The Stability of Compressible Vortex Sheets in Two Space Dimensions

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The Stability of Compressible Vortex Sheets in Two Space Dimensions JEAN-FRANC¸OIS COULOMBEL & PAOLO SECCHI ABSTRACT. We study the linear stability of compressible vortex sheets in two space dimensions. Under a supersonic condition that precludes violent instabilities, we prove an energy estimate for the linearized boundary value problem. Since the problem is characteristic, the estimate we prove exhibits a loss of control on the trace of the solution. Furthermore, the failure of the uni- form Kreiss-Lopatinskii condition yields a loss of derivatives in the energy estimate. 1. INTRODUCTION A velocity discontinuity in an inviscid flow is called a vortex sheet. In three-space dimensions, a vortex sheet has vorticity concentrated along a surface in the space. In two-space dimensions, the vorticity is concentrated along a curve in the plane. The present paper deals with compressible vortex sheets, i.e., vortex sheets in a compressible flow. If the solution is piecewise constant on either side of the interface of discon- tinuity, one has planar vortex sheets in the three dimensional case and rectilinear vortex sheets in the two dimensional case, respectively. The linear stability of pla- nar and rectilinear compressible vortex sheets has been analyzed a long time ago, see [12, 27]. In three space dimensions, planar vortex sheets are known to be vio- lently unstable (see e.

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• compressible vortex


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• kreiss-lopatinskii condition


• variable coefficients


• coefficients linearized


• vortex sheets

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• Publié par: profil-urra-2012
• Nombre de lectures: 38
• Langue: English

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The

Stability of Compressible Vortex in Two Space Dimensions

Sheets

J EAN -F RANC ¸ OIS C OULOMBEL &P AOLO S ECCHI

A BSTRACT . We study the linear stability of compressible vortex sheets in two space dimensions. Under a supersonic condition that precludes violent instabilities, we prove an energy estimate for the linearized boundary value problem. Since the problem is characteristic, the estimate we prove exhibits a loss of control on the trace of the solution. Furthermore, the failure of the uni- form Kreiss-Lopatinskii condition yields a loss of derivatives in the energy estimate.

1. I NTRODUCTION A velocity discontinuity in an inviscid flow is called avortex sheet three-space. In dimensions, a vortex sheet has vorticity concentrated along a surface in the space. In two-space dimensions, the vorticity is concentrated along a curve in the plane. The present paper deals with compressible vortex sheets, i.e., vortex sheets in a compressible flow. If the solution is piecewise constant on either side of the interface of discon- tinuity, one has planar vortex sheets in the three dimensional case and rectilinear vortex sheets in the two dimensional case, respectively. The linear stability of pla- nar and rectilinear compressible vortex sheets has been analyzed a long time ago, see [12,27three space dimensions, planar vortex sheets are known to be vio-]. In lently unstable (see e.g. [30]). In the two dimensional case, subsonic vortex sheets are also violently unstable, while supersonic vortex sheets are neutrally linearly sta- ble, see e.g. [27,30This result formally agrees with the theory of incompressible]. vortex sheets. In fact, in the incompressible limit, the speed of sound tends to in- finity, with the result that two-dimensional vortex sheets are always unstable. This kind of instability is usually referred to as the Kelvin-Helmholtz instability. For the incompressible theory of two-dimensional vortex sheets, we refer the reader to

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J EAN -F RAN ¸ COIS C OULOMBEL & P AOLO S ECCHI

