RECTILINEAR VORTEX SHEETS OF INVISCID LIQUID-GAS TWO-PHASE FLOW: LINEAR STABILITY

Transcription

1 RECTILINEAR VORTEX SHEETS OF INVISCID LIQUID-GAS TWO-PHASE FLOW: LINEAR STABILITY LIZHI RUAN, DEHUA WANG, SHANGKUN WENG, AND CHANGJIANG ZHU Abstract. The vortex sheet solutions are considered for the inviscid liquid-gas twophase flow. In particular, the linear stability of rectilinear vortex sheets in two spatial dimensions is established for both constant and variable coefficients. The linearized problem of vortex sheet solutions with constant coefficients is studied by means of Fourier analysis, normal mode analysis and Kreiss symmetrizer, while the linear stability with variable coefficients is obtained by Bony-Meyer s paradifferential calculus theory. The linear stability is crucial to the existence of vortex sheet solutions of the nonlinear problem. A novel symmetrization and some weighted Sobolev norms are introduced to study the hyperbolic linearized problem with characteristic boundary. Contents. Introduction. Vortex Sheet Problem and Reformulation 4.. Vortex sheet problem 5.. Reformulation 7 3. Linear Stability: Constant Coefficients The linearized equations Main result 3 4. Proof of Theorem Elimination of the front The normal mode analysis Construction of the symmetrizer Derivation of the energy estimate 5. Linear Stability: Variable Coefficients The linearized equations The effective linearized equations The linearized boundary conditions Main result 7 6. Proof of Theorem Some preliminary transformations Paralinearization Estimate for the paralinearized problem and microlocalization The proof of Theorem 5. 4 Appendix A. Derivation of Equations Mathematics Subject Classification. 34B5, 35L5, 35L65, 76T. Key words and phrases. Inviscid liquid-gas two-phase flow, vortex sheet, linear stability.

2 LIZHI RUAN, DEHUA WANG, SHANGKUN WENG, AND CHANGJIANG ZHU Acknowledgements 43 References 43. Introduction Two-phase or multi-phase flows are concerned with flows with two or more components and have a wide range of applications in nature, engineering, and biomedicine. Examples include sediment transport, geysers, volcanic eruptions, clouds, rain in natural and climate system; mixture of oil and natural gas in extraction tubes of oil exploitation, oil transportation, steam generators, cooling systems, mixture of hot water and vapor of water in cooling tubes of nuclear power stations in energy production; bubble columns, aeration systems, tumor biology, anticancer therapies, developmental biology, plant physiology in chemical engineering, medical and genetic engineering, bioengineering, and so on. Multi-phase flow is much more complicated than single-phase flow due to the existence of a moving and deformable interface and its interactions with multi-phases [6, 8, 3, 3]. In this paper, we consider the following system of partial differential equations for the compressible inviscid liquid-gas two-phase flow of drift-flux type: t m + mu =, t n + nu =,. t nu + nu u + pm, n =, where m = α g ρ g and n = α l ρ l denote the gas mass and the liquid mass, respectively, α g, α l [, ] denote the gas and liquid volume fractions satisfying α g + α l =, ρ g and ρ l denote the gas and liquid densities; u denotes the mixed velocity of the liquid and the gas, and p is the common pressure for both phases. We assume that the pressure p is a smooth function of m, n defined on, +, +, and in particular we take the pressure of the following form see e.g. [8]: pm, n = c m + c n P c m + c n, where c, c are positive constants and P = P ρ is a smooth function such as P ρ = ρ γ, γ >. Without loss of generality, we take c = c = and the hence pressure is and pm, n = γ m + n γ,. p m m, n = p n m, n = γγ m + n γ..3 We shall see in the next section that. is a non-strictly hyperbolic system of conservation laws in the region m, n, +, +. The viscous two-phase flows have been investigated extensively, in particular, the existence, uniqueness, regularity, asymptotic behavior, decay rate estimates, and blow-up phenomena of solutions to various one-dimensional and multi-dimensional viscous twophase flows have been studied recently in [3, 4, 8, 9, 3, 4, 6, 35, 5, 5 55] and related references therein. The theory of the inviscid two-phase flow. is comparatively mathematically underdeveloped although there have been many numerical studies;

3 RECTILINEAR VORTEX SHEETS OF TWO-PHASE FLOW 3 see [3,4,6,9,5 8,] and their references. In this paper, we are concerned with the rectilinear vortex sheet problem for the two-phase flow. in the two-dimensional space R. A velocity discontinuity in an inviscid flow is called a vortex sheet, which yields a concentration of vorticity along the discontinuity front see []. In the three-dimensional space, a vortex sheet has vorticity concentrated along a surface in the space. In two-dimensional space, the vorticity is concentrated along a curve in the plane. Vortex sheets occur commonly in nature, sciences and engineering, and have attracted enormous studies. Some early studies on the linear stability of planar and rectilinear compressible vortex sheets can be found in [, 43]. In three spatial dimensions, it is known that the planar vortex sheets is unstable see e.g. [45]. In the two-dimensional case, subsonic vortex sheets are also unstable, but the supersonic vortex sheets are linearly stable; see e.g. [43,45]. For the incompressible theory of vortex sheets, we refer the readers to [, 7, 9, 3, 36, 37, 4, 47, 49, 5] and the references theirin. Our work of this paper is inspired by [8,,, 44, 48] on the stability of planar and rectilinear vortex sheets for the compressible isentropic [, ] and non-isentropic Euler equations [44], and the ideal compressible magnetohydrodynamics MHD [8, 48]. The linear stability of compressible vortex sheets for the isentropic Euler equations in two spatial dimensions was studied under a supersonic condition, and an energy estimate for the linearized boundary value problem was proved in []. The nonlinear stability was analyzed in [] based on the linear analysis in [] and the Nash-Moser iteration. The result on linear stability for the isentropic case in [] was also extended to the nonisentropic case see [44]. The linear and nonlinear stability of current-vortex sheets for the ideal compressible MHD was studied in [8, 48]. As mentioned in [8,,, 44, 48], the existence of compressible vortex sheets is a nonlinear hyperbolic problem with free boundaries. Since the vortex sheet is a contact discontinuity, the free boundary is characteristic, thus we have a hyperbolic initial-boundary value problem with a characteristic boundary, violating the uniform Kreiss-Lopatinskii condition, and causing loss of derivatives with respect to the source terms for energy estimates as well as loss of control of the tangential velocity the characteristic part of the solution on the boundary. Thanks to the ellipticity of the boundary conditions for the unknown front, we will be able to gain one derivative as for shock waves in Majda [39]. The purpose of this paper is to establish the linear stability with both constant and variable coefficients of rectilinear vortex sheets for the two-phase flow. in two spatial dimensions. To this end, we organize the analysis as follows. In Section, we first set up the vortex sheet problem as a free boundary problem by analyzing the Rankine-Hugoniot conditions corresponding to the vortex sheets of the equations., and then reformulate this problem into a fixed boundary problem by employing the standard partial hodograph transformation [8,, 48] and tedious calculations. We note that a natural approach of introducing the Lagrangian coordinates to fix the boundary does not seem to work for the vortex sheet problem. In Section 3, in order to obtain the a priori energy estimates mentioned above, we first in Subsection 3. introduce a good symmetric form of the linearized version of the liquid-gas two-phase flow system., which plays a crucial role in our analysis. To our best knowledge this good symmetric form is new and does not follow directly from any known symmetrization. As emphasized in [8,, 48], a good symmetric form is very important although it is easy to perform a trivial symmetrization of the system..

