Two-Dimensional Vortex Sheets for the Nonisentropic Euler Equations: Nonlinear Stability
aa r X i v : . [ m a t h . A P ] O c t Two-Dimensional Vortex Sheets for theNonisentropic Euler Equations: NonlinearStability
Alessandro Morando ∗ DICATAM, University of Brescia, Via Valotti 9, 25133 Brescia, Italy
Paola Trebeschi † DICATAM, University of Brescia, Via Valotti 9, 25133 Brescia, Italy
Tao Wang ‡ School of Mathematics and Statistics, Wuhan University, Wuhan 430072, ChinaHubei Key Laboratory of Computational Science, Wuhan University, Wuhan 430072, China
Abstract
We show the short-time existence and nonlinear stability of vortex sheets for the non-isentropic compressible Euler equations in two spatial dimensions, based on the weaklylinear stability result of Morando–Trebeschi (2008) [20]. The missing normal derivatives arecompensated through the equations of the linearized vorticity and entropy when derivinghigher-order energy estimates. The proof of the resolution for this nonlinear problem followsfrom certain a priori tame estimates on the effective linear problem in the usual Sobolevspaces and a suitable Nash–Moser iteration scheme.
Keywords:
Nonisentropic fluid, compressible vortex sheet, characteristic boundary, exis-tence, nonlinear stability, Nash–Moser iteration.
Mathematics Subject Classification:
Contents L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3 A priori tame estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.4 Proof of Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 ∗ e-mail: [email protected] † e-mail: [email protected] ‡ e-mail: [email protected] A. Morando , P. Trebeschi & T. Wang
The motion of a perfect polytropic ideal gas in the whole plane R is described by thecompressible Euler equations ( ∂ t + u · ∇ ) p + γ p ∇ · u = 0 ,ρ ( ∂ t + u · ∇ ) u + ∇ p = 0 , ( ∂ t + u · ∇ ) s = 0 , (1.1)where the pressure p = p ( t, x ) ∈ R , velocity u = ( v ( t, x ) , u ( t, x )) T ∈ R , and entropy s = s ( t, x ) ∈ R are unknown functions of time t and position x = ( x , x ) T ∈ R . The density ρ obeys theconstitutive law ρ = ρ ( p, s ) := Ap γ e − s γ , (1.2)where A > γ > p, u , s ) of (1.1) that is smooth on either side ofa smooth surface Γ( t ) := { x = ϕ ( t, x ) } is said to be a vortex sheet (even called a contactdiscontinuity ) provided that it is a classical solution to (1.1) on each side of Γ( t ) and the followingRankine–Hugoniot conditions hold at each point of Γ( t ): ∂ t ϕ = u + · ν = u − · ν, p + = p − . (1.3)Here ν := ( − ∂ x ϕ, T is a spatial normal vector to Γ( t ). As usual, u ± , p ± , and s ± denotethe restrictions of u , p , and s to both sides {± ( x − ϕ ( t, x )) > } of Γ( t ), respectively. Recallthat conditions (1.3) are derived from the conservative form of the compressible Euler equations.These conditions yield that the normal velocity and pressure are continuous across Γ( t ). Hencethe only possible jumps displayed by a vortex sheet concern the tangential velocity and entropy.Remark also that the first two identities in (1.3) are the eikonal equations: ∂ t ϕ + λ ( p + , u + , s + , ∂ x ϕ ) = 0 , ∂ t ϕ + λ ( p − , u − , s − , ∂ x ϕ ) = 0 , where λ ( p, u , s, ξ ) := u · ( ξ, − T denotes the second characteristic field of system (1.1).We are interested in the structural stability of vortex sheets to nonisentropic compressibleEuler equations (1.1) with the initial data being a perturbation of piecewise constant vortexsheets: ( p, u , s ) = ( (¯ p r , ¯ v r , ¯ u r , ¯ s r ) if x > σt + nx , (¯ p l , ¯ v l , ¯ u l , ¯ s l ) if x < σt + nx . (1.4)Here, ¯ p r,l , ¯ v r,l , ¯ u r,l , ¯ s r,l , σ, and n are constants and satisfy σ + ¯ v r n − ¯ u r = 0 , σ + ¯ v l n − ¯ u l = 0 , ¯ p r = ¯ p l =: ¯ p > . -D Nonisentropic Vortex Sheets u r = ¯ u l = σ = n = 0 , ¯ v r = − ¯ v l = ¯ v > . In such nonlinear stability problem, interface Γ( t ) (namely, function ϕ ) is a part of unknowns.The usual approach consists of straightening unknown interface Γ( t ) by means of a suitable changeof coordinates in R , in order to reformulate the free boundary problem in a fixed domain.Precisely, unknowns ( p, u , s ) are replaced by functions( p ± ♯ , u ± ♯ , s ± ♯ )( t, x , x ) := ( p ± , u ± , s ± )( t, x , Φ ± ( t, x , x )) , where Φ ± are smooth functions satisfyingΦ ± ( t, x ,
0) = ϕ ( t, x ) and ± ∂ x Φ( t, x , x ) ≥ κ > x ≥ . (1.5)Hereafter we drop the “ ♯ ” index and set U := ( p, v, u, s ) T for convenience. Then the constructionof vortex sheets for system (1.1) amounts to proving the existence of smooth solutions ( U ± , Φ ± )to the following initial-boundary value problem: L ( U + , Φ + ) := L ( U + , Φ + ) U + = 0 if x > , (1.6a) L ( U − , Φ − ) := L ( U − , Φ − ) U − = 0 if x > , (1.6b) B ( U + , U − , ϕ ) = 0 if x = 0 , (1.6c)( U ± , ϕ ) | t =0 = ( U ± , ϕ ) , (1.6d)where B denotes the boundary operator B ( U + , U − , ϕ ) := ( v + − v − ) | x =0 ∂ x ϕ − ( u + − u − ) | x =0 ∂ t ϕ + v + | x =0 ∂ x ϕ − u + | x =0 ( p + − p − ) | x =0 . (1.7)Owing to transformation (1.5), differential operator L ( U, Φ) takes the form L ( U, Φ) := I ∂ t + A ( U ) ∂ x + e A ( U, Φ) ∂ x , (1.8)where I is the 4 × e A ( U, Φ) := 1 ∂ x Φ ( A ( U ) − ∂ t Φ I − ∂ x Φ A ( U )) , and A ( U ) := v γ p /ρ v v
00 0 0 v , A ( U ) := u γ p u /ρ u
00 0 0 u . Observe that equations (1.5)–(1.6) are not enough to determine functions Φ ± . Majda’s pio-neering work in [14, 15] specifies Φ ± ( t, x , x ) = ± x + ϕ ( t, x ), which turns out to be appropriatein the study of shock waves. However, this particular choice does not work well for vortex sheetswhere boundary { x = 0 } becomes characteristic. As in Francheteau–M´etivier [10], we requirefunctions Φ ± to satisfy the following eikonal equations: ∂ t Φ ± + λ ( p ± , u ± , s ± , ∂ x Φ ± ) = 0 if x ≥ . (1.9)This choice of Φ ± has the advantage to simplify much the expression of equations (1.6a)–(1.6b).More importantly, the rank of the boundary matrix for problem (1.6) keeps constant on the wholedomain { x ≥ } , which allows the application of the Kreiss symmetrizer technique to problem(1.6) in the spirit of Majda–Osher [16]. A. Morando , P. Trebeschi & T. Wang
Remark 1.1.
It is worthwhile noticing that in interior equations (1.6a)–(1.6b), the “+” and“ − ” states are decoupled, whereas in boundary conditions (1.6c) the coupling of the two statesoccurs, cf. (1.7).In the new variables, piecewise constant state (1.4) corresponds to the following trivial solutionof (1.5)–(1.6c) and (1.9) U ± = (¯ p, ± ¯ v, , ¯ s ± ) T , Φ ± ( t, x , x ) = ± x , (1.10)with ¯ p > v >
0. Let us denote by ¯ c ± = c (¯ p, ¯ s ± ) the sound speeds corresponding to theconstant states U ± , where for the polytropic gas (1.2), c ( p, s ) := q p ρ ( ρ, s ) = s γe s/γ Ap γ − . We aim to show the short-time existence of solutions to nonlinear problem (1.5)–(1.6) and(1.9) provided the initial data is sufficiently close to (1.10). Our main result is stated as follows.
Theorem 1.1.
Let
T > and µ ∈ N with µ ≥ . Assume that background state (1.10) satisfiesthe stability conditions v > (¯ c + + ¯ c − ) , v = √ c + + ¯ c − ) . (1.11) Assume further that the initial data U ± and ϕ satisfy the compatibility conditions up to order µ in the sense of Definition 4.1, and that ( U ± − U ± , ϕ ) ∈ H µ +1 / ( R ) × H µ +1 ( R ) has a compactsupport. Then there exists δ > such that, if k U ± − U ± k H µ +1 / ( R ) + k ϕ k H µ +1 ( R ) ≤ δ , thenthere exists a solution ( U ± , Φ ± , ϕ ) of (1.5) – (1.6) and (1.9) on the time interval [0 , T ] satisfying ( U ± − U ± , Φ ± − Φ ± ) ∈ H µ − ((0 , T ) × R ) , ϕ ∈ H µ − ((0 , T ) × R ) . Compressible vortex sheets, along with shocks and rarefaction waves, are fundamental wavesthat play an important role in the study of general entropy solutions to multidimensional hyper-bolic systems of conservation laws. The stability of one uniformly stable shock and multidimen-sional rarefaction waves has been obtained in [14, 15] and Alinhac [1].It was observed long time ago in Miles [18] ( cf . Coulombel–Morando [6] for using only algebraictools) that for two-dimensional nonisentropic Euler equations (1.1), piecewise constant vortexsheets (1.10) are violently unstable unless the following stability criterion is satisfied:2¯ v ≥ (¯ c + + ¯ c − ) , (1.12)while vortex sheets (1.10) are linearly stable under condition (1.12). In the seminal work ofCoulombel–Secchi [8], building on their linear stability results in [7] and a modified Nash–Moseriteration, the short-time existence and nonlinear stability of compressible vortex sheets are es-tablished for the two-dimensional isentropic case under condition (1.12) (as a strict inequality).These results have been generalized very recently by Chen–Secchi–Wang [4] to cover the rela-tivistic case.As for three-dimensional gas dynamics, vortex sheets are showed in Fejer–Miles [9] to be al-ways violently unstable, which is analogous to the Kelvin–Helmholtz instability for incompress-ible fluids. In contrast, Chen–Wang [3] and Trakhinin [26] proved independently the nonlinearstability of compressible current-vortex sheets for three-dimensional compressible magnetohydro-dynamics (MHD). This result indicates that non-paralleled magnetic fields stabilize the motionof three-dimensional compressible vortex sheets. -D Nonisentropic Vortex Sheets L –estimates for the lin-earized problems of (1.5)–(1.6) and (1.9) around background state (1.10) under condition (1.12)(as a strict inequality), and that around a small perturbation of (1.10) under (1.11). The maingoal of the present paper is to prove the structurally nonlinear stability of two-dimensional non-isentropic vortex sheets by adopting the Nash–Moser iteration scheme developed in [8, 11] ,whichhas been successfully applied to the plasma-vacuum interface problem [25], three-dimensionalcompressible steady flows [27], and MHD contact discontinuities [22]. To work in the usualSobolev spaces, we compensate the missing normal derivatives through the equations for thelinearized vorticity and entropy when deriving the higher-order energy estimates.It is worth noting that in the statement of Theorem 1.1, the inequality2¯ v = √ c + + ¯ c − ) (1.13)is required in addition to stability condition (1.12) (with strict inequality). This is due to thefact that the linearized problem about piece-wise constant basic state (1.10) with ¯ v taking thecritical value in (1.13) satisfies an a priori estimate with additional loss of regularity from thedata, which is related to the presence of a double root of the associated Lopatinski˘ı determinant( cf. [20, Theorem 3.1]). At the subsequent level of the variable coefficient linearized problemabout a perturbation of (1.10), the authors in [20] were not able to handle this further loss ofregularity, thus the case of ¯ v = (¯ c + + ¯ c − ) / √ c + = ¯ c − = ¯ c ), value (¯ c + + ¯ c − ) coincides with √ c + + ¯ c − ) and condition (1.11) reducesto the supersonic condition ¯ v > √ c studied in Coulombel–Secchi [8].The plan of this paper is as follows. In §
2, we introduce the effective linear problem andits reformulation. Section 3 is devoted to the proof of a well-posedness result of the effectivelinear problem in the usual Sobolev space H s with s large enough. In particular, the weightedSobolev spaces and norms are introduced in § §
4, by imposingnecessary compatibility conditions on the initial data, we introduce the smooth “approximatesolution”, which reduces problem (1.5)–(1.6) and (1.9) into a nonlinear one with zero initialdata. In §
5, we use a modification of the Nash–Moser iteration scheme to derive the existenceof solutions to the reduced problem and conclude the proof of our main result, namely Theorem1.1.
