Two dimensional subsonic and subsonic-sonic spiral flows outside a porous body
aa r X i v : . [ m a t h . A P ] F e b Two dimensional subsonic and subsonic-sonic spiral flowsoutside a porous body
Shangkun Weng ∗ Zihao Zhang † Abstract
In this paper, we investigate two dimensional subsonic and subsonic-sonic spiral flowsoutside a porous body. The existence and uniqueness of the subsonic spiral flows areobtained via variational formulation. The optimal decay rate at far field is also derivedby the Kelvin’s transformation and some elliptic estimates. By extracting spiral subsonicsolutions as the approximate sequences, we obtain the spiral subsonic-sonic limit solution.The main ingredients of our analysis are methods of calculus of variations, the theory ofsecond-order quasilinear equations and the compactness framework.
Mathematics Subject Classifications 2010: Primary 35B40, 35Q31; Secondary35J25, 76N15.Key words: subsonic spiral flows, Euler equations, subsonic-sonic limit, a porousbody.
In this paper, we are concerned with two-dimensional subsonic and subsonic-sonic spiralflows outside a porous body D (Γ), which are governed by the following two-dimensional Eulersystem: ∂ x ( ρu ) + ∂ x ( ρu ) = 0 ,∂ x ( ρu ) + ∂ x ( ρu u ) + ∂ x p = 0 ,∂ x ( ρu u ) + ∂ x ( ρu ) + ∂ x p = 0 , (1.1)where u = ( u , u ) is the velocity field, ρ is the density, and p is pressure. Here we onlyconsider the polytropic gas, therefore p = Aρ γ , where A is a positive constant and γ is theadiabatic constant with γ > ∂ x u − ∂ x u = 0 . (1.2)Then it follows from (1.1) and (1.2) that the flow satisfies the Bernoulli’s law q h ( ρ ) = C , (1.3) ∗ School of mathematics and statistics, Wuhan University, Wuhan, Hubei Province, 430072, People’s Re-public of China. Email: [email protected] † School of mathematics and statistics, Wuhan University, Wuhan, Hubei Province, 430072, People’s Re-public of China. Email: [email protected] h ( ρ ) is the enthalpy satisfying h ′ ( ρ ) = p ′ ( ρ ) ρ , q = p u + u is the flow speed and C isa constant depending on the flow. We normalize the flow as in [7], such that p ( ρ ) = ρ γ /γ isthe pressure for the polytropic gas, and the Bernoulli’s law (1.3) reduces to q Z ρ p ′ ( ρ ) ρ d ρ = γ + 12( γ − . (1.4)So (1.4) yields a representation of the density ρ = g ( q ) = (cid:18) γ + 1 − ( γ − q (cid:19) / ( γ − . (1.5)The sound speed c is defined as c = p ′ ( ρ ). At the sonic point q = c , (1.5) implies q = 1.We define the critical speed q cr as q cr = 1. Thus the flow is subsonic when q <
1, sonic when q = 1 and supersonic when q > ψ as follows ∂ x ψ = − ρu , ∂ x ψ = ρu . (1.6)Obviously, |∇ ψ | = ρq . Then it follows from the normalized Bernoulli’s law (1.4) that ρq isa nonnegative function of q , which is increasing for q ∈ (0 ,
1) and decreasing for q ≥
1, andvanishes at q = 0. So ρq attains its maximum at q = 1. Therefore ρ is a two-valued functionof |∇ ψ | . Subsonic flows correspond to the branch where ρ > |∇ ψ | ∈ [0 , ρ = H ( |∇ ψ | ) (1.7)such that ρ > |∇ ψ | ∈ [0 , H is a positive decreasing function defined on[0 , , H (1) = 1. Then (1.1) reduces to a singleequation div (cid:18) ∇ ψH ( |∇ ψ | ) (cid:19) = 0 . (1.8)By using the hodograph method, Courant and Friedrichs in Section 104 of [7] obtainedsome particular planar radially symmetric flows including circulatory flows and purely radialflows. Their superpositions are called spiral flows. Weng-Xin-Yuan [23] gave a completeclassification of the radially