4 4 LIZHI RUAN, DEHUA WANG, SHANGKUN WENG, AND CHANGJIANG ZHU However, a good symmetric form is required to separate the characteristic part and non-characteristic part of solutions so that one can get rid of the singularity from the boundary matrix and reduce the symbolic characteristic case to the non-characteristic case. Fortunately, we find a good transformation see 3.5, 3.6 which leads to a good symmetric form of the linearized equations. We remark that, as pointed out in [5] that most hyperbolic operators are not symmetrizable in d d spatial dimensions although the physical systems of compressible Euler equations, magnetohydrodynamics and liquidgas two-phase flows can be reduced to symmetric hyperbolic systems of conservation laws in time-space dimension + d. Then, in Subsection 3., some weighted Sobolev norms are introduced since there is a loss of control on derivatives in the normal direction for the hyperbolic problem with characteristic boundary. The main result on the linear stability for constant coefficients is stated. In Section 4, we shall prove the main theorem on the linearized problem with constant coefficients by the normal mode analysis of the linearized problem and constructing a degenerate Kreiss symmetrizers in order to derive our energy estimate. As in [, 39, 4], the linearized Rankine-Hugoniot conditions form an elliptic system for the unknown front, which is very important when eliminating the unknown front and considering a standard boundary value problem with a symbolic boundary condition. Based on our new observation and the property that no jump occurs in the sum of mass of both liquid and gas even though each has jump individually, we introduce an appropriate C smooth mapping Q to obtain the elliptic estimate on the corresponding symbol and formally cast our problem in the framework of [44], which allows us directly employ their construction of symmetrizers to simplify our calculation and mathematical analysis. However, the construction of the symmetrizers is microlocal. Near the neighborhood of the poly point, the construction of symmetrizers is different from those in [] and [44] for the original symbolic algebraic-differential equations. In Section 5, we discuss and formulate the linearized problem of vortex sheets with variable coefficients, and state the main result on the linear stability. In Section 6, we shall prove the main theorem on the linearized problem with variable coefficients by paralinearized techniques and Bony-Meyer s paradifferential calculus theory [5,4]. The key point is to freeze the coefficients to turn the variable coefficient problem into the constant coefficient problem. The critical set of Lopatinski determinants will be constructed in details. This is a key point of microlocal analysis in the neighborhood of bicharacteristic curve along which singularities propagate. In the analysis, we need some precise calculations which will be collected in the Appendix. We conclude the introduction by remarking that the linear stability studied in this paper is a crucial step towards the local-in-time existence of vortex sheet solutions of the nonlinear problem. This nonlinear stability will be investigated in a forthcoming paper based on the linear stability obtained in this paper and the Nash-Moser iteration method.. Vortex Sheet Problem and Reformulation In this section, we first set up the vortex sheet problem as a free boundary problem and then reformulate it into an initial-boundary value problem with fixed boundaries.

5 RECTILINEAR VORTEX SHEETS OF TWO-PHASE FLOW 5.. Vortex sheet problem. We will consider the liquid-gas two-phase flow. in the whole space R and present the analysis which leads to a vortex sheet problem. Let x = x, x be the space variable in R, v and u be the first and second components of the velocity field and thus u = v, u R. Then, for U = m, n, u, +, + R, we define the following matrices: v m u m v n u n A U = p mn p nn, A U =.. v u p v mn p nn u Denote the spatial partial derivatives by = x, = x. In the region where m, n, u is smooth e.g. following quasilinear form: The eigenvalues of the matrix are given by differentiable,. is equivalent to the t U + A U U + A U U =.. AU, ξ = Am, n, u, ξ = ξ A U + ξ A U, ξ = ξ, ξ R, λ U, ξ = ξ u ξ mpm+np n n with multiplicity m =, λ U, ξ = ξ u with multiplicity m =, λ 3 U, ξ = ξ u + ξ mpm+np n n with multiplicity m 3 =,.3 in the region U, +, + R. The eigenvector corresponding to the second eigenvalue field λ U, ξ is given by r U, ξ = ξ, ξ, ξ p n, ξ p m. If p n = p m for the pressure p, the second characteristic field of the system. or. is linearly degenerate, which leads us to consider the vortex sheet contact discontinuity solutions for the two-phase flow. In fact, a vortex sheet contact discontinuity solution is a weak solution with possible strong discontinuities. Definition. Weak solution. Let m, n, u be a smooth function of t, x, x on either side of a smooth surface Γ := {x = ϕt, x, t >, x R}. Then m, n, u is a weak solution of. if and only if m, n, u is a classical solution of. on both sides of Γ and the Rankine-Hugoniot conditions hold at each point of Γ: t ϕ[m] [mu ν] =, t ϕ[n] [nu ν] =,.4 t ϕ[nu] [nu νu] [p]ν =,

6 6 LIZHI RUAN, DEHUA WANG, SHANGKUN WENG, AND CHANGJIANG ZHU where ν := ϕ, is a space normal vector to Γ and t ϕ, ϕ, = t ϕ, ν is a time-space conormal vector to Γ. As usual, [q] = q + q denotes the jump of a quantity q across the interface Γ. Moreover, vortex sheets have continuous normal velocity and possible jump of tangential velocity, yielding a concentration of vorticity along the discontinuity front. The velocity of the front t ϕ satisfies t ϕ = u + ν = u ν, which means that the first two equations in.4 are automatically satisfied and the third one gives p + = p. We are now in a position to give the rigorous definition of a vortex sheet solution contact discontinuity in sense of Lax [33] of the liquid-gas two-phase flow. Definition. Vortex sheet solution. A piecewise smooth vector-function m, n, u is called a rectilinear vortex sheet solution of. if m, n, u is a classical solution of. on either side of the smooth surface Γ and the Rankine-Hugoniot conditions.4 are satisfied at each point of Γ in the following way: We note that.5 implies t ϕ = u + ν = u ν, p + = p..5 t ϕ = u + ν = u ν, m + + n + = m + n..6 Then the problem of existence and stability of vortex sheets solution can be formulated as the following free boundary problem: determine U ± t, x, x = m ±, n ±, v ±, u ± t, x, x, +, + R and a free boundary Γ := {x = ϕt, x, t >, x R} such that t U + + A U + U + + A U + U + =, x > ϕt, x, t U + A U U + A U U =, x < ϕt, x, { U + x, x, x > ϕ x, U, x, x = satisfying the jump conditions on Γ: U x, x, x < ϕ x,.7 t ϕ = v + ϕ + u + = v ϕ + u, m + + n + = m + n,.8 where ϕ x = ϕ, x. Note that the first two equalities in.8 are just the eikonal equations: ϕ t + λ m +, n +, u +, ϕ = and ϕ t + λ m, n, u, ϕ =,.9 on {x = }, where the eigenvalue λ m, n, u, ξ is defined in.3. To prove the existence of tangential discontinuities vortex sheets for the free boundary problem.7 and.8, one needs to find a solution U, ϕt, x, x of the problem.7 and.8 at least locally in time. More precisely, we need to prove the local in time well-posedness of the problem.7 and.8. Our goal in this paper is to establish the well-posedness of the linearized problem resulting from the linearization of.7 and.8 around a background vortex sheet piecewise constant solution. As discussed in [], for the isentropic Euler equations., these solutions are exactly the contact discontinuities in the sense of Lax [33].