In this section, we employ the so-called “good unknowns” of Alinhac [1] to obtain the effec-tive linear problem and transform it into an equivalent problem with a constant and diagonalboundary matrix.
We are going to linearize the nonlinear problem (1.5)–(1.6) and (1.9) around a basic state ( U r,l , Φ r,l ) := ( p r,l , v r,l , u r,l , s r,l , Φ r,l ) T given by a perturbationof the stationary solution (1.10). The index r (resp. l ) denotes the state on the right (resp. onthe left) of the interface (after change of variables). More precisely, the perturbation( ˙ U r,l ( t, x , x ) , ˙Φ r,l ( t, x , x )) := ( U r,l ( t, x , x ) , Φ r,l ( t, x , x )) − ( U ± , Φ ± ) (2.1)is assumed to satisfysupp (cid:0) ˙ U r,l , ˙Φ r,l (cid:1) ⊂ {− T ≤ t ≤ T, x ≥ , | x | ≤ R } , (2.2)˙ U r,l ∈ W , ∞ (Ω) , ˙Φ r,l ∈ W , ∞ (Ω) , (cid:13)(cid:13) ˙ U r,l (cid:13)(cid:13) W , ∞ (Ω) + (cid:13)(cid:13) ˙Φ r,l (cid:13)(cid:13) W , ∞ (Ω) ≤ K, (2.3) A. Morando , P. Trebeschi & T. Wang where T , R , and K are positive constants and Ω denotes the half-space { ( t, x , x ) ∈ R : x > } .Moreover, we assume that ( ˙ U r,l , ˙Φ r,l ) satisfies constraints (1.5), (1.9), and Rankine–Hugoniotconditions (1.6c), that is, ∂ t Φ r,l + v r,l ∂ x Φ r,l − u r,l = 0 if x ≥ , (2.4a) ± ∂ x Φ r,l ≥ κ > x ≥ , (2.4b)Φ r = Φ l = ϕ if x = 0 , (2.4c) B (cid:0) U r , U l , ϕ (cid:1) = 0 if x = 0 , (2.4d)for a suitable positive constant κ .The linearized problem of (1.5)–(1.6) and (1.9) around the basic state ( U r,l , Φ r,l ) is given by L ′ ( U r,l , Φ r,l )( V ± , Ψ ± ) := dd θ L ( U r,l + θV ± , Φ r,l + θ Ψ ± ) (cid:12)(cid:12)(cid:12)(cid:12) θ =0 = f ± , x > , B ′ ( U r,l , Φ r,l )( V + , V − , ψ ) := dd θ B ( U r + θV + , U l + θV − , ϕ + θψ ) (cid:12)(cid:12)(cid:12)(cid:12) θ =0 = g, x = 0 , with given source terms f ± = f ± ( t, x ) and boundary data g = g ( t, x ), where ψ denotes thecommon trace of Ψ ± on the boundary { x = 0 } .In order to get rid of the first order terms in Ψ ± involved in the expression of linear operators L ′ ( U r,l , Φ r,l )( V ± , Ψ ± ), we utilize the “good unknowns” of Alinhac [1]:˙ V + := V + − Ψ + ∂ x Φ r ∂ x U r , ˙ V − := V − − Ψ − ∂ x Φ l ∂ x U l , (2.5)to get ( cf. M´etivier [17, Proposition 1.3.1]) L ′ ( U r,l , Φ r,l )( V ± , Ψ ± ) = L ( U r,l , Φ r,l ) ˙ V ± + C ( U r,l , Φ r,l ) ˙ V ± + Ψ ± ∂ x Φ r,l ∂ x { L ( U r,l , Φ r,l ) } , (2.6)where differential operators L ( U r,l , Φ r,l ) are defined in (1.8), while C ( U r,l , Φ r,l ) X := d A ( U r,l ) X ∂ x U + (cid:8) d A ( U r,l ) X − ∂ x Φ r,l d A ( U r,l ) X (cid:9) ∂ x U r,l ∂ x Φ r,l , (2.7)for all X ∈ R . Notice that matrices C ( U r,l , Φ r,l ) are C ∞ –functions of ( ˙ U r,l , ∇ ˙ U r,l , ∇ ˙Φ r,l ) thatvanish at the origin.Let us denote ˙ V := ( ˙ V + , ˙ V − ) T , ∇ t,x ψ = ( ∂ t ψ, ∂ x ψ ) T . In terms of new unknowns (2.5), adirect computation yields B ′ ( U r,l , Φ r,l )( ˙ V | x =0 , ψ ) = b ∇ t,x ψ + b ♯ ψ + M ˙ V | x . (2.8)Coefficients b , b ♯ , and M are defined by b ( t, x ) := v r − v l ) | x =0 v r | x =0 , b ♯ ( t, x ) := M ( t, x ) ∂ x U r ∂ x Φ r ∂ x U l ∂ x Φ l (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x =0 , (2.9) M ( t, x ) := ∂ x ϕ − − ∂ x ϕ ∂ x ϕ − − . (2.10) -D Nonisentropic Vortex Sheets b ♯ is a C ∞ –function of ( ∂ x ˙ U r,l | x =0 , ∂ x ϕ, ∂ x ˙Φ r,l | x =0 ) vanishing at the origin.In view of the nonlinear results obtained in [1, 8, 10], we neglect the last term in (2.6), whichis the zero-th order term in Ψ ± , and consider the following effective linear problem : L ′ e ( U r , Φ r ) ˙ V + := L ( U r , Φ r ) ˙ V + + C ( U r , Φ r ) ˙ V + = f + if x > , (2.11a) L ′ e ( U l , Φ l ) ˙ V − := L ( U l , Φ l ) ˙ V − + C ( U l , Φ l ) ˙ V − = f − if x > , (2.11b) B ′ e ( U r,l , Φ r,l )( ˙ V, ψ ) := b ∇ t,x ψ + b ♯ ψ + M ˙ V | x = g if x = 0 , (2.11c)Ψ + = Ψ − = ψ if x = 0 , (2.11d)where B ′ e ≡ B ′ defined in (2.8) (differently from interior equations (2.11a)–(2.11b), in linearizedboundary conditions (2.11c) all terms involved in (2.8) are considered). It follows from (2.3) that b, M ∈ W , ∞ ( R ), b ♯ ∈ W , ∞ ( R ), C ( U r,l , Φ r,l ) ∈ W , ∞ (Ω), and the coefficients of the operators L ( U r,l , Φ r,l ) are in W , ∞ (Ω). We will consider the dropped terms in (2.11a)–(2.11b) as errorterms at each Nash–Moser iteration step in the subsequent nonlinear analysis.We observe that linearized boundary conditions (2.11c) depend on the traces of ˙ V ± onlythrough the vector P ( ϕ ) ˙ V := ( P ( ϕ ) ˙ V + , P ( ϕ ) ˙ V − ) T with P ( ϕ ) ˙ V ± := ( ˙ V ± , ˙ V ± − ∂ x ϕ ˙ V ± ) T . (2.12)In view of (2.4a), the coefficients for the normal derivative ∂ x in operators L ′ e ( U r,l , Φ r,l ) takethe form: e A ( U r,l , Φ r,l ) = 1 ∂ x Φ r,l − γ p r,l ∂ x Φ r,l γ p r,l − ∂ x Φ r,l /ρ r,l /ρ r,l , (2.13)with ρ r,l = ρ ( p r,l , s r,l ). Hence the boundary { x = 0 } is characteristic for problem (2.11) and weexpect to control only the traces of the noncharacteristic components of unknown ˙ V ± , namelythe vector P ( ϕ ) ˙ V ± . It is more convenient to rewrite the effective linear problem (2.11) into anequivalent problem with a constant and diagonal boundary matrix. This can be achieved becausethe boundary matrix for (2.11) has constant rank on the whole closed half-space { x ≥ } .Let us consider the coefficient matrices of ∂ x ˙ V ± in (2.11a)–(2.11b) ( cf. (2.13)). It is easilyverified that the eigenvalues are λ ∗ = 0 with multiplicity 2 , λ = c ( p, s ) ∂ x Φ h ∂ x Φ i , λ = − c ( p, s ) ∂ x Φ h ∂ x Φ i , where the notation h ∂ x Φ i = p ∂ x Φ) has been used and we drop the subscripts r, l fromthe basic state ( U r,l , Φ r,l ) for simplicity. Define T ( U, Φ) := h ∂ x Φ i h ∂ x Φ i − c γ p ∂ x Φ cγp ∂ x Φ 0 ∂ x Φ c γ p − c γ p
00 0 0 1 , (2.14) A. Morando , P. Trebeschi & T. Wang and A ( U, Φ) = diag (1 , λ − , λ − , A ( U, Φ) T − ( U, Φ) e A ( U, Φ) T ( U, Φ) = I := diag (0 , , , . In view of this identity, we perform the transformation: W + := T − ( U r , Φ r ) ˙ V + , W − := T − ( U l , Φ l ) ˙ V − , (2.15)and set for brevity T r,l := T ( U r,l , Φ r,l ) , A r,l := A ( U r,l , Φ r,l ) . Multiplying (2.11a)–(2.11b) by A r,l T − r,l yields the equivalent system of (2.11a)–(2.11b): A r,l ∂ t W ± + A r,l ∂ x W ± + I ∂ x W ± + C r,l W ± = F ± , (2.16)where we have set F ± = A r,l T − r,l f ± , A r,l := A r,l T − r,l A ( U r,l ) T r,l , and C r,l := A r,l T − r,l h ∂ t T r,l + A ( U r,l ) ∂ x T r,l + e A ( U r,l , Φ r,l ) ∂ x T r,l + C ( U r,l , Φ r,l ) T r,l i . The coefficient matrices A r,lj ∈ W , ∞ (Ω) ( j = 0 ,
1) and C r,l ∈ W , ∞ (Ω). Moreover, A r,lj are C ∞ –functions of their arguments ( U r,l , ∇ Φ r,l ), and C r,l are C ∞ –functions of their arguments( U r,l , ∇ U r,l , ∇ Φ r,l , ∇ Φ r,l ). In terms of the vector W := ( W + , W − ) T as defined by (2.15), theboundary conditions (2.11c)–(2.11d) become equivalent to B ( W, ψ ) := b ∇ t,x ψ + b ♯ ψ + M W | x =0 = g if x = 0 , (2.17a)Ψ + = Ψ − = ψ if x = 0 , (2.17b)where b , b ♯ are given by (2.9), and M := M (cid:18) T r T l (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) x =0 . (2.18)It is clear that matrix M ∈ W , ∞ ( R ) is a C ∞ –function of ( U r,l , ∇ Φ r,l ) | x =0 . Remark 2.1.
We find that the trace of vector M W involved in boundary conditions (2.17a)depends only on the trace of the noncharacteristic part of vector W ± , i.e. sub-vector W nc :=( W +2 , W − , W +3 , W − ) T . Moreover, from (2.15) and (2.12), one can compute explicitly P ( ϕ ) ˙ V ± | x =0 = h ∂ x ϕ i ( W ± + W ± ) c r,l γ p h ∂ x ϕ i ( W ± − W ± ) !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x =0 , (2.19)where p := p r,l | x =0 and c r,l = c ( p r,l , s r,l ). In the following we shall feel free to rewrite theproduct M W | x =0 as M W nc | x =0 with a slight abuse of notation. In this section, we are going to establish a well-posedness result for the effective linear problem(2.11) in the usual Sobolev space H s with s large enough. The essential point is to deducecertain a priori tame estimates in H s . For a general hyperbolic problem with a characteristicboundary, there is a loss of control near the boundary of normal derivatives in a priori energyestimates, and hence Sobolev spaces with conormal regularity are required (see Secchi [24] and -D Nonisentropic Vortex Sheets a priori estimates in the usual Sobolev spaces. This is achievedby employing the idea in [8] and estimating the missing derivatives through the equations of thelinearized vorticity and entropy. For all notation used below, we address the reader to Section3.1.The main result in this section is stated as follows: Theorem 3.1.