symmetric flow with or without shocks by prescribing suitableboundary conditions on the inner and outer circle of an annulus and also analyzed the de-pendence of the solutions on the boundary data. However, there are very few papers workingon subsonic spiral flows outside a body. This motivate us to investigate the subsonic spiralflows whose asymptotic state is the radially symmetric subsonic spiral flow. To this end, wefirst describe the subsonic spiral flow in the exterior domain B c = { ( x , x ) : r = | x | > } . Itis convenient to use the polar coordinate ( r, θ ): ( x = r cos θ,x = r sin θ, (1.9)and decompose u = U e r + U e θ with e r = (cos θ, sin θ ) T , e θ = ( − sin θ, cos θ ) T . (1.10)2hen the flow is described by radially symmetric smooth functions of the form u ( x ) = U ( r ) e r + U ( r ) e θ , ρ ( x ) = ρ b ( r ) and p ( x ) = p b ( r ), which solves the following system ( ρ b U ) ′ + r ρ b U = 0 , r > ,U U ′ + ρ b p ′ b − U r = 0 , r > ,U U ′ + U U r = 0 , r > ρ b (1) = ρ > , U (1) = κ > , U (1) = κ . Here ρ , κ and κ are chosen to satisfy (1.4), that is12 ( κ + κ ) + ρ γ − γ − γ + 12( γ − . Hence it is easy to see that ρ b U = ρ κ r , U = κ r . (1.12)Therefore u ( x ) = ρ − b ρ κ x − κ x r , ρ − b ρ κ x + κ x r ! . (1.13)Furthermore, by (1.6), the corresponding stream functions ψ , ψ and ψ are ψ ( θ ) = ρ κ θ, ψ ( r ) = − κ Z r ρ b ( s ) 1 s d s,ψ = ψ + ψ = ρ κ θ − κ Z r ρ b ( s ) 1 s d s. (1.14)Denote M = U c , M = U c and M = M + M . Then by simple calculations, we have dd r ( M ) = − (2(1 + M ) + ( γ − M ) M r (1 − M ) , dd r ( M ) = − (2(1 − M ) + ( γ − M ) M r (1 − M ) , dd r ( M ) = − [( γ − M + 2)] M r (1 − M ) . (1.15)Assume that κ + κ > κ <
1. Then there exists a smooth transonic spiral flowsfor all r >
1. The structural stability of this special transonic flows was investigated byWeng-Xin-Yuan in [24]. Assume that κ + κ <
1. Then it follows from (1.15) that M < M < r >
1, which means that the flow is uniformly subsonic.Let D (Γ) is a porous body in R , which contains the domain B (0). Γ is a boundedand connected C ∞ smooth surface describing the boundary of D . Suppose Ω is the exteriordomain of D (Γ), i.e., Ω := R \D , which is connected and filled with compressible and invisiblefluid. 3n this paper, we aim to construct a smooth subsonic spiral flows outside D (Γ) whichtends to the radially symmetric subsonic spiral flow described above. Since D (Γ) is a porousbody, which means that on the boundary, we impose ρ u · ~n = ρ b ( r ) U ( r ) e r · ~n, (1.16)where ~n stands for the unit inward normal of domain D (Γ). It follows from (1.16) that ψ = ψ on Γ.Moreover, the radially symmetric subsonic spiral flow u ( x ) decays like O ( | x | − ), thenwe may have the decay rate | u ( x ) − u ( x ) | = O ( | x | − ) as | x | → ∞ . In terms of the streamfunction, we expect |∇ ( ψ − ψ ) | = O ( | x | − ) . (1.17)Within this paper, we will consider the following problem: Problem ( κ , κ ): Find function ψ which solving the following system: div (cid:16) ∇ ψH ( |∇ ψ | ) (cid:17) = 0 , in Ω ,ψ = ψ , on Γ , |∇ ( ψ − ψ ) | = O ( | x | − ) . (1.18)The main results of this paper can be stated as follows: Theorem 1.1.
For any fixed κ ∈ (0 , , there exists a positive number ˆ κ such that for | κ | ∈ [0 , ˆ κ ) , there exists a unique uniformly subsonic spiral flows to Problem ( κ , κ ) .More precisely, there exists a unique smooth solution ψ ∈ C ∞ (Ω) to (1.18) such that sup x ∈ Ω |∇ ψ | < . (1.19) Theorem 1.2.