7 RECTILINEAR VORTEX SHEETS OF TWO-PHASE FLOW 7.. Reformulation. We now reformulate the free-boundary problem into a fixed-boundary problem. To straighten the unknown front we employ the standard partial hodograph transformation see, e.g. [8,, 48]: with some smooth functions Φ ± satisfying t = t, x = x, x = Φ ± t, x, x ± x Φ ± t, x, x κ >,. Φ + t, x, = Φ t, x, = ϕ t, x,. for some constant κ >. Under., the domains are transformed into { x > } and the free boundary Γ into the fixed boundary { x = }. More precisely, the unknowns m ±, n ±, u ± t, x, x, that are smooth on either side of {x = ϕt, x }, are replaced by the functions m ±, n±, u± t, x, x = m ±, n ±, u ± t, x, x = m ±, n ±, u ± t, x, Φ ± t, x, x. which are smooth on the fixed domain { x > }. From now on we drop the tildes for simplicity of notation. Let us denote by v and u the two components of the velocity, that is, u ± = v ±, u±. Then the smooth solutions m ±, n±, v±, u±, Φ± satisfy the following initial-boundary value problem with the fixed boundary x = : t m ± + v± m ± + u ± tφ ± v ± Φ ± m ± Φ ± +m ± v ± + m ± u ± Φ ± m ± Φ ± Φ ± v ± =,.3 t v ± + v ± v ± + u ± t n ± + v± n ± + u ± p m ± + n ± tφ ± v ± Φ ± n ± Φ ± +n ± v ± + n ± tφ ± v ± Φ ± v ± Φ ± m ± p m ± n ± Φ + Φ ± m ± u ± Φ ± n ± p + n ± n ± Φ ± Φ ± v ± =, n ± p n ± n ± Φ ± Φ ± n ± =,.4.5 t u ± + v± u ± + u ± tφ ± v ± Φ ± u ± p m ± m ± + Φ ± n ± + Φ ± p n ± n ± n ± Φ ± =,.6

8 8 LIZHI RUAN, DEHUA WANG, SHANGKUN WENG, AND CHANGJIANG ZHU in the fixed domain {x > }, together with the boundary conditions from. and.: Φ + t, x, x = x = Φ t, x, x = ϕt, x, x = v + v t, x, x ϕt, x u + x = u t, x, x =, x =.7 t ϕt, x + v + t, x, x ϕt, x u + t, x, x =, x = x = m + + n+ t, x, x m + x = n t, x, x =. x = For contact discontinuities, one can choose the change of variables Φ ± satisfying the eikonal equations: t Φ ± + λ m ±, n±, u±, Φ ± = t Φ ± + v ± Φ ± u ± =,.8 in the whole closed half-space {x }. We know from.7 that.8 is satisfied on the boundary {x = }. Again for the sake of simplicity of notations, we shall drop the symbol in.3-.6 and denote U := m, n, v, u, then the nonlinear equations.3-.6 read t U + + A U + U + t U + A U U + Φ + A U + t Φ + I 4 4 Φ + A U + U + =,.9 + Φ A U t Φ I 4 4 Φ A U U =. Here A U, A U are defined by. and I 4 4 is the 4 4 identity matrix. Define the differential operator L as the left side of.9, then the system.9 becomes LU +, Φ + U + =, LU, Φ U =,. where Φ ± = t Φ ±, Φ ±, Φ ±. With a slight abuse of notations we also write this system as LU, ΦU =,. where U denotes the vector U +, U and Φ for Φ +, Φ. The two equations in.9 are decoupled in the interior of the domain, and their coupling is made through the boundary conditions.7. There exist many simple solutions of corresponding to rectilinear vortex sheets in the original variables. In the new variables, they are regular solutions of.-.7-.: U r = with the relation m r n r v r, U l = m l n l v l, Φ r,lt, x, x ±x, ϕ,. m r + n r m l + n l =, v r + v l =..3

9 RECTILINEAR VORTEX SHEETS OF TWO-PHASE FLOW 9 We only consider the case v r, and without loss of generality we assume v r >. We also assume m r, m l, n r, n l >. In the next section, we study the linearized equations around the special background solution defined by.. 3. Linear Stability: Constant Coefficients In this section, we formulate the linearized problem with constant coefficients, and state the main result on the linear stability with constant coefficients. 3.. The linearized equations. Denote by U ± := ṁ ±, ṅ ±, v ±, u ±, Ψ ±, the small perturbations of the exact solution given by., that is, U ± = U r,l + U ±, Φ ± = Φ r,l + Ψ ±. Up to second order, the perturbations U ± = ṁ ±, ṅ ±, v ±, u ± satisfy the following linearized equations: t U + + A U r U + + A U r U + =, 3. t U + A U l U A U l U =, in the domain {x > }, as well as the linearized Rankine-Hugoniot relations: Ψ + = Ψ = ψ, v r v l ψ u + u =, t ψ + v r ψ u + =, ṁ + + ṅ + ṁ + ṅ =, on the boundary {x = }. Rewrite the equations as L U =, if x >, B U, ψ =, if x =, where U = U +, U, and U + L U = t U A U r U + + A U l U A U r + A U l U + U, B U, ψ = v r v l ψ u + u t ψ + v r ψ u + ṁ + + ṅ + ṁ + ṅ.

10 LIZHI RUAN, DEHUA WANG, SHANGKUN WENG, AND CHANGJIANG ZHU We remark that the interior equations do not involve the perturbation ψ, thus the operator L only acts on U. Generally energy estimates for the linearized equations depend on the source terms both in the interior domain and on the boundary. We now consider the linear equations: L U = f = f +, f, if x >, B U, 3.4 ψ = g = g, g, g 3, if x =, and the goal is to establish the estimates of U and ψ in terms of f and g in appropriate functional spaces. To simplify the calculations, we introduce some new unknown functions by performing the linear and invertible changes of variables: ṁ + ṅ + v + u + = n r m r m r n r n r n r c r c r W W W 3 W and ṁ ṅ v u where c r,l are defined by = n l m l m l n l n l n l c l c l c r,l = + m r,l n r,l W 5 W 6 W 7 W 8, 3.6 p n 3.7 and p n = p n m r, n r = p n m l, n l given in.3. We also define the following vectors: W := W, W, W 3, W 4, W 5, W 6, W 7, W 8, W c := W, W, W 5, W 6, W nc := W 3, W 4, W 7, W 8. The notations W c and W nc are introduced in order to separate the characteristic part of the vector W and the noncharacteristic part of W. It is obvious that estimating W is equivalent to estimating U. The vector W satisfies LW := t W + Ā W + Ā W = f, if x >, t ψ 3.8 BW nc, ψ := MW nc x = + b = ḡ, if x =, ψ

11 with new f and ḡ and RECTILINEAR VORTEX SHEETS OF TWO-PHASE FLOW Ā := v r v r c r c r 4 v r 4 v r O O v l v l c l c l 4 v l 4 v l, 3.9 Ā := c r c r O O c l 3. c l as well as b = M = v r v l v r = v r v r, c r c r c l c l c r c r m r + n r m r + n r m l + n l m l + n l. 3. Hereafter O stands for the 4 4 zero matrix.