Let
T > and s ∈ [3 , ˜ α ] ∩ N with any integer ˜ α ≥ . Assume that the stationarysolution (1.10) satisfies (1.11) , and that perturbations ( ˙ U r,l , ˙Φ r,l ) belong to H s +3 γ (Ω T ) for all γ ≥ and satisfy (2.2) – (2.4) , and k ( ˙ U r,l , ∇ ˙Φ r,l ) k H γ (Ω T ) + k ( ˙ U r,l , ∂ x ˙ U r,l , ∇ ˙Φ r,l ) | x =0 k H γ ( ω T ) ≤ K. (3.1) Assume further that ( f ± , g ) ∈ H s +1 (Ω T ) × H s +1 ( ω T ) vanish in the past. Then there exists apositive constant K , which is independent of s and T , and there exist two constants C > and γ ≥ , which depend solely on K , such that, if K ≤ K , then problem (2.11) admits a uniquesolution ( ˙ V ± , ψ ) ∈ H s (Ω T ) × H s +1 ( ω T ) that vanishes in the past and obeys the following tameestimate: k ˙ V k H sγ (Ω T ) + k P ( ϕ ) ˙ V | x =0 k H sγ ( ω T ) + k ψ k H s +1 γ ( ω T ) ≤ C (cid:8) k f k H s +1 γ (Ω T ) + k g k H s +1 γ ( ω T ) + (cid:0) k f k H γ (Ω T ) + k g k H γ ( ω T ) (cid:1) k ( ˙ U r,l , ˙Φ r,l ) k H s +3 γ (Ω T ) (cid:9) , (3.2) where ˙ V := ( ˙ V + , ˙ V − ) , P ( ϕ ) ˙ V := ( P ( ϕ ) ˙ V + , P ( ϕ ) ˙ V − ) , f := ( f + , f − ) and we have used definition (3.7) . We consider the particular case where the source terms f ± and g vanish in the past, whichcorresponds to the nonlinear problem (1.5)–(1.6) and (1.9) with zero initial data. The general caseis postponed to the nonlinear analysis which involves the construction of a so-called approximatesolution . Remark 3.1.
Let us notice that in inequality (3.2) the involved norms are the natural onesin the exponentially weighted Sobolev spaces H sγ (Ω T ) and H sγ ( ω T ) for a suitable fixed γ (seeSection 3.1), even though the functions considered in the statement of Theorem 3.1 are takenin the usual Sobolev spaces H s (Ω T ), H s ( ω T ). However the functions involved in the statementabove vanish for negative time. This guarantees that they also belong to the weighted Sobolevspaces. To be definite, we introduce certain notationson weighted Sobolev spaces and norms. For every real number s , H s ( R ) denotes the usualSobolev space of order s , which is endowed with the γ − weighted norm k u k s,γ := 1(2 π ) Z R ( γ + | ξ | ) s | b u ( ξ ) | d ξ, (3.3)where γ ≥ b u is the Fourier transform of any distribution u in R . We will abbreviate theusual norm of L ( R ) as k · k := k · k ,γ . Throughout the paper, we introduce the notation: A . B ( B & A ) if A ≤ CB holds uniformlyfor some positive constant C that is independent of γ . The notation, A ∼ B , means that both A . B and B . A are satisfied. Then, for k ∈ N , one has k u k k,γ ∼ X | α |≤ k γ k −| α | ) k ∂ α u k for all u ∈ H k ( R ) , (3.4)0 A. Morando , P. Trebeschi & T. Wang where ∂ α := ∂ α t ∂ α x ∂ α x with α = ( α , α , α ) ∈ N and | α | := α + α + α . For s ∈ R and γ ≥
1, we denote the γ − weighted Sobolev space H sγ ( R ) by H sγ ( R ) := (cid:8) u ∈ D ′ ( R ) : e − γt u ( t, x ) ∈ H s ( R ) (cid:9) , and its norm by k u k H sγ ( R ) := k e − γt u k s,γ . We write L γ ( R ) := H γ ( R ) and k u k L γ ( R ) := k e − γt u k for short. We define L ( R + ; H sγ ( R )), briefly denoted by L ( H sγ ), as the space of distributionswith finite L ( H sγ )–norm, where k u k L ( H sγ ) := Z R + k u ( · , x ) k H sγ ( R ) d x . Let us recall that Ω denotes the half-space { ( t, x , x ) ∈ R : x > } and let R := { ( x , x ) ∈ R : x > } . The boundaries ∂ Ω and ∂ R will be respectively identified to R and R . We set L γ (Ω) := L ( H γ ) and k u k L γ (Ω) . For all k ∈ N and γ ≥
1, the weighted Sobolev space H kγ (Ω) isdefined as H kγ (Ω) := (cid:8) u ∈ D ′ (Ω) : e − γt u ∈ H k (Ω) (cid:9) . For any real number T , we denote Ω T := ( −∞ , T ) × R and ω T := ( −∞ , T ) × R ≃ ∂ Ω T . Forall k ∈ N and γ ≥
1, we define the weighted space H kγ (Ω T ) as H kγ (Ω T ) := (cid:8) u ∈ D ′ (Ω T ) : e − γt u ∈ H k (Ω T ) (cid:9) . In view of relation (3.4), the norm on H kγ (Ω T ) is defined as k u k H kγ (Ω T ) := X | α |≤ k γ k −| α | ) k e − γt ∂ α u k L (Ω T ) . (3.5)The norm on H kγ ( ω T ) is defined in the same way. For all k ∈ N and γ ≥
1, we define thespace L ( R + ; H kγ ( ω T )), briefly denoted by L ( H kγ ( ω T )), as the space of distributions with finite L ( H kγ ( ω T ))–norm, where k u k L ( H kγ ( ω T )) := Z R + k u ( · , x ) k H kγ ( ω T ) d x = X α + α ≤ k γ k − α − α ) k e − γt ∂ α t ∂ α u k L (Ω T ) . (3.6)We write L γ (Ω T ) := L ( H γ ( ω T )) and k u k L γ (Ω T ) := k u k L ( H γ ( ω T )) for brevity.Since the most of functions we are dealing with have double ± states, throughout the paperwe will make use of the following general shortcut notation: for any function space X and everyfunction W = ( W + , W − ), with scalar/vector ± states W ± = W ± ( t, x , x ), we set k W k X := X ± k W ± k X . (3.7) L . In this subsection, we apply the well-posedness result in L ofCoulombel [5] to the effective linear problem (2.11). We recall that system (2.11a)–(2.11b) issymmetrizable hyperbolic with Friedrichs symmetrizer ∫ r,l := diag (1 /ρ r,l , γ p r,l , γ p r,l , γ p r,l ), andobserve that the coefficients of the linearized operators satisfy the regularity assumptions of [5].The first two authors prove in [20, Theorem 4.1] that problem (2.11) satisfies a basic L – apriori estimate with a loss of one tangential derivative as follows. -D Nonisentropic Vortex Sheets Theorem 3.2 ([20]) . Assume that the stationary solution (1.10) satisfies (1.11) , and that thebasic state ( U r,l , Φ r,l ) satisfies (2.2) – (2.4) . Then there exist constants K > and γ ≥ suchthat, if K ≤ K and γ ≥ γ , then, for all ( ˙ V ± , ψ ) ∈ H γ (Ω) × H γ ( R ) , the following estimateholds: γ k ˙ V k L γ (Ω) + k P ( ϕ ) ˙ V | x =0 k L γ ( R ) + k ψ k H γ ( R ) . γ − (cid:13)(cid:13) L ′ e ˙ V (cid:13)(cid:13) L ( H γ ) + γ − (cid:13)(cid:13) B ′ e ( U r,l , Φ r,l )( ˙ V, ψ ) (cid:13)(cid:13) H γ ( R ) , (3.8) where L ′ e ˙ V := ( L ′ e ( U r , Φ r ) ˙ V + , L ′ e ( U l , Φ l ) ˙ V − ) . In view of the result in [5], we only need to find for (2.11) a dual problem that obeys anappropriate energy estimate. To this end, we introduce the following matrices: M := − ℓ l ℓ r ℓ l ℓ r ℓ r − ℓ l − ℓ l (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x =0 , N := ℓ r ℓ r ℓ l ℓ l (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x =0 , (3.9)and N := ∂ x ϕ − ∂ x ϕ − ∂ x ϕ − , where ℓ r,l := − γ p r,l ∂ x Φ r,l , ℓ r,l := − ∂ x ϕ ρ r,l ∂ x Φ r,l , ℓ r,l := 12 ρ r,l ∂ x Φ r,l . By virtue of (2.13), we compute that these matrices satisfy the following relation:diag (cid:0) e A ( U r , Φ r ) , e A ( U l , Φ l ) (cid:1)(cid:12)(cid:12) x =0 = M T M + N T N, (3.10)where matrix M = M ( t, x ) was defined in (2.10). Moreover, we infer from (2.3) that all matrices M , N , N , and M belong to W , ∞ ( R ).Then we define the dual problem for (2.11), posed on Ω, as follows: L ′ e ( U r , Φ r ) ∗ U + = e f + , x > , L ′ e ( U l , Φ l ) ∗ U − = e f − , x > ,N U = 0 , x = 0 , div( b T M U ) − b T ♯ M U = 0 , x = 0 , (3.11)where b , b ♯ , M , and N are defined in (2.9) and (3.9), dual operators L ′ e ( U r,l , Φ r,l ) ∗ are theformal adjoints of L ′ e ( U r,l , Φ r,l ), and div denotes the divergence operator in R with respect to( t, x ). We refer to [17, § N is one, the dual problem (3.11) has exactly two independent scalarboundary conditions, which is compatible with the number of the incoming characteristics , i.e.,negative eigenvalues of the boundary matrix for (3.11).Assuming (2.2)–(2.3) hold for a sufficiently small K , we can repeat the same analysis as in[20] to show that the dual problem (3.11) satisfies the backward Lopatinski˘ı condition and thatthe roots of the associated Lopatinski˘ı determinant are simple and coincide with those for the2 A. Morando , P. Trebeschi & T. Wang original problem (2.11). Then we can find the desired a priori estimate with the loss of onetangential derivative for problem (3.11). The effective linear problem (2.11) thus satisfies all theassumptions ( i.e. , symmetrizability, regularity, and weak stability) listed in [5]. We thereforeobtain the following well-posedness result.
Theorem 3.3.
Let
T > . Assume that all the hypotheses of Theorem 3.2 hold. Then thereexist positive constants K > and γ ≥ , independent of T , such that, if K ≤ K , then, forsource terms f ± ∈ L ( H ( ω T )) and g ∈ H ( ω T ) that vanish for t < , the problem : L ′ e (cid:0) U r , Φ r (cid:1) ˙ V + = f + , for t < T, x > , L ′ e (cid:0) U l , Φ l (cid:1) ˙ V − = f − , for t < T, x > , B ′ e (cid:0) U r,l , Φ r,l (cid:1) ( ˙ V, ψ ) = g for t < T, x = 0 , has a unique solution ( ˙ V ± , ψ ) ∈ L (Ω T ) × H ( ω T ) that vanishes for t < and satisfies P ( ϕ ) ˙ V ± | x =0 ∈ L ( ω T ) . Moreover, the following estimate holds for all γ ≥ γ and for all t ∈ [0 , T ]: γ k ˙ V k L γ (Ω t ) + k P ( ϕ ) ˙ V | x =0 k L γ ( ω t ) + k ψ k H γ ( ω t ) . γ − k f k L ( H γ ( ω t )) + γ − k g k H γ ( ω t ) . (3.12)Theorem 3.3 shows the well-posedness of linearized problem (2.11) in L when source terms( f ± , g ) belong to L ( H ) × H . We now turn to the derivation of energy estimates for thehigher-order derivatives of solutions. A priori tame estimate.