the books [7,22]. Moreover, we refer to [14] for the study of the instability of vortex sheets when heat conduction is taken into account. However, the normal modes analysis performed to derive the linear stability of supersonic vortex sheets is by far not sufficient to guarantee the existence of nonconstant vortex sheets (that is, contact discontinuities) solutions to the com- pressible isentropic Euler equations. In this paper, we first show that supersonic constant vortex sheets are linearly stable, in the sense that the linearized system (around these particular piecewise constant solutions) obeys an energy estimate. Then we consider the linearized equations around a perturbation of a constant vortex sheet, and we show that these linearized equations obey the same energy estimate. This is a first crucial step towards proving the existence of nonconstant compressible vortex sheets. Several points need to be highlighted. First of all, the existence of compress- ible vortex sheets is a free boundary nonlinear hyperbolic problem. Moreover, the free boundary is characteristic with respect to both left and right states since we deal with contact discontinuities. This is one of the reasons why one can not apply Majdas analysis on shock waves (see [20,21]), that are noncharacteristic interfaces. In some previous works devoted to weakly stable shock waves, see [10,11], the first author has considered noncharacteristic hyperbolic Initial Boundary Value Prob- lems that did not meet the uniform Kreiss-Lopatinskii condition. In the case of vortex sheets, the analysis is closely related, with the additional difficulty that the boundary is characteristic (the present analysis thus relies more on the work of Majda and Osher [23] rather than on the work of Kreiss [6,17]). The connection with [10,11] is that in both cases, the analogue of the Kreiss-Lopatinskii condition is fulfilled but not in a uniform way. Furthermore, in the case of vortex sheets as in the case of shock waves, the linearized Rankine-Hugoniot conditions form an elliptic system for the unknown front. This property is a key point in our work since it allows toeliminatethe unknown front and to consider a standard Bound- ary Value Problem with a symbolic boundary condition (this ellipticity property is also crucial in Majdas analysis on shock waves [20,21]). Regarding the energy estimates for the linearized problems, the failure of the uniform Kreiss-Lopatinskii condition yields a loss of derivatives with respect to the source terms. Furthermore, because the boundary is characteristic, we expect to lose some control on the trace of the solution at the boundary. As a matter of fact, we shall see that the only loss of control is on the tangential velocity (which corresponds to the “characteristic part” of the solution). The good point is that the ellipticity of the boundary conditions for the unknown front enables us to gain one derivative for it, as in Majdas work on shock waves [20 slightly]. Going more into the details, we shall prove that the only frequencies for which we lose some control on the solution correspond to bicharacteristic curves. Those curves originate from those points at the boundary of the space domain where the so- called Lopatinskii determinant vanishes. In the interior of the space domain, these singularities propagate along two bicharacteristics associated with theincoming modes.

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Let us now describe the content of the paper. In Section2, we present the nonlinear equations describing the evolution of compressible vortex sheets and introduce some notations. Then, in Section3, we shall consider the linearized equations around a constant (stationary) vortex sheet. The main result for the constant coefficient linearized problem is given in Theorems3.1and3.2. After several reductions, we shall detail in Section4the normal modes analysis of the linearized problem and construct adegenerate in order to de-Kreiss symmetrizers rive our energy estimate. In Section5, we first present the variable coefficients linearized problem and introduce Alinhac sgood unknown. Then we paralinearize the equations, in order to use symbolic calculus and derive the energy estimate. A precise estimate of the paralinearization errors is given. Eventually, we show how to control the different pieces of the solution, depending on their microlocalization. The main result for the variable coefficients linearized problem is given in Theo- rem5.1. In Section6, we make some remarks about possible future achievements. AppendixAis devoted to the proof of several technical lemmas and Appendix Bgathers the main results on paradifferential calculus that are used throughout Section5.

2. T HE N ONLINEAR E QUATIONS 2 We consider Euler equations of isentropic gas dynamics in the whole plane R . Denoting by u the velocity of the fluid and ν the density, the equations read: ( (2.1) ϑ ϑ e e kν ν+ u ∙ + rk r ν ∙ u νk  u 0 ψ ⊗u+ ra 0 ψ where aakν is the pressure law. In all this paper, a is assumed to be a strictly increasing function of ν , defined on ‡ 0 ψ+∞ . We also assume that a is a C ∞ function of ν . The speed of sound Pkν in the fluid is then defined by the relation

q Pkν : a 0 kνχ Let kνψukeψ i 1 ψ i 2  on either side of a smooth hyper-be a smooth function surface  {keψ i 1 } . Then kνψu is a (weak) solution of (2.1) if and only if kνψu is a classical solution of (2.1) on both sides of Rankine- an Hugoni  dnoc : to i it 2 io  ns ' hold at each point of  :  d the