12 LIZHI RUAN, DEHUA WANG, SHANGKUN WENG, AND CHANGJIANG ZHU Let us further define the following 8 8 symmetric matrices: 4 O c r c A := r 4 O c l, 3. A := v r 4 v r c r c r c r c rv r c r c rv r A := O c 3 r c 3 r O c l O v l 4 v l c l c l c l c l v l c l c l v l O c 3 l, c 3 l Then, the linear problem 3.8 becomes equivalently the following symmetric system with A definite positive: { LW := A t W + A W + A W = f, if x >, BW nc, ψ BW 3.5 nc, ψ = g, if x =, with new f and g.

13 RECTILINEAR VORTEX SHEETS OF TWO-PHASE FLOW 3 We remark that the kernel of A consists exactly of those W satisfying W nc = and W c is arbitrary. Thus the boundary {x = } is characteristic with multiplicity. As noted in earlier works see e.g. [, 34, 38], we expect to lose control of the trace of W c. However, we expect to have control of the trace of W nc on {x = }, that is, W nc x = in 4.4 later. 3.. Main result. Before stating our energy estimate for the system 3.5, we need to introduce some weighted Sobolev norms since 3.5 is a hyperbolic problem with characteristic boundary and there is a loss of control on derivatives in the normal direction x. First define the half-space Ω := { t, x, x R 3 : x > } = R R +. For all real number s and all λ, define the weighted Sobolev space which is equipped with the norm Letting ũ := exp λtu, one has where H s λ R := { u D R : exp λtu H s R }, u H s λ R := exp λtu H s R. u H s λ R ũ s,λ, v s,λ := λ π + ξ s ˆvξ dξ, R and ˆv is the Fourier transform of a function v defined on R. For all integers k and real λ, one can define the space Hλ k Ω as follows: } Hλ {u k Ω := D Ω : exp λtu H k Ω. For all s > r, it holds that Hλ s R Hλ r R, v r,λ λ s r v s,λ. The space L R + ; H s λ R is equipped with the norm Our first main result is stated as follows. v L Hλ s := v, x Hλ sr dx. Theorem 3.. Let U r,l, Φ r,l be a solution to.,.7 and.8 defined by. and.3. i If v r v l > c 3r + c 3l 3 and v r v l c r + c l, 3.6

14 4 LIZHI RUAN, DEHUA WANG, SHANGKUN WENG, AND CHANGJIANG ZHU then there exists a positive constant C such that for all λ and for all solutions W, ψ H λ Ω H λ R to 3.5, the following estimate holds: λ W L λ Ω + W nc x = L λ R + ψ Hλ R C LW λ 3 L Hλ + BW, ψ. λ Hλ R 3.7 ii If v r v l = c r + c l, 3.8 then there exists a positive constant C such that for all λ and for all solutions W, ψ Hλ 3Ω H3 λ R to 3.5, the following estimate holds: Here c r,l are defined by 3.7. λ W L λ Ω + W nc x = L λ R + ψ Hλ R C LW λ 5 L Hλ + BW, ψ. λ 4 Hλ R 3.9 We shall prove Theorem 3. by a normal mode analysis of 3.5. Introduce the new unknown functions: W := exp λtw, ψ := exp λtψ. Then 3.5 is equivalent to L λ W := λa W + L W = exp λtf, if x >, B λ W nc, ψ = M W nc x = + b λ ψ + t ψ 3. = exp λtg, if x =. ψ We can rewrite Theorem 3. equivalently as the following: Theorem 3.. i Assume that 3.6 holds. Then there exists a positive constant C such that for all λ and for all W, ψ H Ω H R, the following estimate holds: λ W + W nc x = + ψ,λ C L λ W λ 3,λ + B λ W, ψ λ,λ. 3. ii Assume that 3.8 holds. Then there exists a positive constant C such that for all λ and for all W, ψ H 3 Ω H 3 R, the following estimate holds: λ W + W nc x = + ψ,λ C L λ W λ 5,λ + B λ W, ψ λ 4,λ. In 3. and 3., we have used the following notations for v L R + ; H s R : + v s,λ := v, x s,λ dx. 3. For instance,,λ is the usual norm on L Ω and does not involve λ, so we shall denote it by. The norm,λ is the weighted norm on L R + ; H R.

15 RECTILINEAR VORTEX SHEETS OF TWO-PHASE FLOW 5 4. Proof of Theorem 3. This section is devoted to the proof of Theorem 3.. To simplify the notations, we shall drop the tildes from the unknowns W, ψ. As in [] by introducing an auxiliary problem with maximally dissipative boundary conditions, one can show that it is sufficient to prove estimates 3. and 3. for the system with zero interior source term: λa W + A t W + A W + A W = 4. in the interior domain Ω, as well as the following boundary conditions: λψ + t ψ MW nc x = + b = g, on x =. 4. ψ Recall that all matrices A j j =,, are symmetric, and that A is positive definite see 3.. Taking the scalar product of 4. with W and integrating over Ω yield the following inequality: λ W C W nc x =. 4.3 Consequently, in order to obtain 3. and 3., it is sufficient to derive the following estimates: W nc x = + ψ,λ C λ g,λ, 4.4 W nc x = + ψ,λ C λ 4 g,λ, 4.5 which can be further reduced to estimate the L -norm of the trace of W nc in the Sobolev norm of g by eliminating the front ψ in the boundary conditions 4. as in []. In the next subsections, we shall perform the normal mode analysis in detail and construct a symbolic symmetrizer. 4.. Elimination of the front. First we apply the Fourier transform in t, x on Denote the dual variables by δ, η and define τ := λ + iδ. Then we obtain the following system of ordinary differential equations ODEs: where bτ, η is simply defined by τa + iηa Ŵ + A dŵ dx =, x >, bτ, η ψ + MŴ nc = ĝ, bτ, η := b τ iη = iv r η τ + iv r η Recall that b and M are defined by 3.. Observe that bτ, η is homogeneous of degree with respect to τ, η. Define the hemisphere Σ := {τ, η C R : τ + v rη = and Rτ }, where Rτ is the real part of τ and denote by Ξ the set Ξ := {λ, δ, η [, + R : λ, δ, η,, } =, + Σ.