In order to obtain the estimates for higher-order derivativesof solutions to (2.11), it is convenient to deal with the reformulated problem (2.16) and (2.17)for new unknowns W . Until the end of this section, we always assume that γ ≥ γ and K ≤ K ,where γ and K are given by Theorem 3.3. Then estimate (3.12) can be rewritten as √ γ k W k L γ (Ω T ) + k W nc | x =0 k L γ ( ω T ) + k ψ k H γ ( ω T ) . γ − / k F k L ( H γ ( ω T )) + γ − k g k H γ ( ω T ) . (3.13)We first derive the estimate for tangential derivatives. Let k ∈ [1 , s ] be a fixed integer. Apply-ing the tangential derivative ∂ α tan = ∂ α t ∂ α x with | α | = k to system (2.16) yields the equations for ∂ α tan W ± that involve the linear terms of the derivatives, i.e. , ∂ α − β tan ∂ t W ± and ∂ α − β tan ∂ x W ± , with | β | = 1. These terms cannot be treated simply as source terms, owing to the loss of derivativesin energy estimate (3.13). To overcome this difficulty, we adopt the idea of [8], which is to dealwith a boundary value problem for all the tangential derivatives of order equal to k , i.e. , for W ( k ) := { ∂ α t ∂ α x W ± , α + α = k } . Such a problem satisfies the same regularity and stabilityproperties as the original problem (2.16) and (2.17). Repeating the derivation in [20, § W ( k ) satisfies an energy estimate similar to (3.13) with new source terms F ( k ) and G ( k ) . Then we can employ the Gagliardo–Nirenberg and Moser-type inequalities ( cf . [8, Theo-rems 8–10]) to derive the following estimate for tangential derivatives (see [8, Proposition 1] forthe detailed proof). Lemma 3.1 (Estimate of tangential derivatives) . Assume that the hypotheses of Theorem hold. Then there exist constants C s > and γ s ≥ , independent of T , such that, for all γ ≥ γ s and for all ( W ± , ψ ) ∈ H s +2 γ (Ω T ) × H s +2 γ ( ω T ) that are solutions of problem (2.16) and (2.17) , -D Nonisentropic Vortex Sheets the following estimate holds : √ γ k W k L ( H sγ ( ω T )) + k W nc | x =0 k H sγ ( ω T ) + k ψ k H s +1 γ ( ω T ) ≤ C s (cid:8) γ k g k H s +1 γ ( ω T ) + 1 γ / (cid:13)(cid:13) F (cid:13)(cid:13) L ( H s +1 γ ( ω T )) + 1 γ / k W k W , ∞ (Ω T ) (cid:13)(cid:13)(cid:0) ˙ U r,l , ∇ ˙Φ r,l (cid:1)(cid:13)(cid:13) H s +2 γ (Ω T ) + 1 γ (cid:0) k W nc | x =0 k L ∞ ( ω T ) + k ψ k W , ∞ ( ω T ) (cid:1)(cid:13)(cid:13)(cid:0) ˙ U r,l , ∂ x ˙ U r,l , ∇ x ˙Φ r,l (cid:1) | x =0 (cid:13)(cid:13) H s +1 γ ( ω T ) (cid:9) . Since the boundary matrix for our problem (2.16) and (2.17) is singular, there is no hope toestimate all the normal derivatives of W directly from equations (2.16) by applying the standardapproach for noncharacteristic boundary problems as in [21, 23]. However, for our problem (2.11),we can obtain the estimate of missing normal derivatives through the equations of the “linearizedvorticity” and entropy, where the linearized vorticity has been introduced in [8] and defined as:˙ ξ ± := ∂ x ˙ u ± − ∂ x Φ r,l (cid:0) ∂ x Φ r,l ∂ x ˙ u ± + ∂ x ˙ v ± (cid:1) , (3.14)where ˙ u ± and ˙ v ± are the second and third components of good unknowns ˙ V ± ( cf. (2.5)), respec-tively.Plugging (3.14) into (2.11a)–(2.11b), we can compute( ∂ t + v r,l ∂ x ) ˙ ξ ± = (cid:18) ∂ x − ∂ x Φ r,l ∂ x Φ r,l ∂ x (cid:19) F ± − ∂ x F ± ∂ x Φ r,l + Λ r,l ∂ x ˙ V ± + Λ r,l ∂ x ˙ V ± , (3.15)where F ± and F ± are given by F ± := (cid:0) f ± − C ( U r,l , Φ r,l ) ˙ V ± (cid:1) , F ± := (cid:0) f ± − C ( U r,l , Φ r,l ) ˙ V ± (cid:1) , and vectors Λ r,l , are C ∞ –functions of ( ˙ U r,l , ∇ ˙ U r,l , ∇ ˙Φ r,l , ∇ ˙Φ r,l ) vanishing at the origin.Applying a standard energy method to the transport equations (3.15), we can utilize theGagliardo–Nirenberg and Moser-type inequalities to derive the following estimate of ˙ ξ ± : Lemma 3.2 (Estimate of vorticity) . Assume that the hypotheses of Theorem hold. Thenthere exist constants C s > and γ s ≥ , independent of T , such that, for all γ ≥ γ s and for all ( W ± , ψ ) ∈ H s +2 γ (Ω T ) × H s +2 γ ( ω T ) that are solutions of problem (2.16) and (2.17) , functions ˙ ξ ± defined by (3.14) satisfy the following estimate: γ (cid:13)(cid:13) ˙ ξ ± (cid:13)(cid:13) H s − γ (Ω T ) ≤ C s (cid:8)(cid:13)(cid:13) f ± (cid:13)(cid:13) H sγ (Ω T ) + (cid:13)(cid:13) f ± (cid:13)(cid:13) L ∞ (Ω T ) (cid:13)(cid:13) ∇ ˙Φ r,l (cid:13)(cid:13) H sγ (Ω T ) + (cid:13)(cid:13) ˙ V ± (cid:13)(cid:13) H sγ (Ω T ) + (cid:13)(cid:13) ˙ V ± (cid:13)(cid:13) W , ∞ (Ω T ) (cid:0)(cid:13)(cid:13) ˙ U r,l (cid:13)(cid:13) H s +1 γ (Ω T ) + (cid:13)(cid:13) ∇ ˙Φ r,l (cid:13)(cid:13) H sγ (Ω T ) (cid:1)(cid:9) . We turn to the estimate for normal derivatives ∂ x ( W ± , W ± , W ± ) by expressing them interms of tangential derivatives ∂ t W ± , ∂ x W ± , and vorticity ˙ ξ ± . Since I = diag (0 , , , W nc ± = ( W ± , W ± ) T are recovereddirectly from (2.16) as: I ∂ x W ± = F ± − A r,l ∂ t W ± − A r,l ∂ x W ± − C r,l W ± . (3.16)The “missing” normal derivatives ∂ x W ± can be expressed by ˙ ξ ± and equations (2.16). Fromtransformation (2.15) and definition (2.14), we have ∂ x ˙ v ± = ∂ x W ± − c r,l γ p r,l ∂ x Φ r,l (cid:0) ∂ x W ± − ∂ x W ± (cid:1) + (cid:0) ( ∂ x T r,l ) W ± (cid:1) ,∂ x ˙ u ± = ∂ x Φ r,l ∂ x W ± + c r,l γ p r,l (cid:0) ∂ x W ± − ∂ x W ± (cid:1) + (cid:0) ( ∂ x T r,l ) W ± (cid:1) . A. Morando , P. Trebeschi & T. Wang
In view of definition (3.14), we obtain h ∂ x Φ r,l i ∂ x W ± = ∂ x Φ r,l ( ∂ x ˙ u ± − ˙ ξ ± ) − ∂ x Φ r,l (cid:0) ( ∂ x T r,l ) W ± (cid:1) − (cid:0) ( ∂ x T r,l ) W ± (cid:1) . (3.17)Gathering (3.16) and (3.17) yields ∂ x W ± W ± W ± = F ± F ± − e A r,l ∂ t W ± − e A r,l ∂ x W ± − e C r,l W ± − ∂ x Φ r,l h ∂ x Φ r,l i ˙ ξ ± , (3.18)with new matrices e A r,l , and e C r,l , where e A r,l , are C ∞ –functions of ( ˙ U r,l , ∇ ˙Φ r,l ) and e C r,l are C ∞ –functions of ( ˙ U r,l , ∇ ˙ U r,l , ∇ ˙Φ r,l , ∇ ˙Φ r,l ) that vanish at the origin.From (3.18), by an induction argument similar to Coulombel–Secchi [8, Proposition 3], we canget an estimate of the L γ (Ω T )–norm of all normal derivatives up to order s of ( W ± , W ± , W ± ).More precisely, we have the following lemma. Lemma 3.3.
Assume that the hypotheses of Theorem hold. Then there exist constants C s > and γ s ≥ , which are independent of T , such that, for all γ ≥ γ s and solutions ( W ± , ψ ) ∈ H s +2 γ (Ω T ) × H s +2 γ ( ω T ) of problem (2.16) and (2.17) , the following estimate holdsfor all integer k ∈ [1 , s ]: (cid:13)(cid:13) ∂ kx W ± (cid:13)(cid:13) L ( H s − kγ ( ω T )) ≤ C s n(cid:13)(cid:13)(cid:0) F ± , W ± , ˙ ξ ± (cid:1)(cid:13)(cid:13) H s − γ (Ω T ) + (cid:13)(cid:13) W ± (cid:13)(cid:13) L ( H sγ ( ω T )) + (cid:13)(cid:13) ˙ ξ ± (cid:13)(cid:13) L ∞ (Ω T ) (cid:13)(cid:13) ∇ ˙Φ r,l (cid:13)(cid:13) H s − γ (Ω T ) + (cid:13)(cid:13) W ± (cid:13)(cid:13) L ∞ (Ω T ) (cid:13)(cid:13)(cid:0) ˙ U r,l , ∇ ˙Φ r,l (cid:1)(cid:13)(cid:13) H sγ (Ω T ) o . It remains to obtain an estimate for the normal derivatives of the last components W ± of W ± .To this end, it is sufficient to notice from (2.14) that W ± = ˙ s ± (namely the last componentsof W ± coincides with those of original unknowns ˙ V ± ). According to the equations for ˙ s ± in(2.11a)–(2.11b), we have ∂ t W ± + v r,l ∂ x W ± + ( C r,l ˙ V ± ) = f ± , which is a transport equation. Arguing on it by the standard energy method leads to the followinglemma. Lemma 3.4 (Estimate of entropy) . Assume that the hypotheses of Theorem hold. Then thereexist constants C s > and γ s ≥ , which are independent of T , such that, for all γ ≥ γ s andsolutions ( W ± , ψ ) ∈ H s +2 γ (Ω T ) × H s +2 γ ( ω T ) of problem (2.16) and (2.17) , the following estimateholds: γ k W ± k H sγ (Ω T ) ≤ C s n k f ± k H sγ (Ω T ) + k ˙ V k H sγ (Ω T ) + k ( ˙ U r,l , ∇ ˙Φ r,l ) k H s +1 γ (Ω T ) k ˙ V k L ∞ (Ω T ) o . In light of definition (3.5), we see that, for all s ∈ N and θ ∈ H sγ (Ω T ), k θ k H sγ (Ω T ) = s X k =0 k ∂ kx θ k L ( H s − kγ ( ω T )) , γ k θ k H s − γ (Ω T ) ≤ k θ k H sγ (Ω T ) . Thanks to these identities, we combine Lemmas 3.1–3.4 and use the Moser-type inequalities todeduce the following a priori estimates on the H sγ –norm of solution ˙ V ± to the linearized problem(2.11). -D Nonisentropic Vortex Sheets Proposition 3.5.