(2.2a) ϑ e 'ν‡−νu∙κ ‡ 0 ψ (2.2b) ϑ e 'νu‡−kνu∙κ u‡−κa‡ 0 ψ where κ : k−ϑ i 1 'ψ 1  is a (space) normal vector to b + −b − Following Lax [18], we shall say that kνψu is a   actdiscocont .Asusual,  n b‡ tinuity if the denotes the jump of a quantity b  [ (seeacross the interface29]). Rankine-Hugoniot conditions (2.2) are satisfied in the following way: ϑ e 'u + ∙κu − ∙ aκ ψ + a − χ

4 J EAN -F RANC ¸ OIS C OULOMBEL & P AOLO S ECCHI Because a is monotone, the previous equalities read (2.3) ϑ e 'u + ∙κu − ∙ νκ ψ + ν − χ Since n t ly he ju d m en p si e t x y p a e n ri d m t e h n e te n d or by ma t l he ve s l o o l c u i t t i y on ar i e s c o o n nt t i h n e u t o a u n s g a e c n r t o ia s l s v t e h l e oc i i n t t y e . rf ( a H ce ere  , , the o r n e o a r so m n a , l a an co d n t t a a n ct ge d n i t s i c a o l nt m i e n a u n ity n i o s r a ma v l or a t n e d xs t h a e n e g t e a nt n i d al w w e it s h ha r l e l sp m e a c k t e to no  dsiitcnitno shirtFo). in the terminology we use. Note that the first two equalities above are nothing but eikonal equations: if i 2 'keψ i 1  is the equation of the interface  , then ' satisfies ϑ e '+η 2 kν + ψu + ψ ϑ i 1 ' 0 and ϑ e '+η 2 kν − ψu − ψ ϑ i 1 ' 0 ψ on {i 2  0 } , where ! η 2 kνψuψ λ : u∙
− λ 1 ψ λ∈Rψ

is the second characteristic field of the system (2.1). It is linearly degenerate since the corresponding eigenvector, in the quasilinear form of (2.1), is given by c 2 kνψuψ λ :  0 B @ λ 0 1 C A 1 χ

Recall that the space dimension equals 2. , or ion ' , is part of the unknowns of thep T ro h b e l i e n m t . erf W ac e e th  iuqeefthctunlevalynt htninommsim.leobprcoisitAsrfeetiahadyrobnuealwusd kind of situation, we first straighten the unknown front in order to work in a fixed domain. More precisely, the unknowns kνψu , that are smooth on either side of {i 2 'keψ i 1 } , are replaced by the functions kν + ψu + keψ i 1 ψ i 2  : kνψukeψ ψ 1 ψ i 2 ψ ] ] where Ø is a sm k o ν ] o − t ψ h u f ] − u  n k c e t ψ io i n 1 ψ sa i t 2 is  iyf :  n k g νψukeψi i  1 1 ψØ Ø kψei ψike 1 ψ−i 2 ψ

m W a i e t i x n h p t { h o i e n 2 se en > qer 0 ts. } W . u ϑ e i i F r 2 a o e Ø m ls r k e o e n c ψ t d o i n s e 1 v f fi ψ e o n i in r e 2 e t  Ø , eh n ≥ al e, f c ζ nu l > 0 c c ito dwe unf ψ toi p n n o s r Ø s t k h ν e e ψ ] ± ] ψ i 1 u i ψ ] n ± 0 d  e a x r  e ' a k s n e m ψ o d i o l th no 1 y χ epth o k n e thefix e e + -o da d n d − Ø ± keψ i 1 ψ i 2  : Økeψ i 1 ψ±i 2 ψ