16 6 LIZHI RUAN, DEHUA WANG, SHANGKUN WENG, AND CHANGJIANG ZHU We always identify λ, δ R with τ = λ + iδ C. We remark that a symbolic symmetrizer rτ, η of 4.5 as a homogeneous function of degree zero with respect to τ, η Ξ will be constructed. In fact it is enough to construct rτ, η in the unit hemisphere Σ. Since Σ is a compact set, by a smooth partition of unity, it can be reduced to construct rτ, η in a neighborhood of each point of Σ. One crucial note is that the symbol bτ, η is elliptic, that is, it does not vanish on the closed hemisphere Σ. Similar to [], we can choose the c mapping Q on Σ as follows: τ, η Σ, Qτ, η := m+ n τ + iv rη iv r η iv r η τ iv r η, 4.8 which is homogeneous of degree zero. For reducing the boundary matrix βτ, η in 4.3 later to the same form as that of [44] and directly employing their parts of construction on symmetrizer, we may choose m + n = m r + n r = m l + n l 4.9 in the first row of Qτ, η and 4.9 can truely be satisfied due to.3. Then one can easily check Qτ, ηbτ, η =. θτ, η Here θ is C, homogeneous of degree and given by satisfying the lower bound: θτ, η := τ + iv r η + 4v rη, τ, η Σ, min θτ, η >. 4. τ,η Σ Note that the last row of Qτ, η is nothing but bτ, η, when τ, η Σ and Q can be extended to Ξ by homogeneity. Let us multiply the boundary conditions in 4.6 by the matrix Qτ, η. By repeating the arguments of [], the elliptic estimate 4. for θτ, η yields the control of the front: ψ,λ C W nc x = + g C W nc x = + g λ,λ 4. C W nc x = + g λ 4,λ. Therefore, in order to obtain 4.4 and 4.5, it is sufficient to derive an estimate on the trace of W nc. Consequently, we focus on the reduced algebraic-differential equation problem: τa + iηa Ŵ + A dŵ dx =, x >, βτ, ηŵ nc = ĥ, 4.

17 RECTILINEAR VORTEX SHEETS OF TWO-PHASE FLOW 7 and shall derive an estimate for Ŵ nc. Here the source term ĥ C can be estimated by ĝ and βτ, η = c r τ iv r η c r τ iv r η c l τ + iv r η c l τ + iv r η 4.3 as in [44] because of our choice of Q in 4.8, but the precise expressions of c r,l defined by 3.7 in this paper are different from those in [44]. Next we shall recall that under the assumption made in Theorems 3. and 3., the above problem satisfies the Kreiss-Lopatinskii condition but violates the uniform Kreiss- Lopatinskii condition. 4.. The normal mode analysis. Due to the singularity of the matrix A, some equations in 4. do not involve derivation with respect to the normal variable x. The second and sixth equations in 4. read: 4 τ + iv rηŵ ic rηŵ3 + ic rηŵ4 =, 4 τ + iv lηŵ6 ic l ηŵ7 + ic l ηŵ8 = 4.4 based on the expression τa + iηa = τa + iηa r τa + iηa l and τa + iηa r,l = τ + iv r,l η 4 τ + iv r,lη ic r,l η ic r,l η ic r,l η c r,l τ + iv r,lη ic r,l η c r,l τ + iv r,lη. Thus we obtain an expression for Ŵ and Ŵ6 that we can substitute in the third, fourth, seventh and eighth equations in 4.. This operation yields a system of ordinary differential equations of the following form: dŵ nc dx = Aτ, ηŵ nc, x >, βτ, ηŵ nc = ĥ, x =. 4.5

18 8 LIZHI RUAN, DEHUA WANG, SHANGKUN WENG, AND CHANGJIANG ZHU The matrix Aτ, η in 4.5 is given by the same form as in [, 44]: µ r m r m r µ r Aτ, η :=, with µ l m l m l µ l 4.6 µ r,l := /c r,lτ + iv r,l η + c r,l /η, τ + iv r,l η m r,l := c r,l/η τ + iv r,l η, where c r,l are defined by 3.7. By computing the eigenvalues and the stable subspace of Aτ, η, the theoretical results in [5, 7, 39] apply. The following lemma of [] gives an expression of the stable subspace. Lemma 4. [], Lemma 4.. Let τ C and η R, with Rτ > and τ, η Σ. The eigenvalues of Aτ, η are the roots ω of the dispersion relations: ω r = µ r m r = c r τ + iv r η + η, ωl = µ l m l = τ + iv c l η + η. l 4.7 In particular, 4.7 resp. 4.7 admits a unique root ω r resp. ω l of negative real part. The other root of 4.7 resp. 4.7 is ω r resp. ω l, and has positive real part. The stable subspace E τ, η of Aτ, η has dimension, and is spanned by the following two vectors: E r τ, η := cr η, c τ + iv r rη + cr η τ + iv r ηω r,,, 4.8 E l τ, η :=,, c l τ + iv r η + c l η τ + iv r ηω r, c l η. Both ω r and ω l admit a continuous extension to any point τ, η such that Rτ = and τ, η Σ. This allows to extend both vectors E r and E l in 4.8 to the whole hemisphere Σ. Those two vectors are linearly independent for any value of τ, η Σ. The symbol Aτ, η is diagonalizable as long as both ω r and ω l do not vanish, that is, when τ ±v r ± c r,l iη. Away from such points, A admits a C basis of eigenvectors. Following Majda and Osher [, 38], we define the Lopatinskii determinant associated with the boundary conditions β in the following way: τ, η := det [βτ, η E r τ, η E l τ, η], 4.9 with β defined by 4.3 and E r, E l defined by 4.8. We emphasize that the Lopatinskii determinant is defined on the whole hemisphere Σ and is continuous with respect to τ, η. The first step in proving an energy estimate for 4.5 is to determine whether vanishes on Σ. The answer is given in the following result.

19 RECTILINEAR VORTEX SHEETS OF TWO-PHASE FLOW 9 Proposition 4. cf. [44], Proposition 3.4. Assume that the condition v r v l > c 3r + c 3 3l holds. a If c r = c l := c, then there exists a positive number V such that for any τ, η Σ, one has τ, η = if and only if τ = or τ = ±iv η. Each of the preceding roots of τ, η = is simple; namely if τ, η is any of the points above, there exists an open neighborhood V of τ, η in Σ and a C function h defined on V such that for all τ, η V one has τ, η = τ τ hτ, η and hτ, η. b If c r c l, then there exist two positive numbers X, X 3 satisfying c r v r < c r X < c r X 3 < c l + v r, such that τ, η = for τ, η Σ, if and only if c r+c l. τ = iqv r η or τ = ic r X q or τ = ic r X 3 q, where q := cr c l For v r cr+c l, each of the preceding roots of τ, η = is simple. When v r = cr+c l one and only one of the two identities below holds true qv r = c r X or qv r = c r X 3. Hence each of the roots iqv r η, η Σ of τ, η = is quadratic. This means that to every point iqv r η, η Σ there correspond an open neighborhood V in Σ and a C function h on V such that τ, η = τ iqv r η hτ, η, τ, η V, and hiqv r η, η. The other root of τ, η = is simple Construction of the symmetrizer. This subsection will be entirely devoted to constructing a symbolic symmetrizer rτ, η, which is a homogeneous function of degree zero with respect to τ, η Ξ. As remarked earlier it is sufficient to construct rτ, η in a neighborhood of each point of Σ. The analysis performed in Section 4. shows that we have to consider five different classes of frequencies τ, η Σ in the construction of rτ, η: C The interior points τ, η of Σ such that Rτ > ; C The boundary points τ, η of Σ where Aτ, η is diagonalizable and the Lopatinskii condition is satisfied at τ, η namely τ, η ; C3 The boundary points τ, η where Aτ, η is diagonalizable but the Lopatinskii condition breaks down at τ, η i.e., τ, η = ; C4 Those points τ, η where Aτ, η is not diagonalizable, that is, τ = ±v r ± c r,l iη η, and in this case, Proposition 4. asserts that the Lopatinskii condition is satisfied at τ, η ; C5 Those points τ, η that are the poles of A, that is, τ = ±iv r η, and at those points, the Lopatinskii condition is satisfied.