Assume that the hypotheses of Theorem hold. Then there exists a constant K > independent of s and T ) and constants C s > and γ s ≥ depending on s , butindependent of T ) such that, if K ≤ K , then, for all γ ≥ γ s and solutions ( ˙ V ± , ψ ) ∈ H s +2 γ (Ω T ) × H s +2 γ ( ω T ) of problem (2.11) , the following estimate holds : √ γ (cid:13)(cid:13) ˙ V (cid:13)(cid:13) H sγ (Ω T ) + (cid:13)(cid:13) P ( ϕ ) ˙ V | x =0 (cid:13)(cid:13) H sγ ( ω T ) + k ψ k H s +1 γ ( ω T ) ≤ C s n √ γ k f k H sγ (Ω T ) + 1 γ / k f k L ( H s +1 γ ( ω T )) + 1 γ k g k H s +1 γ ( ω T ) + 1 γ (cid:0)(cid:13)(cid:13) P ( ϕ ) ˙ V | x =0 (cid:13)(cid:13) L ∞ ( ω T ) + k ψ k W , ∞ ( ω T ) (cid:1) k ( ˙ U r,l , ∂ x ˙ U r,l , ∇ ˙Φ r,l ) | x =0 k H s +1 γ ( ω T ) + 1 γ / (cid:0) k f k L ∞ (Ω T ) + (cid:13)(cid:13) ˙ V (cid:13)(cid:13) W , ∞ (Ω T ) (cid:1)(cid:13)(cid:13)(cid:0) ˙ U r,l , ∇ ˙Φ r,l (cid:1)(cid:13)(cid:13) H s +2 γ (Ω T ) o . (3.19) According to Theorem 3.3, the effective linear problem (2.11) iswell-posed for sources terms ( f ± , g ) ∈ L ( H ( ω T )) × H ( ω T ) that vanish in the past. Following[2, 23], we can use Proposition 3.5 to convert Theorem 3.3 into a well-posedness result for(2.11) in H s . More precisely, we can prove that, under the assumptions of Theorem 3.1, if( f ± , g ) ∈ H s +1 (Ω T ) × H s +1 ( ω T ) vanish in the past, then there exists a unique solution ( ˙ V ± , ψ ) ∈ H s (Ω T ) × H s +1 ( ω T ) that vanishes in the past and satisfies (3.19) for all γ ≥ γ s .It remains to prove the tame estimate (3.2). To this end, we first fix the value of γ such that γ is greater than max { γ , . . . , γ ˜ α } . Using (3.19) with s = 3 and (3.1), we have (cid:13)(cid:13) ˙ V (cid:13)(cid:13) H γ (Ω T ) + (cid:13)(cid:13) P ( ϕ ) ˙ V | x =0 (cid:13)(cid:13) H γ ( ω T ) + k ψ k H γ ( ω T ) . K (cid:0) k f k L ∞ (Ω T ) + (cid:13)(cid:13) ˙ V (cid:13)(cid:13) W , ∞ (Ω T ) + (cid:13)(cid:13) P ( ϕ ) ˙ V | x =0 (cid:13)(cid:13) L ∞ ( ω T ) + k ψ k W , ∞ ( ω T ) (cid:1) + k f k H γ (Ω T ) + k g k H γ ( ω T ) . (3.20)Note that T and γ have been fixed. By virtue of classical Sobolev inequalities k θ k L ∞ (Ω T ) . k θ k H (Ω T ) and k θ k L ∞ ( ω T ) . k θ k H ( ω T ) , we also get k f k L ∞ (Ω T ) . k f k H γ (Ω T ) , hence (cid:13)(cid:13) ˙ V (cid:13)(cid:13) W , ∞ (Ω T ) + (cid:13)(cid:13) P ( ϕ ) ˙ V | x =0 (cid:13)(cid:13) L ∞ ( ω T ) + k ψ k W , ∞ ( ω T ) . (cid:13)(cid:13) ˙ V (cid:13)(cid:13) H γ (Ω T ) + (cid:13)(cid:13) P ( ϕ ) ˙ V | x =0 (cid:13)(cid:13) H γ ( ω T ) + k ψ k H γ ( ω T ) . (3.21)We utilize (3.20) together with (3.21) and take K > (cid:13)(cid:13) ˙ V (cid:13)(cid:13) W , ∞ (Ω T ) + (cid:13)(cid:13) P ( ϕ ) ˙ V | x =0 (cid:13)(cid:13) L ∞ ( ω T ) + k ψ k W , ∞ ( ω T ) . k f k H γ (Ω T ) + k g k H γ ( ω T ) . Plugging these estimates into (3.19) yields (3.2), which completes the proof of Theorem 3.1.
This section is devoted to the construction of the so-called approximate solution, which enablesus to reduce the original problem (1.5)–(1.6) and (1.9) into a nonlinear one with zero initial data.The reformulated problem is expected to be solved in the space of functions vanishing in the past,where we have already established a well-posedness result for the linearized problem, cf.
Theorem3.1. The necessary compatibility conditions have to be prescribed on the initial data ( U ± , ϕ )6 A. Morando , P. Trebeschi & T. Wang for the existence of smooth approximate solutions ( U a ± , Φ a ± ), which are solutions of problem(1.5)–(1.6) and (1.9) in the sense of Taylor’s series at time t = 0.Let s ≥ U ± := U ± − U ± ∈ H s +1 / ( R ) and ϕ ∈ H s +1 ( R ).We also assume without loss of generality that ( ˙ U ± , ϕ ) has the compact support:supp ˙ U ± ⊂ { x ≥ , x + x ≤ } , supp ϕ ⊂ [ − , . (4.1)We extend ϕ from R to R by constructing ˙Φ ± ∈ H s +3 / ( R ) such that˙Φ ± | x =0 = ϕ , supp ˙Φ ± ⊂ { x ≥ , x + x ≤ } , (4.2)and (cid:13)(cid:13) ˙Φ ± (cid:13)(cid:13) H s +3 / ( R ) ≤ C k ϕ k H s +1 ( R ) . (4.3)For problem (1.5) and (1.9), we prescribe the initial data:Φ ± | t =0 = Φ ± := ˙Φ ± + Φ ± , (4.4)where Φ ± are defined in (1.10). The estimate (4.3) and the Sobolev embedding theorem yield ± ∂ x Φ ± ≥ / x ∈ R , (4.5)provided ϕ is sufficiently small in H s +1 ( R ).Let us denote the perturbation by ( ˙ U ± , ˙Φ ± ) := ( U ± − U ± , Φ ± − Φ ± ), and the traces of the k -th order time derivatives on { t = 0 } by˙ U ± k := ∂ kt ˙ U ± | t =0 , ˙Φ ± k := ∂ kt ˙Φ ± | t =0 , k ∈ N . To introduce the compatibility conditions, we need to determine traces ˙ U ± k and ˙Φ ± k in termsof the initial data ˙ U ± and ˙Φ ± through equations (1.6a)–(1.6b) and (1.9). For this purpose,we set W ± := ( ˙ U ± , ∇ x ˙ U ± , ∇ x ˙Φ ± ) ∈ R , and rewrite (1.6a)–(1.6b), (1.9) (for ( U ± , Φ ± ) =( U ± + ˙ U ± , Φ ± + ˙Φ ± )) as ∂ t ˙ U ± = F ( W ± ) , ∂ t ˙Φ ± = F ( W ± ) , (4.6)where F and F are suitable C ∞ –functions that vanish at the origin. Applying operator ∂ kt to(4.6) and taking the traces at time t = 0, one can employ the generalized Fa`a di Bruno’s formula( cf . [19, Theorem 2.1]) to derive˙ U ± k +1 = X α i ∈ N , | α | + ··· + k | α k | = k D α + ··· + α k F ( W ± ) k Y i =1 k ! α i ! (cid:18) W ± i i ! (cid:19) α i , (4.7)˙Φ ± k +1 = X α i ∈ N , | α | + ··· + k | α k | = k D α + ··· + α k F ( W ± ) k Y i =1 k ! α i ! (cid:18) W ± i i ! (cid:19) α i , (4.8)where W ± i denotes trace ( ˙ U ± i , ∇ x ˙ U ± i , ∇ x ˙Φ ± i ) at t = 0. From (4.7)–(4.8), one can determine( ˙ U ± k , ˙Φ ± k ) k ≥ inductively as functions of the initial data ˙ U ± and ˙Φ ± . Furthermore, we have thefollowing lemma (see [17, Lemma 4.2.1] for the proof): -D Nonisentropic Vortex Sheets Lemma 4.1.
Let s ≥ be an integer. Assume that (4.1) – (4.3) and (4.5) hold. Then equations (4.7) – (4.8) determine ˙ U ± k ∈ H s +1 / − k ( R ) for k = 1 , . . . , s , and ˙Φ ± k ∈ H s +3 / − k ( R ) for k =1 , . . . , s + 1 , such that supp ˙ U ± k ⊂ { x ≥ , x + x ≤ } , supp ˙Φ ± k ⊂ { x ≥ , x + x ≤ } . Moreover, s X k =0 (cid:13)(cid:13) ˙ U ± k (cid:13)(cid:13) H s +1 / − k ( R ) + s +1 X k =0 (cid:13)(cid:13) ˙Φ ± k (cid:13)(cid:13) H s +3 / − k ( R ) ≤ C (cid:0)(cid:13)(cid:13) ˙ U ± (cid:13)(cid:13) H s +1 / ( R ) + k ϕ k H s +1 ( R ) (cid:1) , where constant C > depends only on s and k ( ˙ U ± , ˙Φ ± ) k W , ∞ ( R ) . To construct a smooth approximate solution, one has to impose certain assumptions on traces˙ U ± k and ˙Φ ± k . We are now ready to introduce the notion of the compatibility condition . Definition 4.1.
Let s ≥ be an integer. Let ˙ U ± := U ± − U ± ∈ H s +1 / ( R ) and ϕ ∈ H s +1 ( R ) satisfy (4.1) . The initial data U ± and ϕ are said to be compatible up to order s if there existfunctions ˙Φ ± ∈ H s +3 / ( R ) satisfying (4.2) – (4.5) such that functions ˙ U ± , . . . , ˙ U ± s , ˙Φ ± , . . . , ˙Φ ± s +1 determined by (4.7) – (4.8) satisfy: ∂ jx (cid:0) ˙Φ + k − ˙Φ − k (cid:1) | x =0 = 0 for j, k ∈ N with j + k < s + 1 , (4.9a) ∂ jx (cid:0) ˙ p + k − ˙ p − k (cid:1) | x =0 = 0 for j, k ∈ N with j + k < s, (4.9b) and Z R (cid:12)(cid:12) ∂ s +1 − kx (cid:0) ˙Φ + k − ˙Φ − k (cid:1)(cid:12)(cid:12) d x d x x < ∞ for k = 0 , . . . , s + 1 , (4.10a) Z R (cid:12)(cid:12) ∂ s − kx (cid:0) ˙ p + k − ˙ p − k (cid:1)(cid:12)(cid:12) d x d x x < ∞ for k = 0 , . . . , s. (4.10b)It follows from Lemma 4.1 that ˙ p ± , . . . , ˙ p ± s − , ˙Φ ± , . . . , ˙Φ ± s − ∈ H / ( R ) ⊂ W , ∞ ( R ). Thenwe can take the j -th order derivatives of the traces in (4.9). Relations (4.9) and (4.10) enableus to utilize the lifting result in [13, Theorem 2.3] to construct the approximate solution in thefollowing lemma. We refer to [8, Lemma 3] for the proof. Lemma 4.2.
Let s ≥ be an integer. Assume that ˙ U ± := U ± − U ± ∈ H s +1 / ( R ) and ϕ ∈ H s +1 ( R ) satisfy (4.1) , and that the initial data U ± and ϕ are compatible up to order s . If ˙ U ± and ϕ are sufficiently small, then there exist functions U a ± , Φ a ± , and ϕ a such that ˙ U a ± := U a ± − U ± ∈ H s +1 (Ω) , ˙Φ a ± := Φ a ± − Φ ± ∈ H s +2 (Ω) , ϕ a ∈ H s +3 / ( ∂ Ω) , and ∂ t Φ a ± + v a ± ∂ x Φ a ± − u a ± = 0 in Ω , (4.11a) ∂ jt L ( U a ± , Φ a ± ) | t =0 = 0 for j = 0 , . . . , s − , (4.11b)Φ a + = Φ a − = ϕ a on ∂ Ω , (4.11c) B ( U a + , U a − , ϕ a ) = 0 on ∂ Ω . (4.11d) Furthermore, we have ± ∂ x Φ a ± ≥ / for all ( t, x ) ∈ Ω , (4.12)supp (cid:0) ˙ U a ± , ˙Φ a ± (cid:1) ⊂ (cid:8) t ∈ [ − T, T ] , x ≥ , x + x ≤ (cid:9) , (4.13)8 A. Morando , P. Trebeschi & T. Wang and (cid:13)(cid:13) ˙ U a ± (cid:13)(cid:13) H s +1 (Ω) + (cid:13)(cid:13) ˙Φ a ± (cid:13)(cid:13) H s +2 (Ω) + k ϕ a k H s +3 / ( ∂ Ω) ≤ ε (cid:0)(cid:13)(cid:13) ˙ U ± (cid:13)(cid:13) H s +1 / ( R ) + k ϕ k H s +1 ( R ) (cid:1) , (4.14) where we denote by ε ( · ) a function that tends to when its argument tends to . Vectors ( U a ± , Φ a ± ) in Lemma 4.2 are called the approximate solution to problem (1.5)–(1.6)and (1.9). It follows from relations (4.11c) and (4.13) that ϕ a is supported within {− T ≤ t ≤ T, x ≤ } . Since s ≥
3, it follows from (4.14) and the Sobolev embedding theorem that (cid:13)(cid:13) ˙ U a ± (cid:13)(cid:13) W , ∞ (Ω) + (cid:13)(cid:13) ˙Φ a ± (cid:13)(cid:13) W , ∞ (Ω) ≤ ε (cid:0)(cid:13)(cid:13) ˙ U ± (cid:13)(cid:13) H s +1 / ( R ) + k ϕ k H s +1 ( R ) (cid:1) . Let us reformulate the original problem into that with zero initial data by utilizing the ap-proximate solution ( U a ± , Φ a ± ). For this purpose, we introduce f a ± := ( − L ( U a ± , Φ a ± ) if t > , t < . (4.15)Since ( ˙ U a ± , ∇ ˙Φ a ± ) ∈ H s +1 (Ω), taking into account (4.11b) and (4.13), we obtain that f a ± ∈ H s (Ω) and supp f a ± ⊂ (cid:8) ≤ t ≤ T, x ≥ , x + x ≤ (cid:9) . Using the Moser-type inequalities and the fact that f a ± vanish in the past, we obtain from (4.14)the estimate: k f a ± k H s (Ω) ≤ ε (cid:0)(cid:13)(cid:13) ˙ U ± (cid:13)(cid:13) H s +1 / ( R ) + k ϕ k H s +1 ( R ) (cid:1) . (4.16)Let ( U a ± , Φ a ± ) be the approximate solution defined in Lemma 4.2. By virtue of (4.11) and(4.15), we see that ( U ± , Φ ± ) = ( U a ± , Φ a ± ) + ( V ± , Ψ ± ) is a solution of the original problem(1.5)–(1.6) and (1.9) on [0 , T ] × R , if ( V ± , Ψ ± ) solve the following problem: L ( V ± , Ψ ± ) := L ( U a ± + V ± , Φ a ± + Ψ ± ) − L ( U a ± , Φ a ± ) = f a ± in Ω T , E ( V ± , Ψ ± ) := ∂ t Ψ ± + ( v a ± + v ± ) ∂ x Ψ ± + v ± ∂ x Φ a ± − u ± = 0 in Ω T , B ( V, ψ ) := B ( U a + + V + , U a − + V − , ϕ a + ψ ) = 0 on ω T , Ψ + = Ψ − = ψ on ω T , ( V ± , Ψ ± ) = 0 , for t < . (4.17)The initial data (1.6d) and (4.4) have been absorbed into the interior equations. From (4.11a)and (4.11d), we observe that ( V ± , Ψ ± ) = 0 satisfies (4.17) for t <
0. Therefore, the originalnonlinear problem on [0 , T ] × R is now reformulated as a problem on Ω T whose solutions vanishin the past. In this section, we are going to solve the nonlinear problem, stated in the equivalent form(4.17), by a suitable iteration scheme of Nash–Moser type ( cf.