Let κ ǫ → ˆ κ as ǫ → with κ ǫ < ˆ κ . Denote by ( u ǫ , u ǫ ) the uniformlysubsonic spiral flow corresponding to Problem ( κ , κ ǫ ) . Then there exists a subsequence,still labeled by ( u ǫ , u ǫ ) such that u ǫ → ˆ u , u ǫ → ˆ u , (1.20) g (( q ǫ ) ) u ǫ → g (ˆ q )ˆ u , g (( q ǫ ) ) u ǫ → g (ˆ q )ˆ u , (1.21) where ( q ǫ ) = ( u ǫ ) + ( u ǫ ) , ˆ q = ˆ u + ˆ u , and g ( q ) is the function defined by (1.5) . All theabove convergence are almost everywhere convergence. Moreover, this limit yields a subsonic-sonic spiral flow (ˆ ρ, ˆ u , ˆ u ) , where ˆ ρ = g (ˆ q ) , which is a weak solution of Problem ( κ , ˆ κ ) .The limit solution (ˆ ρ, ˆ u , ˆ u ) satisfies (1.1) - (1.2) in the sense of distribution and the bound-ary condition (1.16) as the normal trace of the divergence-measure field (ˆ ρ ˆ u , ˆ ρ ˆ u ) on theboundary. Theorem 1.3.
For any fixed κ ∈ ( − , , there exists a positive number ˜ κ such that for κ ∈ (0 , ˜ κ ) , there exists a unique uniformly subsonic spiral flows to Problem ( κ , κ ) . Moreprecisely, there exists a unique smooth solution ψ ∈ C ∞ (Ω) to (1.18) such that sup x ∈ Ω |∇ ψ | < . heorem 1.4. Let κ ǫ → ˜ κ as ǫ → with κ ǫ < ˜ κ . Denote by ( u ǫ , u ǫ ) the uniformlysubsonic spiral flow corresponding to Problem ( κ ǫ , κ ) . Then there exists a subsequence,still labeled by ( u ǫ , u ǫ ) such that u ǫ → ˜ u , u ǫ → ˜ u , (1.22) g (( q ǫ ) )˜ u → g (˜ q )˜ u , g (( q ǫ ) )˜ u → g (˜ q )˜ u , (1.23) where ( q ǫ ) = ( u ǫ ) + ( u ǫ ) , ˜ q = ˜ u + ˜ u . All the above convergence are almost everywhereconvergence. Moreover, this limit yields a subsonic-sonic spiral flow (˜ ρ, ˜ u , ˜ u ) , where ˜ ρ = g (˜ q ) , which is a weak solution of Problem (˜ κ , κ ) . The limit solution (˜ ρ, ˜ u , ˜ u ) satisfies (1.1) - (1.2) in the sense of distribution and the boundary condition (1.16) as the normal traceof the divergence-measure field (˜ ρ ˜ u , ˜ ρ ˜ u ) on the boundary.Remark . The decay rate |∇ ( ψ − ψ ) | = O ( | x | − ) is optimal as was observed by the resultsin [3], but here we remove the small perturbation conditions in [3]. We employe the Kevin’stransformation and some elliptic estimates to derive it. Remark . We prescribe the boundary condition (1.16) on the boundary Γ, not the one that ρ u · ~n = ( ρ b ( r ) U ( r ) e r + ρ b ( r ) U ( r ) e θ ) · ~n . Otherwise, the solution will be the backgroundradially symmetric solution. One may pose ρ u · ~n = ∇ ⊥ Ψ · ~n on Γ for any given smoothfunction Ψ, our method still works in this case.The research on compressible inviscid flows has a long history, which provides manysignificant and challenging problems. The flow past a body, through a nozzle, and past a wallare typical flows patterns, which have physical significances and physical effects. The firsttheoretical result on the problem for irrotational flows past a body was obtained by Frankl andKeldysh in [14]. The important progress for two dimensional subsonic irrotational flows past asmooth body with a small free stream Mach number was obtained by Shiffman [21]. Later on,Bers [1] proved the existence of two dimensional subsonic irrotational flows around a profilewith a sharp trailing edge and also showed that the maximum of Mach numbers approachesone as the free stream Mach number approaches the critical value. The uniqueness andasymptotic behavior of subsonic irrotational plane flows were studied by Finn and Gilbarg in[12]. The well-posedness theory for two-dimensional subsonic flows past a wall or a symmetricbody was established by Chen, Du, Xie and Xin in [2]. For the three-(or higher-) dimensionalcases, the existence and uniqueness of three dimensional subsonic irrotational flows arounda smooth body were proved by Finn and Gilbarg in [13]. Dong and Ou [9] extended theresults of Shiffman to higher dimensions by the direct method of calculus of variations andthe standard Hilbert space method. The respective incompressible case