20 LIZHI RUAN, DEHUA WANG, SHANGKUN WENG, AND CHANGJIANG ZHU The construction of the symmetrizer is microlocal and is performed near each point τ, η Σ. For a point belonging to the above classes C, C, C4, the construction is very similar to the corresponding one of Sections. 4.4, 4.5, 4.7 in []. It can be proved that if τ, η is a point in one of the above classes, there exist a neighborhood V of τ, η and two C mappings T : V GL 4 C and r : V M 4 4C such that, for all τ, η V, rτ, η is Hermitian, homogeneous of degree zero with respect to τ, η, and the following estimates R rτ, ηt τ, ηaτ, ηt τ, η κi 4 4 κλi 4 4, τ, η V, rτ, η + C βτ, ηt τ, η βτ, ηt τ, η I 4 4, τ, η V, 4. hold, where I 4 4 denotes the identity matrix of order 4, C, κ are suitable positive constants. On the left-hand side of 4., we use the notation RM := M+M for every complex square matrix M. Here M = M T is the conjugate transpose. For the case of points belonging to the class C3, as in [44] the symmetrizer is defined in a neighborhood of τ, η by rτ, η := diag λ ν, λ ν, K, K, where ν = or ν = corresponding to v r cr+c l or v r = cr+c l respectively, and K is a constant to be chosen large enough. The matrix rτ, η above is Hermitian and satisfies R rτ, ηt τ, ηaτ, ηt τ, η κλdiag λ ν, λ ν, K, K, rτ, η + C βτ, ηt τ, η βτ, ηt τ, η λ ν I 4 4, 4. for all τ, η V and suitable positive constants κ, C >. We now consider the last case C5, which is τ, η Σ with τ = iv r η. As in [], we have to go back to the original system: τa + iηa Ŵ + A dŵ dx =, x >, βτ, ηŵ nc = ĥ. 4. Note that the matrices τa + iηa and A are different from those of []. The idea now is to perform some manipulations on the rows of these two matrices so that 4. is transformed into an almost diagonal system of differential equations. By direct calculations similar to those in [], we can choose both C matrices S and T on the whole neighborhood V as follows: Sr τ, η O Tr τ, η O Sτ, η :=, T τ, η :=, 4.3 O S l τ, η O T l τ, η where S r τ, η =, i c 3 rη τ + iv rη c r ω r ξ r c 3 rη c 3 r

21 and RECTILINEAR VORTEX SHEETS OF TWO-PHASE FLOW S l τ, η = iτ+iv lη+c l ω l ξ +, c 3 l ηω lτ+iv l η 4c 3 l ω l lτ+iv l η c 4 l η ω l τ+iv l η iτ+iv lη c l ω l ξ c 3 l ηω lτ+iv l η 4c 3 l ω l lτ+iv l η c 4 l η ω l τ+iv l η ic r ητ + iv r η c r ω r T r τ, η := c r η, τ + iv r ηµ r ω r ic l ητ + iv l η c l ω l ic l ητ + iv l η + c l ω l T l τ, η := τ + iv l ηµ l ω l τ + iv l ηµ l + ω l c l η c l η ξ r : = τ + iv r ηµ r ω r = τ + iv r ηµ r τ + iv r ηω r ξ ± l = c r τ + iv r η + cr η τ + iv r ηω r, : = τ + iv l ηµ l ± ω l = τ + iv l ηµ l ± τ + iv l ηω l 4.4 = c l τ + iv l η + c l η ± τ + iv l ηω l. It is now easy to derive the following equalities for all τ, η in V: Sτ, ηa T τ, η = diag,, c 4 rη,,,,,, Sτ, ητa + iηa T τ, η Sr τ, ητa + iηa r T r τ, η O = O S l τ, ητa + iηa l T l τ, η, 4.5 where = S r τ, ητa + iηa r T r τ, η τ + iv r η ic rη 4 τ + iv rη ic rη ic rη c 4 rη ω r ω r

22 LIZHI RUAN, DEHUA WANG, SHANGKUN WENG, AND CHANGJIANG ZHU and S l τ, ητa + iηa l T l τ, η τ + iv l η 4 = τ + iv lη. ω l ω l Thus we almost diagonalize simultaneously the original system of algebraic-differential equations 4. due to 4.5. This is sufficient to derive energy estimates in such a neighborhood V of the pole τ, η Derivation of the energy estimate. We now derive the estimates 3. and 3.. Thanks to 4.4, 4.5 and 4., it is sufficient to estimate the trace of W nc on {x = }. As in [, 44], the previous analysis shows that for all τ, η Σ, there exists a neighborhood V of τ, η in Σ with desired properties. Because Σ is a C compact manifold, there exists a finite covering V,, V I of Σ by such neighborhoods, and a smooth partition of unity χ,, χ I associated with this covering, that is, the functions χ,, χ I are nonnegative, C, and satisfy I supp χ i V i and χ i. For τ, η belong to the classes C, C, C4, the energy estimate can be obtained in the same way as in []. For τ, η belong to the class C5, the energy estimate can be obtained similarly to [] by employing almost diagonal matrices 4.5. More precisely, we have λχ i τ, η + i= Ŵ nc δ, η, x dx + χ i τ, η Ŵ nc δ, η, C i χ i τ, η ĥ. For τ, η belonging to the class C3, the energy estimate can be obtained as in [44] by using the estimate 4. as the following: λχ i τ, η + Ŵ nc δ, η, x dx + χ i τ, η Ŵ nc δ, η, C i λ ν χ i τ, η ĥ τ + vrη ν. Adding up above two estimates, using the partition of unity, then integrating the resulting inequality with respect to δ, η R and employing Plancherel s theorem yield the desired estimate: W nc x = + W nc x = C λ ν g ν,λ. 4.6 For ν = or ν = corresponding to v r cr+c l or v r = cr+c l respectively, the combination of 4.4, 4.5, 4. with 4.6 leads to the estimates 3. and 3.. This completes the proof of Theorem 3. equivalent to Theorem 3..