H¨ormander [11] and the referencestherein).In order to keep the notation manageable, from now on, it is undestood that functions definedin the interior domain have + and − states. For instance, unless otherwise explicitly written,we write simply ( U, Φ), ( V, Ψ), by meaning ( U ± , Φ ± ), ( V ± , Ψ ± ). According to this shortcutnotation, we shall write L ( U, Ψ) for the pair L ( U ± , Ψ ± ). -D Nonisentropic Vortex Sheets Before describing the iterative scheme for problem (4.17), we recallthe following result for smoothing operators S θ from [8, Proposition 4]. Proposition 5.1.
Let
T > and γ ≥ , and let m ≥ be an integer. Then there exists a family { S θ } θ ≥ of smoothing operators : S θ : F γ (Ω T ) × F γ (Ω T ) −→ \ β ≥ F βγ (Ω T ) × F βγ (Ω T ) , where F βγ (Ω T ) := (cid:8) u ∈ H βγ (Ω T ) : u = 0 for t < (cid:9) is a closed subspace of H βγ (Ω T ) such that k S θ u k H βγ (Ω T ) ≤ Cθ ( β − α ) + k u k H αγ (Ω T ) for all α, β ∈ [1 , m ] , (5.1a) k S θ u − u k H βγ (Ω T ) ≤ Cθ β − α k u k H αγ (Ω T ) for all ≤ β ≤ α ≤ m, (5.1b) (cid:13)(cid:13)(cid:13)(cid:13) dd θ S θ u (cid:13)(cid:13)(cid:13)(cid:13) H βγ (Ω T ) ≤ Cθ β − α − k u k H αγ (Ω T ) for all α, β ∈ [1 , m ] , (5.1c) and k ( S θ u − S θ v ) | x =0 k H βγ ( ω T ) ≤ Cθ ( β +1 − α ) + k ( u − v ) | x =0 k H αγ ( ω T ) for all α, β ∈ [1 , m ] , (5.2) where α, β ∈ N , ( β − α ) + := max { , β − α } , and C > is a constant depending only on m . Inparticular, if u = v on ω T , then S θ u = S θ v on ω T . Furthermore, there exists another family ofsmoothing operators ( still denoted by S θ ) acting on the functions defined on the boundary ω T andsatisfying the properties in (5.1) with norms k · k H αγ ( ω T ) . The proof of (5.2) is based on the following lifting operator, which will be also useful in thesequel of our analysis (see [10, Chapter 5] and [8] for the proof).
Lemma 5.2.
Let
T > and γ ≥ , and let m ≥ be an integer. Then there exists an operator R T , which is continuous from F sγ ( ω T ) to F s +1 / γ (Ω T ) for all s ∈ [1 , m ] , such that, if s ≥ and u ∈ F sγ ( ω T ) , then ( R T u ) | x =0 = u . We are going to follow [8] and describe the iteration scheme for problem (4.17).
Assumption (A-1) : ( V , Ψ , ψ ) = 0 and, for k = 0 , . . . , n , ( V k , Ψ k , ψ k ) are already given andverify ( V k , Ψ k , ψ k ) | t< = 0 , Ψ + k | x =0 = Ψ − k | x =0 = ψ k . (5.3)Let us consider V n +1 = V n + δV n , Ψ n +1 = Ψ n + δ Ψ n , ψ n +1 = ψ n + δψ n , (5.4)where these differences δV n , δ Ψ n , and δψ n will be specified below.First we find ( δ ˙ V n , δψ n ) by solving the effective linear problem: L ′ e ( U a + V n +1 / , Φ a + Ψ n +1 / ) δ ˙ V n = f n in Ω T , B ′ e ( U a + V n +1 / , Φ a + Ψ n +1 / )( δ ˙ V n , δψ n ) = g n on ω T , ( δ ˙ V n , δψ n ) = 0 for t < , (5.5)0 A. Morando , P. Trebeschi & T. Wang where operators L ′ e , B ′ e are defined in (2.11a)–(2.11b), (2.11c), respectively, δ ˙ V n := δV n − ∂ x ( U a + V n +1 / ) ∂ x (Φ a + Ψ n +1 / ) δ Ψ n (5.6)is the “good unknown” ( cf. (2.5)), and ( V n +1 / , Ψ n +1 / ) is a smooth modified state such that( U a + V n +1 / , Φ a + Ψ n +1 / ) satisfies constraints (2.2)–(2.4). The source term ( f n , g n ) will bedefined through the accumulated errors at Step n later on.Define the modified state with V ± n +1 / = ( p ± n +1 / , v ± n +1 / , u ± n +1 / , s ± n +1 / ) T as Ψ ± n +1 / := S θ n Ψ ± n , v ± n +1 / := S θ n v ± n , s ± n +1 / := S θ n s ± n ,p ± n +1 / := S θ n p ± n ∓ R T (cid:0) S θ n p + n | x =0 − S θ n p − n | x =0 (cid:1) ,u ± n +1 / := ∂ t Ψ ± n +1 / + (cid:0) v a ± + v ± n +1 / (cid:1) ∂ x Ψ ± n +1 / + v ± n +1 / ∂ x Φ a ± , (5.7)where S θ n are the smoothing operators defined in Proposition 5.1 with sequence { θ n } given by θ ≥ , θ n = q θ + n, (5.8)and R T is the lifting operator given in Lemma 5.2. In light of Proposition 5.1 and (5.3), weobtain Ψ + n +1 / | x =0 = Ψ − n +1 / | x =0 =: ψ n +1 / ,p + n +1 / | x =0 = p − n +1 / | x =0 , on ω T , E ( V n +1 / , Ψ n +1 / ) = 0 , in Ω T , (cid:0) V n +1 / , Ψ n +1 / , ψ n +1 / (cid:1) | t< = 0 . (5.9)It then follows from (4.11) that ( U a + V n +1 / , Φ a + Ψ n +1 / ) satisfies (2.4a) and (2.4c)–(2.4d).The additional constraint (2.4b) will be obtained by taking the initial data small enough.The errors at Step n are defined from the following decompositions: L ( V n +1 , Ψ n +1 ) − L ( V n , Ψ n )= L ′ ( U a + V n , Φ a + Ψ n )( δV n , δ Ψ n ) + e ′ n = L ′ ( U a + S θ n V n , Φ a + S θ n Ψ n )( δV n , δ Ψ n ) + e ′ n + e ′′ n = L ′ ( U a + V n +1 / , Φ a + Ψ n +1 / )( δV n , δ Ψ n ) + e ′ n + e ′′ n + e ′′′ n = L ′ e ( U a + V n +1 / , Φ a + Ψ n +1 / ) δ ˙ V n + e ′ n + e ′′ n + e ′′′ n + D n +1 / δ Ψ n (5.10)and B ( V n +1 | x =0 , ψ n +1 ) − B ( V n | x =0 , ψ n )= B ′ ( U a + V n , Φ a + Ψ n )( δV n | x =0 , δψ n ) + ˜ e ′ n = B ′ ( U a + S θ n V n , Φ a + S θ n Ψ n )( δV n | x =0 , δψ n ) + ˜ e ′ n + ˜ e ′′ n = B ′ e ( U a + V n +1 / , Φ a + Ψ n +1 / )( δ ˙ V n | x =0 , δψ n ) + ˜ e ′ n + ˜ e ′′ n + ˜ e ′′′ n , (5.11)where we have set D n +1 / := 1 ∂ (Φ a + Ψ n +1 / ) ∂ L ( U a + V n +1 / , Φ a + Ψ n +1 / ) , (5.12) -D Nonisentropic Vortex Sheets e n := e ′ n + e ′′ n + e ′′′ n + D n +1 / δ Ψ n , ˜ e n := ˜ e ′ n + ˜ e ′′ n + ˜ e ′′′ n . (5.13) Assumption (A-2) : f := S θ f a , ( e , ˜ e , g ) := 0 , and ( f k , g k , e k , ˜ e k ) are already given andvanish in the past for k = 0 , . . . , n − . We compute the accumulated errors at Step n , n ≥
1, by E n := n − X k =0 e k , e E n := n − X k =0 ˜ e k . (5.14)Then we compute f n and g n for n ≥ n X k =0 f k + S θ n E n = S θ n f a , n X k =0 g k + S θ n e E n = 0 . (5.15)Under assumptions (A-1) – (A-2) , ( V n +1 / , Ψ n +1 / ) and ( f n , g n ) have been specified from(5.7) and (5.15). Then we can obtain ( δ ˙ V n , δψ n ) as the solution of the linear problem (5.5) byapplying Theorem 3.1.Now we need to construct δ Ψ n = ( δ Ψ + n , δ Ψ − n ) T satisfying δ Ψ ± n | x =0 = δψ n . From the explicitexpression of the boundary conditions in (5.5) ( cf. (2.9)–(2.10) and (5.6)), we observe that δψ n solves ∂ t δψ n + ( v a + + v + n +1 / ) | x =0 ∂ x δψ n + ( ∂ x ( ϕ a + ψ n +1 / ) ∂ x ( v a + + v + n +1 / ) | x =0 ∂ x (Φ a + + Ψ + n +1 / ) | x =0 − ∂ x ( u a + + u + n +1 / ) | x =0 ∂ x (Φ a + + Ψ + n +1 / ) | x =0 ) δψ n + ∂ x ( ϕ a + ψ n +1 / )( δ ˙ v + n ) | x =0 − ( δ ˙ u + n ) | x =0 = g n, ,∂ t δψ n + ( v a − + v − n +1 / ) | x =0 ∂ x δψ n + ( ∂ x ( ϕ a + ψ n +1 / ) ∂ x ( v a − + v − n +1 / ) | x =0 ∂ x (Φ a − + Ψ − n +1 / ) | x =0 − ∂ x ( u a − + u − n +1 / ) | x =0 ∂ x (Φ a − + Ψ − n +1 / ) | x =0 ) δψ n + ∂ x ( ϕ a + ψ n +1 / )( δ ˙ v − n ) | x =0 − ( δ ˙ u − n ) | x =0 = g n, − g n, . These identities suggest us to define δ Ψ ± n as solutions to the following equations: ∂ t δ Ψ + n + ( v a + + v + n +1 / ) ∂ x δ Ψ + n + ( ∂ x (Φ a + + Ψ + n +1 / ) ∂ x ( v a + + v + n +1 / ) ∂ x (Φ a + + Ψ + n +1 / ) − ∂ x ( u a + + u + n +1 / ) ∂ x (Φ a + + Ψ + n +1 / ) ) δ Ψ + n + ∂ x (Φ a + + Ψ + n +1 / ) δ ˙ v + n − δ ˙ u + n = R T g n, + h + n , (5.16) ∂ t δ Ψ − n + ( v a − + v − n +1 / ) ∂ x δ Ψ − n + ( ∂ x (Φ a − + Ψ − n +1 / ) ∂ x ( v a − + v − n +1 / ) ∂ x (Φ a − + Ψ − n +1 / ) − ∂ x ( u a − + u − n +1 / ) ∂ x (Φ a − + Ψ − n +1 / ) ) δ Ψ − n + ∂ x (Φ a − + Ψ − n +1 / ) δ ˙ v − n − δ ˙ u − n = R T ( g n, − g n, ) + h − n , (5.17)2 A. Morando , P. Trebeschi & T. Wang where h ± n are suitable source terms on Ω T , vanishing in the past and with zero traces on theboundary ω T , whose explicit form will be specified below.In order to compute the values of h ± n , let us make for the operator E , defined in (4.17), adecomposition similar to (5.10). Namely we have E ( V n +1 , Ψ n +1 ) − E ( V n , Ψ n ) = E ′ ( V n , Ψ n )( δV n , δ Ψ n ) + ˆ e ′ n = E ′ ( S θ n V n , S θ n Ψ n )( δV n , δ Ψ n ) + ˆ e ′ n + ˆ e ′′ n = E ′ ( V n +1 / , Ψ n +1 / )( δV n , δ Ψ n ) + ˆ e ′ n + ˆ e ′′ n + ˆ e ′′′ n . (5.18)Let us denote ˆ e n := ˆ e ′ n + ˆ e ′′ n + ˆ e ′′′ n , ˆ E n := n − X k =0 ˆ e k . (5.19)From (4.11a), we have E ( V, Ψ) = ∂ t (Φ a + Ψ) + ( v a + v ) ∂ (Φ a + Ψ) − ( u a + u ) . Making the change of good unknown (5.6) implies that E ′ ( V ± n +1 / , Ψ ± n +1 / )( δV ± n , δ Ψ ± n ) are equalto the left-hand sides of (5.16) and (5.17), respectively. Then it follows from (5.16)–(5.18) that E ( V n +1 , Ψ n +1 ) − E ( V n , Ψ n ) = (cid:18) R T g n, + h + n + ˆ e + n R T ( g n, − g n, ) + h − n + ˆ e − n (cid:19) . Summing these relations and using E ( V , Ψ ) = 0, we get E ( V + n +1 , Ψ + n +1 ) = R T (cid:16) n X k =0 g k, (cid:17) + n X k =0 h + k + ˆ E + n +1 , E ( V − n +1 , Ψ − n +1 ) = R T (cid:16) n X k =0 ( g k, − g k, ) (cid:17) + n X k =0 h − k + ˆ E − n +1 . On the other hand, we obtain from (5.5) and (5.11) that g n = B ( V n +1 | x =0 , ψ n +1 ) − B ( V n | x =0 , ψ n ) − ˜ e n . (5.20)In view of (4.17) and (1.7), one obtains the relations: (cid:0) B ( V n +1 | x =0 , ψ n +1 ) (cid:1) = E ( V + n +1 | x =0 , ψ n +1 )= E ( V − n +1 | x =0 , ψ n +1 ) + (cid:0) B ( V n +1 | x =0 , ψ n +1 ) (cid:1) . (5.21)Summing (5.20) and using B ( V | x =0 , ψ ) = 0, we have E ( V − n +1 , Ψ − n +1 ) = R T (cid:16) E (cid:0) V − n +1 | x =0 , ψ n +1 (cid:1) − e E n +1 , + e E n +1 , (cid:17) + n X k =0 h − k + ˆ E − n +1 . (5.22)Similarly, we can also obtain E ( V + n +1 , Ψ + n +1 ) = R T (cid:16) E (cid:0) V + n +1 | x =0 , ψ n +1 (cid:1) − e E n +1 , (cid:17) + n X k =0 h + k + ˆ E + n +1 . (5.23) -D Nonisentropic Vortex Sheets Assumption (A-3) : ( h +0 , h − , ˆ e ) = 0 , and ( h + k , h − k , ˆ e k ) are already given and vanish in the pastfor k = 0 , . . . , n − . Under assumptions (A-1) – (A-3) , taking into account (5.22)–(5.23) and the property of R T ,we compute the source terms h ± n from S θ n (cid:0) ˆ E + n − R T e E n, (cid:1) + n X k =0 h + k = 0 , (5.24a) S θ n (cid:0) ˆ E − n − R T e E n, + R T e E n, (cid:1) + n X k =0 h − k = 0 . (5.24b)By virtue of assumption (A-3) and the properties of S θ , it is clear that h ± n vanish in thepast. As in [10], one can also check that the trace of h ± n on ω T vanishes. Hence, we can find δ Ψ ± n , vanishing in the past and satisfying δ Ψ ± n | x =0 = δψ n , as the unique smooth solutions tothe transport equations (5.16)–(5.17).Once δ Ψ n is specified, we can obtain δV n from (5.6) and ( V n +1 , Ψ n +1 , ψ n +1 ) from (5.4). Thenthe errors: e ′ n , e ′′ n , e ′′′ n , ˜ e ′ n , ˜ e ′′ n , ˜ e ′′′ n , ˆ e ′ n , ˆ e ′′ n , and ˆ e ′′′ n are computed from (5.10)–(5.11) and (5.18),while e n , ˜ e n , and ˆ e n are obtained from (5.13) and (5.19).Using (5.5) and (5.15), we sum (5.10)–(5.20) from n = 0 to m , respectively, to obtain L ( V m +1 , Ψ m +1 ) = m X n =0 f n + E m +1 = S θ m f a + ( I − S θ m ) E m + e m , (5.25) B ( V m +1 | x =0 , ψ m +1 ) = m X n =0 g n + e E m +1 = ( I − S θ m ) e E m + ˜ e m . (5.26)Plugging (5.24) into (5.22)–(5.23), we utilize (5.21) to deduce E ( V − m +1 , Ψ − m +1 ) = R T (cid:0)(cid:0) B ( V m +1 | x =0 , ψ m +1 ) (cid:1) − (cid:0) B ( V m +1 | x =0 , ψ m +1 ) (cid:1) (cid:1) + ( I − S θ m ) (cid:0) ˆ E − m − R T (cid:0) e E m, − e E m, (cid:1)(cid:1) + ˆ e − m − R T (cid:0) ˜ e m, − ˜ e m, (cid:1) , E ( V + m +1 , Ψ + m +1 ) = R T (cid:0)(cid:0) B ( V n +1 | x =0 , ψ n +1 ) (cid:1) (cid:1) + ( I − S θ m ) (cid:0) ˆ E + m − R T e E m, (cid:1) + ˆ e + m − R T ˜ e m, . (5.27)Since S θ m → I as m → ∞ , we can formally obtain the solution to problem (4.17) from L ( V m +1 , Ψ m +1 ) → f a , B ( V m +1 | x =0 , ψ m +1 ) →
0, and E ( V m +1 , Ψ m +1 ) →
0, provided that theerrors: ( e m , ˜ e m , ˆ e m ) → Given a constant ε > α that will be chosenlater on, we assume that (A-1) – (A-3) are satisfied and that the following estimate holds: (cid:13)(cid:13) ˙ U a (cid:13)(cid:13) H ˜ α +3 γ (Ω T ) + (cid:13)(cid:13) ˙Φ a (cid:13)(cid:13) H ˜ α +4 γ (Ω T ) + (cid:13)(cid:13) ϕ a (cid:13)(cid:13) H ˜ α +7 / γ (Ω T ) + (cid:13)(cid:13) f a (cid:13)(cid:13) H ˜ α +2 γ (Ω T ) ≤ ε. (5.28)4 A. Morando , P. Trebeschi & T. Wang
Fixing another integer α , our inductive assumption reads:( H n − ) (a) k ( δV k , δ Ψ k ) k H sγ (Ω T ) + k δψ k k H s +1 γ ( ω T ) ≤ εθ s − α − k ∆ k for all k = 0 , . . . , n − s ∈ [3 , ˜ α ] ∩ N ;(b) kL ( V k , Ψ k ) − f a k H sγ (Ω T ) ≤ εθ s − α − k for all k = 0 , . . . , n − s ∈ [3 , ˜ α − ∩ N ;(c) kB ( V k | x =0 , ψ k ) k H sγ ( ω T ) ≤ εθ s − α − k for all k = 0 , . . . , n − s ∈ [4 , α ] ∩ N ;(d) kE ( V k , Ψ k ) k H γ (Ω T ) ≤ εθ − αk for all k = 0 , . . . , n − , where ∆ k := θ k +1 − θ k with θ k defined by (5.8). Notice that13 θ k ≤ ∆ k = q θ k + 1 − θ k ≤ θ k for all k ∈ N . In particular, sequence (∆ k ) is decreasing and tends to 0. Our goal is to show that, for a suitablechoice of parameters θ ≥ ε >
0, and for f a small enough, ( H n − ) implies ( H n ) and that( H ) holds. Once this goal is achieved, we infer that ( H n ) holds for all n ∈ N , which enables usto conclude the proof of Theorem 1.1.From now on, we assume that ( H n − ) holds. As in [8], assumption ( H n − ) implies thefollowing lemma. Lemma 5.3. If θ is large enough, then, for each k = 0 , . . . , n and each integer s ∈ [3 , ˜ α ] , k ( V k , Ψ k ) k H sγ (Ω T ) + k ψ k k H s +1 γ ( ω T ) ≤ ( εθ ( s − α ) + k if s = α,ε log θ k if s = α, (5.29) k (( I − S θ k ) V k , ( I − S θ k )Ψ k ) k H sγ (Ω T ) ≤ Cεθ s − αk . (5.30) Furthermore, for each k = 0 , . . . , n , and each integer s ∈ [3 , ˜ α + 4] , one has k ( S θ k V k , S θ k Ψ k ) k H sγ (Ω T ) ≤ ( Cεθ ( s − α ) + k if s = α,Cε log θ k if s = α. (5.31) To deduce ( H n ) from ( H n − ), we need to estimate the quadraticerrors e ′ k , ˜ e ′ k , and ˆ e ′ k , the first substitution errors e ′′ k , ˜ e ′′ k , and ˆ e ′′ k , the second substitution errors e ′′′ k , ˜ e ′′′ k , and ˆ e ′′′ k , and the last error D k +1 / δ Ψ k . Recall from (5.10) that e ′ k = L ( V k +1 , Ψ k +1 ) − L ( V k , Ψ k ) − L ′ ( U a + V k , Φ a + Ψ k )( δV k , δ Ψ k ) , which can be rewritten as e ′ k = Z (1 − τ ) L ′′ ( U a + V k + τ δV k , Φ a + Ψ k + τ δ Ψ k ) (cid:0) ( δV k , δ Ψ k ) , ( δV k , δ Ψ k ) (cid:1) d τ, (5.32)where operator L ′′ is defined by L ′′ ( U, Φ) (cid:0) ( V ′ , Ψ ′ ) , ( V ′′ , Ψ ′′ ) (cid:1) := dd τ L ′ ( U + τ V ′′ , Φ + τ Ψ ′′ )( V ′ , Ψ ′ ) (cid:12)(cid:12)(cid:12)(cid:12) τ =0 , -D Nonisentropic Vortex Sheets L ′ given in (2.6). We can also obtain a similar expression for ˜ e ′ k (resp. ˆ e ′ k ) definedby (5.11) (resp. (5.18)) in terms of the second derivative operator B ′′ (resp. E ′′ ).To control the quadratic errors, we need the following estimates for operators L ′′ , B ′′ , and E ′′ ( cf. (5.32)). This can be achieved from the explicit forms of L ′′ , B ′′ , and E ′′ by applying theMoser-type and Sobolev embedding inequalities. We refer to [8, Proposition 5] for the detailedproof which is omitted here for brevity. Proposition 5.4.