is considered in [20].For the subsonic flow problem in nozzles, one may referee to [8, 10, 11, 16, 19, 22, 25, 26].On the other hand, the subsonic-sonic limit solution can be constructed by the compact-ness method. The first compactness framework on sonic-subsonic irrotational flows in twodimension was introduced by [4] and [25] independently. In [4], Chen, Dafermos, Slemrod andWang introduced general compactness framework and proved the existence of two-dimensionalsubsonic-sonic irrotational flows. Xie and Xin [25] established the subsonic-sonic limit of thetwo-dimensional irrotational flows through infinitely long nozzles. Later on, they extendedthe result to the three-dimensional axisymmetric flow through an axisymmetric nozzle in[26]. Furthermore, Huang, Wang and Wang [18] established a compactness framework forthe general multidimensional irrotational case. Chen, Huang and Wang [6] established the5ompactness framework for nonhomentropic and rotational flows and proved the existence ofmultidimensional subsonic-sonic full Euler flows through infinitely long nozzles. Recently, bythe compactness framework in [6], the existence of subsonic-sonic flows with general conser-vative forces in an exterior domain was established by Gu and Wang in [17].The rest of this article is organized as follows. In Section 2, we first introduce the sub-sonic truncation to reformulate the problem into a second-order uniformly elliptic equation,and then establish the existence and uniqueness of the modified spiral flow by a variationalmethod. Finally we remove the truncation and complete the proof of Theorem 1.1 and Theo-rem 1.3. In Section 3, the compensated compactness framework for steady irrotational flowsis employed to establish the existence of weak subsonic-sonic spiral flows. This section is mainly devoted to the proof of Theorem 1.1 and Theorem 1.3. The proofcan be divided into 5 subsections.
By direct calculations, it is easy to see that the derivative of function H ( s ) goes to infinityas s →
1. To control the ellipticity and avoid singularity of H ′ , one may truncate H as follows˜ H ( s ) = H ( s ) , if 0 ≤ s ≤ − ε, smooth and decreasing , if 1 − ε ≤ s ≤ − ε,H (1 − ε ) , if s ≥ − ε, (2.1)where ε is a small positive constant and ˜ H is a smooth decreasing function.Then we consider the modified equation: div (cid:16) ∇ ψ ˜ H ( |∇ ψ | ) (cid:17) = 0 , in Ω ,ψ = ψ , on Γ , |∇ ( ψ − ψ ) | = O ( | x | − ) . (2.2)After the straightforward computation, (2.2) can be rewritten as X i,j =1 a ij ∂ ij ψ = 0 , (2.3)where a ij = ˜ Hδ ij − H ′ ∂ i ψ∂ j ψ ˜ H . Then it is easy to verify that there exist two constants λ and Λ such that λ | ξ | ≤ a ij ξ i ξ j ≤ Λ | ξ | , for ξ ∈ R . (2.4)Here λ and Λ depend on γ and ε .Next, we solve (2.2) by a variational method. We first give the definition of weak solutionto be used in next subsection. 6 efinition 2.1. A function ψ ∈ H loc (Ω) is a weak solution to (2.2) if Z Ω ( ˜ H ( |∇ ψ | )) − ∇ ψ · ∇ v d x = 0holds for any v ∈ C ∞ c (Ω). Define the space V = { φ ∈ L loc (Ω) , ∇ φ ∈ L (Ω) , φ | Γ = 0 } . (2.5)It is easy to see that V is a Hilbert space under the ˚ H norm. One should look for the solution ψ to the problem (2.2) with the form: ψ = φ + ψ , where φ ∈ L loc (Ω) and ∇ φ ∈ L (Ω).It follows from the boundary condition in (2.2) that φ = − ψ on Γ. To homogenize theboundary data on Γ, we should introduce another function ϕ such that φ , φ − ϕ ∈ V .Let ζ ( x ) be a cut-off function such that ζ ≡ ζ ≡ B R with R large enough. Then set ϕ = − ζψ . By simple calculations, − ζψ satisfies all therequirements we imposed.Define ψ = φ + ϕ + ψ , F ( s ) = Z s ( ˜ H ( t )) − d t, and I [ φ ; ϕ , ψ ] = Z Ω [ F ( |∇ φ + ∇ ϕ + ∇ ψ | ) − F ( |∇ ψ | ) − F ′ ( |∇ ψ | ) ∇ ψ · ( ∇ φ + ∇ ϕ )]d x. (2.6)We consider the following variational problem: l = min φ ∈V I [ φ ; ϕ , ψ ] . (2.7)For our variational problem, we have the following theorem: Theorem 2.2.