23 RECTILINEAR VORTEX SHEETS OF TWO-PHASE FLOW 3 5. Linear Stability: Variable Coefficients For the case of variable coefficients, we first present the linearized problem and state the main result in this section, and then prove the main result in the next section. 5.. The linearized equations. We first introduce the linearized equations around a state U r,l t, x, x, Φ r,l t, x, x that are given by a perturbation of the constant solution in.. More precisely, let us consider the functions U r,l t, x, x = m r,l n r,l ± v r + U r,l t, x, x, Φ r,l t, x, x = ±x + Φ r,l t, x, x, where m r,l, n r,l, v r are fixed positive constants and m r,l U r,l t, x, x n r,l v r,l t, x, x, u r,l U r,l t, x, x ṁ r,l ṅ r,l v r,l u r,l t, x, x. 5. The index r resp. l denotes the state on the right resp. on the left of the interface after the change of variables. Notice that v r t, x, v l t, x, here. We assume that U r,l, Φ r,l W, Ω, 5. U r, U l W, Ω + Φ r, Φ l W, Ω K, where K > is constant, and the perturbations U r,l have compact support. The corresponding Rankine-Hugoniot conditions and the continuity condition for the functions Φ r,l can be written in the form of.7 by drop the index as: Φ r t, x, x = Φ lt, x, x = ϕt, x, x = x = v r v l t, x, x ϕt, x u r u l t, x, x =, x = x = 5.3 t ϕt, x + v r t, x, x ϕt, x u r t, x, x =, x = x = m r + n r t, x, x m l + n l t, x, x =. x = x = The functions Φ r and Φ l should also satisfy the eikonal equations: t Φ r + v r Φ u r =, t Φ l + v l Φ u l = 5.4 together with Φ r t, x, x κ, Φ l t, x, x κ 5.5 for a suitable constant κ >, in the whole closed half-space {x }. Let us consider some families U ± s = U r,l + sv ±, Φ ± s = Φ r,l + sψ ±,

24 4 LIZHI RUAN, DEHUA WANG, SHANGKUN WENG, AND CHANGJIANG ZHU where s is a small parameter. U r,l, Φ r,l : and obtain L U r,l, Φ r,l V ±, Ψ ± := L U r, Φ r V +, Ψ + We compute the linearized equations around the state { } d ds LU s ±, Φ ± s U s ± = f ±, 5.6 s= = t V + + A U r V + + Φ r A U r t Φ r I 4 4 Φ r A U r V + +[da U r ]V + U r Ψ + Φ r [A U r t Φ r I 4 4 Φ r A U r ] U r Φ r {d[a U r ]V + t Ψ + I 4 4 ψ + A U r Φ r d[a U r ]V + } U r 5.7 = f + in the domain {x > }, and we also obtain a similar equation for L U l, Φ l V, Ψ with V, Ψ, U l, Φ l, f replacing V +, Ψ +, U r, Φ r, f +. Recall that, according to the definition in.9,., the second row in 5.7 may be simply denoted by LU r, Φ r V + := t V + + A U r V + + Φ r [A U r t Φ r I 4 4 Φ r A U r ] V +. The linearized equation 5.7 and the corresponding one for V, Ψ may be simplified by introducing the good unknown as in []: V + = V + Ψ + Φ r U r, A direct calculation shows that V + and V satisfy V = V Ψ Φ l U l. 5.8 where LU r, Φ r V + + CU r, U r, Φ r V + + Ψ + x Φ r {LU r, Φ r U r } = f +, LU l, Φ l V + CU l, U l, Φ l V + Ψ x Φ l {LU l, Φ l U l } = f, 5.9 CU r, U r, Φ r V + := da U r V + U r with a similar expression for CU l, U l, Φ l V. + { da U r Φ V + Φ r [da U r V } + ] U r, r The effective linearized equations. From [,] we neglect the zeroth order term Ψ +, Ψ in the linearized equations 5.9 and consider the effective linear operators: L r V + := LU r, Φ r V + + CU r, U r, Φ r V + = f +, L l V := LU l, Φ l V + CU l, U l, Φ l V = f. 5.

25 RECTILINEAR VORTEX SHEETS OF TWO-PHASE FLOW 5 We can easily verify, using 5., that the coefficients of the operators LU r, Φ r and LU l, Φ l are in W, Ω, that is, A U r W, Ω, Φ r [A U r t Φ r I 4 4 Φ r A U r ] W, Ω, A U l W, Ω, Φ l [A U l t Φ l I 4 4 Φ l A U l ] W, Ω. Moreover, we have CU r,l, U r,l, Φ r,l W, Ω. We note that the linearized equations 5. form a symmetrizable hyperbolic system. As an example, a Friedrichs symmetrizer for the operator L r,l is S r,l t, x = p n m r,l p n n r,l n r,l n r,l t, x. Using the eikonal equations 5.4, we find recall that A U, A U are defined by.: S r Φ r [A U r t Φ r I 4 4 Φ r A U r ] p n Φ r p n p n Φ r p n, p n Φ r p n Φ r p n p n = Φ r 5. and thus expect to control the traces of the components V +, + V +,, and V+,4 Φ r V+,3 on the boundary {x = }. In the same way, we expect to control the traces of the components V, + V,, and V,4 Φ r,l V,3 on the boundary. These preliminary considerations motivate the introduction of the following trace operator on the boundary: Pϕ V x ± := = V ±, + V ±, V ±,4 Φ r,l V±,3 x =. 5.3 This operator will be used in the energy estimates for the linearized equations. Remark 5.. As in [44], one can check that the rows of 5.3 are just the traces of the noncharacteristic components of V±, after multiplication of the equation 5. by the symmetrizer S r,l.

26 6 LIZHI RUAN, DEHUA WANG, SHANGKUN WENG, AND CHANGJIANG ZHU 5.3. The linearized boundary conditions. We now turn to the linearized boundary conditions. The linearization of 5.3 gives Ψ + t, x, x = x = Ψ t, x, x = ψt, x, x = v r v l ψ + v + v ϕ u + u = g, x = x = x = 5.4 x t ψ + v r x ψ + v + x ϕ u + = g, = = = m + + n + m + n = g 3 x = x = on the boundary {x = }. Let us introduce the matrices v r v l x = bt, x = x v r =, ϕ ϕ Mt, x = ϕ. Denote V = V +, V = m +, n +, v +, u +, m, n, v, u, ψ = t ψ, ψ, g = g, g, g 3. Then the linearized boundary conditions become equivalently Ψ + t, x, x = x = Ψ t, x, x = ψt, x, x = b ψ + MV = g. x = 5.5 In terms of the good unknown V = V+, V defined by 5.8, the linearized boundary conditions read as: Ψ + t, x, x = x = Ψ t, x, x = ψt, x, x = B U r,l, Φ r,l V, ψ x = := b ψ + M V+ + Ψ + Φ r U r, V + Ψ x Φ l U l = = b ψ + M U r Φ r ψ, U l x Φ l ψ + M V = = b ψ + M U r Φ r U l Φ l } {{ } b x = x ψ + M V = g. = x = 5.6