Let
T > and s ∈ N with s ≥ . Assume that ( ˜ U, ˜Φ) ∈ H s +1 γ (Ω T ) and k ( ˜ U, ˜Φ) k H γ (Ω T ) ≤ e K for all γ ≥ . Then there exist two positive constants e K and C , which areindependent of T and γ , such that, if e K ≤ e K , γ ≥ , and ( V , Ψ ) , ( V , Ψ ) ∈ H s +1 γ (Ω T ) , then (cid:13)(cid:13) L ′′ (cid:0) U a + ˜ U, Φ a + ˜Φ (cid:1)(cid:0) ( V , Ψ ) , ( V , Ψ ) (cid:1)(cid:13)(cid:13) H sγ (Ω T ) ≤ C n k ( V , Ψ ) k W , ∞ (Ω T ) k ( V , Ψ ) k W , ∞ (Ω T ) (cid:13)(cid:13)(cid:0) ˜ U + ˙ U a , ˜Φ + ˙Φ a (cid:1)(cid:13)(cid:13) H s +1 γ (Ω T ) + X i = j k ( V i , Ψ i ) k H s +1 γ (Ω T ) k ( V j , Ψ j ) k W , ∞ (Ω T ) o , and (cid:13)(cid:13) E ′′ (cid:0) U a + ˜ U, Φ a + ˜Φ (cid:1)(cid:0) ( V , Ψ ) , ( V , Ψ ) (cid:1)(cid:13)(cid:13) H sγ (Ω T ) ≤ C X i = j n k V i k L ∞ (Ω T ) k Ψ j k H s +1 γ (Ω T ) + k V i k H sγ (Ω T ) k Ψ j k W , ∞ (Ω T ) o . Moreover, if ( W , ψ ) , ( W , ψ ) ∈ H sγ ( ω T ) × H s +1 γ ( ω T ) , then (cid:13)(cid:13) B ′′ (cid:0) U a + ˜ U, Φ a + ˜Φ (cid:1)(cid:0) ( W , ψ ) , ( W , ψ ) (cid:1)(cid:13)(cid:13) H sγ ( ω T ) ≤ C X i = j n k W i k L ∞ ( ω T ) k ψ j k H s +1 γ ( ω T )) + k W i k H sγ ( ω T ) k ψ j k W , ∞ ( ω T ) o . Using Proposition 5.4, we can obtain the following result ( cf . [8, Lemma 8]). Lemma 5.5 (Estimate of the quadratic errors) . Let α ≥ . Then there exist ε > sufficientlysmall and θ ≥ sufficiently large such that, for all k = 0 , . . . , n − , and all integers s ∈ [3 , ˜ α − , k ( e ′ k , ˆ e ′ k ) k H sγ (Ω T ) + k ˜ e ′ k k H sγ ( ω T ) ≤ Cε θ L ( s ) − k ∆ k , where L ( s ) := max { ( s + 1 − α ) + + 4 − α, s + 2 − α } . Now we estimate the first substitution errors e ′′ k , ˜ e ′′ k , and ˆ e ′′ k given in (5.10)–(5.11), and (5.18)by rewriting them in terms of L ′′ , B ′′ , and E ′′ . For instance, ˜ e ′′ k can be rewritten as˜ e ′′ k = Z B ′′ (cid:0) U a + S θ k V k + τ ( I − S θ k ) V k , Φ a + S θ k Ψ k + τ ( I − S θ k )Ψ k (cid:1)(cid:0) ( δV k | x =0 , δψ k ) , (( I − S θ k ) V k | x =0 , ( I − S θ k )Ψ k | x =0 ) (cid:1) d τ. (5.33)Then we have the following lemma ( cf . [8, Lemma 9]). Lemma 5.6 (Estimate of the first substitution errors) . Let α ≥ . Then there exist ε > sufficiently small and θ ≥ sufficiently large such that, for all k = 0 , . . . , n − , and all integers s ∈ [3 , ˜ α − , k ( e ′′ k , ˆ e ′′ k ) k H sγ (Ω T ) + k ˜ e ′′ k k H sγ ( ω T ) ≤ Cε θ L ( s ) − k ∆ k , where L ( s ) := max { ( s + 1 − α ) + + 6 − α, s + 5 − α } . A. Morando , P. Trebeschi & T. Wang
Now we estimate the second substitution errors e ′′′ k , ˜ e ′′′ k , and ˆ e ′′′ k given in (5.10)–(5.11) and(5.18) by rewriting them in terms of L ′′ , B ′′ , and E ′′ . For instance, ˆ e ′′′ k can be rewritten asˆ e ′′′ k = Z E ′′ (cid:0) V k +1 / + τ ( S θ k V k − V k +1 / ) , Ψ k +1 / (cid:1)(cid:0) ( δV k , δ Ψ k ) , ( S θ k V k − V k +1 / , (cid:1) d τ. (5.34)Here we have used relation Ψ k +1 / = S θ k Ψ k ( cf. (5.7)). Then one can prove the following resultsimilar to [8, Lemma 10]. Lemma 5.7 (Estimate of the second substitution errors) . Let α ≥ . Then there exist ε > sufficiently small and θ ≥ sufficiently large such that, for all k = 0 , . . . , n − , and all integers s ∈ [3 , ˜ α − , one has ˜ e ′′′ k = 0 , ˆ e ′′′ k = 0 and k e ′′′ k k H sγ (Ω T ) ≤ Cε θ L ( s ) − k ∆ k , where L ( s ) := max { ( s + 1 − α ) + + 8 − α, s + 5 − α } . We now estimate the last error term (5.12): D k +1 / δ Ψ k = δ Ψ k ∂ (Φ a + Ψ k +1 / ) R k , where R k := ∂ L ( U a + V k +1 / , Φ a + Ψ k +1 / ). This error term results from the introduction ofthe good unknown in decomposition (5.10). Note from (5.7), (5.28), and (5.31) that | ∂ (Φ a + Ψ k +1 / ) | = (cid:12)(cid:12) ∂ Φ + ∂ (cid:0) ˙Φ a + Ψ k +1 / (cid:1)(cid:12)(cid:12) ≥ , provided that ε is small enough. Then we have the following estimate. Lemma 5.8.
Let α ≥ and ˜ α ≥ α + 2 . Then there exist ε > sufficiently small and θ ≥ sufficiently large such that, for all k = 0 , . . . , n − , and for all integers s ∈ [3 , ˜ α − , we have k D k +1 / δ Ψ k k H sγ (Ω T ) ≤ Cε θ L ( s ) − k ∆ k , (5.35) where L ( s ) := max { ( s + 2 − α ) + + 8 − α, ( s + 1 − α ) + + 9 − α, s + 6 − α } . From Lemmas 5.5–5.8, we can immediately obtain the following estimate for e k , ˜ e k , and ˆ e k defined in (5.13) and (5.19). Lemma 5.9.
Let α ≥ . Then there exist ε > sufficiently small and θ ≥ sufficiently largesuch that, for all k = 0 , . . . , n − , and for all integers s ∈ [3 , ˜ α − , we have k e k k H sγ (Ω T ) + k ˆ e k k H sγ (Ω T ) + k ˜ e k k H sγ ( ω T ) ≤ Cε θ L ( s ) − k ∆ k , (5.36) where L ( s ) is defined in Lemma . Lemma 5.9 yields the estimate of the accumulated errors E k , e E k , and ˆ E k that are defined in(5.14) and (5.19). Lemma 5.10.
Let α ≥ and ˜ α = α + 4 . Then there exist ε > sufficiently small and θ ≥ sufficiently large such that k ( E n , ˆ E n ) k H α +2 γ (Ω T ) + k e E n k H α +2 γ ( ω T ) ≤ Cε θ n . (5.37) -D Nonisentropic Vortex Sheets Proof.
Observe that L ( α + 2) ≤ α ≥
7. Thanks to (5.36), we get k ( E n , ˆ E n ) k H α +2 γ (Ω T ) + k e E n k H α +2 γ ( ω T ) ≤ n − X k =0 (cid:8) k ( e k , ˆ e k ) k H α +2 γ (Ω T ) + k ˜ e k k H α +2 γ ( ω T ) (cid:9) ≤ n − X k =0 Cε ∆ k ≤ Cε θ n , provided that α ≥ α + 2 ∈ [3 , ˜ α − α is α + 4. To show the main result, we first derive the estimates for sourceterms f n , g n , and h ± n defined in (5.15) and (5.24); see [8] or [4, Lemma 8.11] for the proof. Lemma 5.11.
Let α ≥ and ˜ α = α + 4 . Then there exist ε > sufficiently small and θ ≥ sufficiently large such that, for all integers s ∈ [3 , ˜ α + 1] , k f n k H sγ (Ω T ) ≤ C ∆ n (cid:8) θ s − α − n ( k f a k H α +1 γ (Ω T ) + ε ) + ε θ L ( s ) − n (cid:9) , (5.38) k g n k H sγ ( ω T ) ≤ Cε ∆ n (cid:0) θ s − α − n + θ L ( s ) − n (cid:1) , (5.39) and for all integers s ∈ [3 , ˜ α ] , k h ± n k H sγ (Ω T ) ≤ Cε ∆ n (cid:0) θ s − α − n + θ L ( s ) − n (cid:1) . (5.40)Similar to [8, Lemma 16] (see also [4, Lemma 8.12]), we can obtain the following lemma forthe solution to problem (5.5) by employing tame estimate (3.2). Lemma 5.12.
Let α ≥ . If ε > and k f a k H α +1 γ (Ω T ) /ε are sufficiently small, and if θ ≥ issufficiently large, then, for all integers s ∈ [3 , ˜ α ] , k ( δV n , δ Ψ n ) k H sγ (Ω T ) + k δψ n k H s +1 γ ( ω T ) ≤ εθ s − α − n ∆ n . (5.41)Estimate (5.41) is point (a) of ( H n ). One can show the other inequalities in ( H n ); see [8,Lemmas 17-18] or [4, Lemma 8.13] for the proof. Lemma 5.13.
Let α ≥ . If ε > and k f a k H α +1 γ (Ω T ) /ε are sufficiently small, and θ ≥ issufficiently large, then, for all integers s ∈ [3 , ˜ α − , kL ( V n , Ψ n ) − f a k H sγ (Ω T ) ≤ εθ s − α − n . (5.42) Moreover, for all integers s ∈ [4 , α ] , kB ( V n | x =0 , ψ n ) k H sγ ( ω T ) ≤ εθ s − α − n (5.43) and kE ( V n , Ψ n ) k H γ (Ω T ) ≤ εθ − αn . (5.44)Gathering Lemmas 5.12–5.13, we have deduced ( H n ) from ( H n − ), provided that α ≥ α = α + 4, (5.28) holds, ε > k f a k H α +1 γ (Ω T ) /ε are sufficiently small, and θ ≥ α , ˜ α , ε > θ ≥
1, we can prove ( H ) as in [8].8 A. Morando , P. Trebeschi & T. Wang
Lemma 5.14. If k f a k H α +1 γ (Ω T ) /ε is sufficiently small, then ( H ) holds. Proof of Theorem 1.1.
We consider initial data ( U ± , ϕ ) satisfying all the assumptions ofTheorem 1.1. Let ˜ α = µ − α = ˜ α − ≥
7. Then the initial data U ± and ϕ are compatibleup to order µ = ˜ α + 2. From (4.14) and (4.16), we obtain (5.28) and all the requirements ofLemmas 5.12–5.14, provided that ( ˜ U ± , ϕ ) is sufficiently small in H µ +1 / ( R ) × H µ +1 ( R ) with˜ U ± := U ± − U ± . Hence, for small initial data, property ( H n ) holds for all integers n . Inparticular, we have ∞ X k =0 (cid:16) k ( δV k , δ Ψ k ) k H sγ (Ω T ) + k δψ k k H s +1 γ ( ω T ) (cid:17) ≤ C ∞ X k =0 θ s − α − k < ∞ for s ∈ [3 , α − . Thus, sequence ( V k , Ψ k ) converges to some limit ( V, Ψ) in H α − γ (Ω T ), and sequence ψ k convergesto some limit ψ in H αγ (Ω T ). Passing to the limit in (5.42)–(5.43) for s = α − µ −
7, and in(5.44), we obtain (4.17). Therefore, ( U, Φ) = ( U a + V, Φ a + Ψ) is a solution on [0 , T ] × R of theoriginal problem (1.5)–(1.6) and (1.9). This completes the proof. Acknowledgement.
The research of Alessandro Morando and Paola Trebeschi was supportedin part by the grant from Ministero dell’Istruzione, dell’Universit`a e della Ricerca under contractPRIN 2015YCJY3A-004. The research of Tao Wang was supported in part by the grants fromNational Natural Science Foundation of China under contracts 11601398 and 11731008.
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