Stability of Compressible Vortex Sheets5 which are both smooth on the half-space {i 2 > 0 } . Let us denote by g and f the two components of the velocity, that is, u kg ψ f  the existence of compressible vortex sheets amounts to proving the. Then existence of smooth solutions to the following system: (2.4a) ϑ e ν + +g + ϑ i 1 ν + +kf + −ϑ e Ø + −g + ϑ i 1 Ø +  ϑ ϑ i i 2 ν Ø + + + +ν + ϑ i 1 g + +ν + ϑ ϑ i 2 f Ø + + −ν + ϑ ϑ i 2 Ø Ø + ϑ i 2 g +  0 ψ

+ −ϑ i 1 +  ϑ i g + ϑ + (2.4b) ϑ e g + +g + ϑ i 1 g + +kf + + − a ϑ e 0 k ν Ø ν + +  g ϑ i + 1 ν + Ø − a 0 k ν i ν 2 + + Ø ϑ  ϑ i 2 Ø Ø + ϑ i 2 ν +  0 ψ +

(2.4c) ϑ e f + +g + ϑ i 1 f + +kf + −ϑ e Ø + −g + ϑ i 1 Ø +  ϑ i f + + ϑ i 2 + Ø ν + ϑ i 2 Ø +  0 ψ a 0 kν +  ϑ i ν +

− g ϑ i 1 ν − +kf − −ϑ e − − 2 (2.4d) ϑ e ν − + + ν − ϑ i 1 Ø g − − + g ν − − ϑ i ϑ ϑ 1 i i Ø 2 f Ø −  − − ϑ ϑ i i − Ø ν ν − − ϑ ϑ i i 2 Ø Ø − − ϑ i 2 g −  0 ψ

− − (2.4e) ϑ e g − +g − ϑ i 1 g − +kf − + − a ϑ e 0 Ø kν − − −g  ϑ i − 1 ϑ ν i − 1 Ø − ak ν ϑ  ϑ i i 2 ν − Ø g − −  ϑ ϑ i Ø − − ϑ i 2 ν −  0 ψ 0 ν − i 2 Ø

(2.4f) ϑ e f − +g f − −ϑ e − − ϑ i f − − − ϑ i 1 f − +kØg − ϑ i 1 Ø − ϑ  i 2 + Ø a 0 kν − ϑ  ϑ i i 2 ν Ø − −  0 ψ ν −

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J EAN -F RANC ¸ OIS C OULOMBEL & P AOLO S ECCHI

in the fixed domain {i 2 > 0 } , together with the boundary conditions ' Ø |+ i 2  0 Ø |− i 2  0 ψ − − ψ ϑ e ' −g |+ i 2  0 ϑ i 1 '+f |+ i 2  0  −g | i 2  0 ϑ i 1 '+f | i 2  0 + − ν | i 2  0 ν | i 2  0 χ For convenience, we rewrite the boundary conditions in the following way:

(2.5a) Ø |+ i 2  0 Ø |− i 2  0 'ψ (2.5b) kg + −g −  | i 2  0 ϑ i 1 '−kf + −f −  | i 2  0  0 ψ (2.5c) ϑ e '+g |+ i 2  0 ϑ i 1 '−f |+ i 2  0  0 ψ − (2.5d) kν + ν −  | i 2  0  0 χ ( f T 2 o . h r 6 e a ) f s u u n it c a ti b o le ns co Ø n + ϑ s i a t 2 n a Ø d tn + k ζ Ø > e − ψ i 0 hs 1 ψ . o i ul 2 d  a ≥ ls ζ o ψ sati ϑ sf i y 2 Ø − keψ i 1 ψ i 2 ≤ −ζψ In [20,21], Majda makes the particular choice ± 1 ψ i 2  :  ± This choice is appropriat Ø e i k n eψ th i e study of sh i o 2 c + k 'k w e a ψ v i es b 1 e χ cause these are nonchar- acteristic discontinuities. In the study of contact discontinuities, it seems rather natural to ch ϑ e oo Ø s + e t + he η 2 c k h ν a + n ψ g u e + o ψ f ϑ v i a 1 ri Ø a + b  les  Ø ϑ e ± such that the eikonal equations + − ed in the whole closed 1 h Ø al − f  -sp  ac ϑ e e Ø Ø { − + i + + 2 g g ≥ − + 0 ϑ ϑ } i i . 1 1 Ø − Th − − is f f c + − ho   cie 0 0 ψ ψ , that is in-are satisfi ϑ e Ø − +η 2 kν ψu − ψ ϑ i Ø spired from [13 it simplifies much the expression], has several advantages. First, of the nonlinear equations (2.4 it also implies that the so-called). Butboundary matrixhas constant rank in the whole space domain {i 2 ≥ 0 } , and not only on the boundary {i 2  0 }  symmetrizers. This will enable us to develop a Kreiss technique, in the spirit of [23We shall go back to this feature later on.]. The problem is thus the construction of (local in time) smooth solutions to (2.4)–(2.5)–(2.6 Of course, such initial), once initial data have been prescribed. data will have to fulfill a certain number of compatibility conditions. The first step in proving such an existence result is the study of the linearized problem around a particular constant solution, and this is our first main result, see Theorems3.1 and3.2. The second step is the study of the linearized problem around a (variable

Stability of Compressible Vortex Sheets

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coeffi Thecients) perturbation of the constant solution. extension to the variable coeffi Ourcients linearized problem is addressed in the second part of the paper. second main result states that the constant coefficients energy estimate still holds when one considers a variable coefficients linearized problem, see Theorem5.1. To avoid overloading the paper, we introduce some compact notations for the nonlinear equations (2.4). For all I : kνψ g ψ f H , we define 1 1 0 f 0 ν ϕ 1 kI  :  @ B 0 a 0 εννk νg g 0 C ψ ϕ 2 kI  :  @ B 0 0 f 0 C 0 0 g A a 0 kνεν 0 f A χ Then the nonlinear equations (2.4) read (2.7a) ϑ e I + +ϕ 1 kI + ϑ i 1 I + + ϑ i 2 1 Ø + kϕ 2 kI + −ϑ e Ø + −ϑ i 1 Ø + ϕ 1 kI +  ϑ i 2 I +  0 ψ

(2.7b) ϑ e I − +ϕ 1 kI − ϑ i 1 I − − − With an obvious defini + tio ϑ n i 2 f 1 Ø or − k th ϕ e 2 k d I iff − e  re − n ϑ ti e a Ø l oper ϑ a i t 1 o Ø r − A ϕ , 1 t k h I e − s  y  st ϑ e i m 2 I ( − 2.7  )a 0 χ lso reads

( W 2. h 8 e ) nnoconfusio A n k i I s + p ψ os r si Ø b + le  , I w + e a  0 ols ψ wri A t k e I th − i ψ s r sy Ø st − e  m I − un  re 0 d χ the form

AkI ψrØI 0 ψ

r b w e e h t m e w e r e e m e b n I reeithatthreioerntuq htro e c r e t v h e h a t t i s g f d s l d a n nstfeetat rt s o k a i I it s + o m ψ sn a I d ( − e 2  . b 7 an yth d e era) Ø e bo n u t n ir rfo k e d Ø ly a + r ψ d y Ø c ec − o o  sho nd . up O l ne e s . n dh ( T tioi2 e u . l 5 awsy c d u l o a lpnig .) There exist many simple solutions of (2.8)–(2.5)–(2.6), that correspond, in the original variables, to rectilinear vortex sheets: ( kνψu ψν kν k ψ u u c l   fi if i i 2 >ξ eωξ e + + ni in 1 1 χ ψ 2 Here above, u c , u l are fixed vectors in R 2, and ν > 0, ξ and n are fixed real numbers. These quantities are linked by the Rankine-Hugoniot conditions:

ξ −g c n+f c  −g l n+f l χ