27 RECTILINEAR VORTEX SHEETS OF TWO-PHASE FLOW Main result. We observe that, from 5.5, the linearized boundary conditions only involve P V x ±, where P is defined by 5.3. With this notation, we can state our main = result the norms are the weighted norms defined in Section 3 as follows. Theorem 5.. Assume that the particular solution defined by 5. satisfies v r v l > c 3r + c 3 3l, vr v l c r + c l, 5.7 and that the perturbations U r,l, Φ r,l have compact support and are small enough in W, Ω. Then there exist some constants C and λ depending on K and κ defined in 5., 5.5, such that, for all λ λ, the solution V, ψ H λ Ω H λ R to the linearized problem 5. and 5.6 satisfies the following estimates: λ V L λ Ω + P V x = L λ R + ψ Hλ R C L V λ 3 L Hλ + B V, ψ λ Hλ R := C f +, f λ 3 L Hλ + g, λ Hλ R 5.8 where the linearized operators L and B are defined in 5. and 5.6. Remark 5.. Theorem 5. is counterpart of Theorem 3. for variable coefficients. 6. Proof of Theorem 5. This section is devoted to the proof of Theorem 5. on the linear stability with variable coefficients. 6.. Some preliminary transformations Preliminary transformations of the interior equations. For the linearized equations 5., from multiplication by the Friedrichs symmetrizer defined in the previous Section and an integration by parts, one has the following lemma: Lemma 6.. There exist two constants C > and λ such that for all λ λ, the following estimates hold: and thus λ V ± L λ Ω C λ L r V ± L λ Ω + P V ± x = L λ R, where the operator P is defined in 5.3. λ V L λ Ω C λ L V L λ Ω + P V x = L λ R, As in the case of constant coefficients, we only need estimate the traces P V ± x = and the front function ψ in terms of the source terms in the interior domain and on the boundary. For this purpose we shall reformulate further the interior equations 5. to deal with the matrix coefficient of in the differential operators L r,l, noticing that the boundary matrix

28 8 LIZHI RUAN, DEHUA WANG, SHANGKUN WENG, AND CHANGJIANG ZHU has constant rank in the whole closed half-space. Thus we first consider the coefficients of V± in 5. which are equal to Φ [A U t ΦI 4 4 ΦA U], 6. where we drop the indices r, l. Under the assumption 5.4, 6. reduces to the matrix which has eigenvalues: Ā U, Φ = Φ [A U t ΦI 4 4 ΦA U] m Φ m = n Φ n Φ pn Φ n pn Φ, n p nn p nn λ = with multiplicity, and λ,3 = ± cm, n Φ. Φ Here cm, n is defined by 3.7 and we denote Φ := + Φ. We now diagonalize the above matrix. The eigenvectors associated with the eigenvalues are,,,,,,, Φ, for λ, m n Φ, Φ, cm,n n m n Φ, Φ, cm,n n Φ, cm,n n Φ, cm,n n, for λ,, for λ 3. Observe that these eigenvectors are not orthonormal. Thus, we may define the following non orthogonal matrix m n m Φ n Φ Φ Φ T U, Φ : = cm,n n Φ cm,n, 6. n Φ Φ cm,n n then by direct calculations, the inverse T U, Φ is n m+n cm,n n m m+n Φ Φ Φ n m+n Φ n m+n Φ n m+n Φ n cm,n n m+n Φ Φ n Φ cm,n Φ n Φ cm,n n Φ cm,n Φ, 6.3

29 RECTILINEAR VORTEX SHEETS OF TWO-PHASE FLOW 9 which allows one to diagonalize the matrix ĀU, Φ as: T U, ΦĀU, ΦT U, Φ = λ. λ 3 In order to obtain a constant boundary matrix in the differential operators, we also introduce the matrix A U, Φ : = λ λ 3 = Φ. cm,n Φ Φ It follows that A T Ā T = I := diag,,,. Let us define the new unknown functions and set cm,n Φ W + := T U r, Φ r V +, W := T U l, Φ l V 6.4 T r,l := T U r,l, Φ r,l, A r,l := A U r,l, Φ r,l. From the multiplication on the left side of the equations in 5. by A r,l W ± solve the equations see the Appendix A for details: T r,l, we see that A r tw + + A r W + + I W + + A r Cr W + = F +, A l tw + A l W + I W + A l Cl W = F, 6.5 where we have set with slight abuse of notation: A r,l := Ar,l T A T U r,l, Φ r,l, C r,l := [ T t T + T A T + T Ā T + T CT ] U r,l, U r,l, Φ r,l, F ± = A r,l T r,l f ±,

30 3 LIZHI RUAN, DEHUA WANG, SHANGKUN WENG, AND CHANGJIANG ZHU with v r c A r = v r c r r n r Φ r n r Φ r n r Φ r vr c r Φ r c r Φ r Φ r, Φ r Φ r nr Φ r c r Φ r vr c r + Φ r Φ r Φ r Φ r The matrix A l is similar by changing index r by l. Notice that the matrix coefficient of in 6.5 is the constant and diagonal boundary matrix I. The above equations 6.5 are equivalent to the linearized equations 5.. Introducing W ± = e λt W ±, one can rewrite the equations 6.5 equivalently as: L λ r W + : = λa r W + + A r t W + +A r W + + I W + + A r Cr W + = e λt F +, L λ r W : = λa r W + A r t W 6.6 Recall that we have A r,l j +A r W + I W + A r Cr W = e λt F. W, Ωj =,, and C r,l W, Ω Preliminary transformations of the boundary conditions. Denote the vector W = W +, W = Tr V +, T l V, then the boundary conditions 5.6 become Ψ + t, x, x = x = Ψ t, x, x = ψt, x, x = Tr T r V + b ψ + b ψ + M T l T x l V = 6.7 Tr = b ψ + b ψ + M W T = g. x = l Introducing W ± = e λt W ±, Ψ ± := e λt Ψ, ψ ± := e λt ψ and b := b, =,,, thus be λt ψ = b e λt ψ + λbe λt ψ t = b ψ + λ ψb, = b ψ + λb ψ, then the equations 6.7 are also equivalent to Ψ + t, x, x = Ψ t, x, x = ψt, x, x = x = B λ W, ψ Tr := λb ψ + b ψ + b ψ + M T l W = x = e λt g. 6.8

31 RECTILINEAR VORTEX SHEETS OF TWO-PHASE FLOW 3 From 5. we have b W, R, b W, R, M W, R, T r,l x = W, R A priori estimate for the weighted linearized problem. We now derive an a priori estimate of the solution to the weighted linearized problem 6.6 and 6.8. By Lemma 6., we are looking for an estimate of P V + and P V using the new function W. From the relations one has P V + x = = P V x = = + mr n r ϕ W W + 4 x = c r n r ϕ W + 3 W + 4 x = + m l n l ϕ W 3 + W 4 x = c l n l ϕ W 3 W 4 x =,, 6. P V x + = L λ R + P V x = L λ R 6. C W 3 +, W 4 + x = L λ R + W 3, W 4 x = L λ R. We need to estimate the trace of the vector W +, W 3 +, W, W 3 for a solution W to the weighted linearized equations 6.6 and 6.8. From now on, for the sake of simplicity, we drop the tildes and write W ±, Ψ ±, ψ instead of W ±, Ψ ±, ψ. We note that Ψ ±, ψ are coupled with W ± only through the boundary conditions. 6.. Paralinearization. We now apply paralinearization see Bony [5] and Meyer [4] to reduce the problem to the case of constant coefficients. The Fourier dual variables of t, x are δ, η. Denote τ = λ + iδ the Laplace dual variable of t. We recall that the positive constants K, κ were introduced in 5., The boundary conditions. Define the following symbols: b :=, 6. b t, x := v r v l v r bt, x, δ, η, λ := τb + iηb t, x. t, x,, 6.3