann invariants

(˜u,˜v), that is, the gradient of each invariant is proportional to the left eigenvector of the coef-

ﬁcient matrix. Then, Lax was able to give a suﬃcient and necessary condition for the system

to admit a shock or not. However, for general system with n>2, Lax’s method may fail since

the system may not admit a coordinate system of Riemann invariants. In 1974, F.John in [14]

achieved a remarkable result for the shock formation to general n×nhyperbolic systems of

conservation laws in one space dimension. He considered the following Cauchy problem:

∂tu+A(u)∂xu=0,

u(x,0) =f(x),(1.6)

where A(u) is an n×nsmooth coeﬃcient matrix with real distinct eigenvalues. F.John showed

that if the system (1.6) is genuinely nonlinear and the initial data f(x) is small with compact

support, then any classical solution to (1.6) must form shock in ﬁnite time, which means that

the ﬁrst order derivatives of ublow up in ﬁnite time while itself remains bounded. However,

F.John’s result failed to apply to the Euler equations in one space dimension since the entropy

wave is linearly degenerate, and later T.P.Liu generalized F.John’s result in [18]. T.P.Liu proved

that if each characteristic family for the system (1.6) is either genuinely nonlinear or linearly

degenerate, then under certain condition, any classical solution to (1.6) must develop shock in

ﬁnite time.

In multi-dimensional case, the singularity formation problem for (1.5) is much more compli-

cated than in 1D case. The ﬁrst general result for the singularity formation to the compressible

Euler equations in three spatial dimensions was obtained by Sideris [30] for polytropic gases. In

particular, he exhibited an open set of small initial data for which the corresponding solutions

cease to be C1in ﬁnite time by using dissipative energy estimates. However, his results did

not provide any information for the mechanism of the breakdown of the solutions. Later, Alin-

hac studied the two-dimensional compressible isentropic Euler equations with radial symmetry

in [1]. He showed that a large class of small radially symmetric data leading to the ﬁnite time

blow up of the solutions and gave a precise estimate for the blow up time. Later, in a series

of works [2–5], Alinhac proved the shock formation to the 2Dquasilinear wave equations. He

constructed a class of initial data which lead to the break down of the solutions in ﬁnite time

and gave precise estimates of the solutions up to the ﬁrst singularity.

For the system(1.1), there are many studies both in 1Dand multi-dimensions. For the one-

dimensional Euler equations with damping, the global existence of smooth solutions with small

data was proved by Nishida [24] and Slemrod [32] showed that for small data (L∞sense), the

1DEuler equations with damping admit a global smooth solution while for large data, the equa-

tions can develop a shock in ﬁnite time. Later, these results were generalized by many authors,

see [13, 21–23, 25, 26] and the references therein.

In multi-dimensional case, the global existence and L2estimates for the solutions to the 3D

isentropic Euler system with damping was obtained by Kawashima [16] and the long time be-

havior of the solutions was obtained and generalized by many authors, see [15, 27, 31, 33, 34]

and the references therein.

1.1.1 Christodoulou’ s theory of shock formation

A major breakthrough in understanding the shock mechanism for the hyperbolic systems(1.5)

in multi-dimensions has been made by Christodoulou in a series of works [10–12]. In [12], the

classical, non-relativistic, isentropic compressible Euler¡¯s equations in three spatial dimen-

sions with initial irrotational data were studied. Starting from the short pulse data, the authors

gave a detailed analysis of the solutions near the singularity (shock) and showed a complete

geometric structure for the shock development. Note that under the isentropic and irrotational

assumptions, the 3D Euler system can be rewritten as the following quasilinear wave equation:

g(ϕ)ϕ=0,(1.7)

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Formationofshiftedshockforthe3DcompressibleEulerequationswithdampingChenZhendong*11InstituteofMathematicalScience,TheChineseUniversityofHongKong,Shatin,NT,HongKong.AbstractInthispaper,weshowtheshockformationofthesolutionstothe3-dimensional(3D)compressibleisentropicandirrotationalEulerequationswithda...

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