The functional I has uniquely one minimizer φ , i.e. I [ φ ; ϕ , ψ ] = l , then ψ = φ + ϕ + ψ satisfies ( div (cid:16) ∇ ψ ˜ H ( |∇ ψ | ) (cid:17) = 0 , in Ω ,ψ = ψ , on Γ , (2.8) in the weak sense. In addition, there is a constant C such that Z Ω |∇ φ | d x ≤ C. (2.9) The constant C here and in the rest of this paper depends on Ω , γ , ε , κ and κ . We willnot repeat the dependence every time. Proof . Step 1 . I [ φ ; ϕ , ψ ] is coercive on V . Let p = ( p , p ), F ( p ) = F ( | p | ). It followsfrom (2.4) that ∀ ξ ∈ R , λ | ξ | ≤ ∂ p i p j F ( p ) ξ i ξ j ≤ Λ | ξ | . (2.10)7hen by direct calculation, we have F ( |∇ φ + ∇ ϕ + ∇ ψ | ) − F ( |∇ ψ | ) − F ′ ( |∇ ψ | ) ∇ ψ · ( ∇ φ + ∇ ϕ )= F ( ∇ ψ ) − F ( ∇ ψ ) − ∂ p F ( ∇ ψ ) · ( ∇ φ + ∇ ϕ )= Z t∂ p i p j F ( ∇ ψ + (1 − t )( ∇ φ + ∇ ϕ )) ∂ i ( φ + ϕ ) ∂ j ( φ + ϕ )d t. This implies Λ k φ + ϕ k V ≥ I ≥ λ k φ + ϕ k V ≥ λ k φ k V − λ Z Ω |∇ ϕ | d x. (2.11)Therefore we obtain I ≥ λ k φ k V − C. (2.12)From (2.12), one can conclude that the energy functional is coercive on V . Step 2 . The existence of the minimizer φ ∈ V . By coerciveness of the functional I [ · ; ϕ , ψ ] on V , we know that l = inf φ ∈V I [ φ ; ϕ , ψ ] exists. Since I [0; ϕ , ψ ] ≤ Λ k ϕ k V ≤ C ,we have l ≤ C . By definition, there is a sequence { φ k } ∞ k =1 ⊂ V such that lim k →∞ I [ φ k ; ϕ , ψ ] = l . For sufficiently large k , C ≥ I [ φ k ; ϕ , ψ ] ≥ λ k φ k k V − C , hence k φ k k V ≤ Cλ . Since V isa Hilbert space, there exists a subsequence of { φ k } (still denoted by { φ k } ), that convergesweakly to some φ ∈ V .Next, we need to show that the functional I [ · ; ϕ , ψ ] is lower semi-continuous with respectto the weak convergence of V , that is I [ φ ; ϕ , ψ ] ≤ lim inf k →∞ I [ φ k ; ϕ , ψ ] , if φ k ⇀ φ in V . (2.13)It follows from φ k ⇀ φ in V that ∇ φ k ⇀ ∇ φ in L loc (Ω). Furthermore, one has 2 F ′ ( |∇ ψ | ) ∇ ψ ∈ L loc (Ω). Denote Ω R j = Ω T B R j (0), where { R j } ∞ j =1 ∈ R is an increasing sequence and R j → ∞ as j → ∞ . Thenlim k →∞ Z Ω Rj F ′ ( |∇ ψ | ) ∇ ψ · ( ∇ φ k + ∇ ϕ )d x = Z Ω Rj F ′ ( |∇ ψ | ) ∇ ψ · ( ∇ φ + ∇ ϕ )d x. (2.14)For ψ k = φ k + ϕ + ψ , a simple calculation yields that F ( |∇ ψ k | ) − F ( |∇ ψ | ) − F ′ ( |∇ ψ | ) ∇ ψ · ( ∇ φ k − ∇ φ )= Z t∂ p i p j F ( ∇ ψ + (1 − t ) ∇ ( φ k − φ )) ∂ i ( φ k − φ ) ∂ j ( φ k − φ )d t ≥ . (2.15)Similar to (2.14), we havelim k →∞ Z Ω Rj F ′ ( |∇ ψ | ) ∇ ψ · ( ∇ φ k − ∇ φ ) = 0 . So integrating both sides of (2.15) over Ω R j leads tolim inf k →∞ Z Ω Rj F ( |∇ ψ k | ) − F ( |∇ ψ | )d x ≥ . (2.16)8ollecting (2.14)-(2.15) and (2.16) together giveslim inf k →∞ Z Ω Rj [ F ( |∇ ψ k | ) − F ( |∇ ψ | ) − F ′ ( |∇ ψ | ) ∇ ψ · ( ∇ φ k + ∇ ϕ )]d x ≥ Z Ω Rj [ F |∇ ψ | ) − F ( |∇ ψ | ) − F ′ ( |∇ ψ | ) ∇ ψ · ( ∇ φ + ∇ ϕ )]d x. (2.17)It follows from (2.17) that we deducelim inf k →∞ I [ φ k , ϕ , ψ ] ≥ Z Ω Rj F ( |∇ ψ | ) − F ( |∇ ψ | ) − F ′ ( |∇ ψ | ) ∇ ψ · ( ∇ φ + ∇ ϕ )]d x. (2.18)This inequality holds for each R j . Let R j → ∞ and use the monotone convergence theoremto conclude lim inf k →∞ I [ φ k ; ϕ , ψ ] ≥ I [ φ ; ϕ , ψ ] . Therefore we proved (2.13). φ ∈ V is actually a minimizer. Step 3 . The uniqueness of the minimizer φ . For any φ , φ ∈ V , we derive that I [ φ ; ϕ , ψ ] + I [ φ ; ϕ , ψ ] − I [ φ + φ ϕ , ψ ]= Z Ω F ( ∇ ψ ) − F ( ∇ ψ ) − F (cid:18) ∇ ψ + ∇ ψ (cid:19) d x ≥ λ k ψ − ψ k V = λ k φ − φ k V . (2.19)For the uniqueness, suppose φ and φ are two minimizers. Then φ + φ ∈ V , and I [ φ ; ϕ , ψ ] + I [ φ ; ϕ , ψ ] ≤ I (cid:20) φ + φ ϕ , ψ (cid:21) . (2.20)Using (2.19), one has0 ≥ I [ φ ; ϕ , ψ ] + I [ φ ; ϕ , ψ ] − I (cid:20) φ + φ ϕ , ψ (cid:21) ≥ λ k φ − φ k V , (2.21)which implies that the minimizer is unique in V . Step 4 . ψ = φ + ϕ + ψ satisfies (2.8) in the weak sense. It follows from Step 1-3 thatthe functional I has a unique minimizer φ . For any v ∈ C ∞ c (Ω), define I [ φ + τ v ; ϕ , ψ ] for τ >
0. Then I ( τ ) − I (0) = Z Ω [ F ( ∇ φ + τ ∇ v + ∇ ϕ + ∇ ψ | ) − F ( |∇ φ + ∇ ϕ + ∇ ψ | )]d x − τ (cid:18)Z Ω F ′ ( |∇ ψ | ) ∇ ψ · ∇ v d x (cid:19) = V + τ V . F ′ ( |∇ ψ | ) ∇ ψ ) = 0, so V = 0 for any v ∈ C ∞ c (Ω). Hence we deduce that I ( τ ) − I (0) τ = 1 τ Z Ω [ F ( |∇ φ + τ ∇ v + ∇ ϕ + ∇ ψ | ) − F ( |∇ φ + ∇ ϕ + ∇ ψ | )]d x = Z Ω Z F ′ ( |∇ ψ + sτ ∇ v | )( ∇ ψ + sτ ∇ v ) · ∇ v d s d x = Z Ω Z ( F ′ ( |∇ ψ + sτ ∇ v | ) − F ′ ( |∇ ψ | )) ∇ ψ · ∇ v d s d x + τ Z Ω Z F ′ ( |∇ ψ + sτ ∇ v | ) s |∇ v | d s d x + Z Ω F ′ ( |∇ ψ | ) ∇ ψ · ∇ v d s d x = W + W + W . Due to truncation, F ′ is bounded. Then W ≤ τ Λ k v k V , so lim τ → W = 0. By Lebesgueconvergence theorem, we get lim τ → W = 0. Since I ′ (0) = 0, this gives W = 0. Therefore ψ = φ + ϕ + ψ is a weak solution to (2.8). In this section, we verify that the weak solution obtained from the variation in subsection2.2 is classical solution.
Lemma 2.3.
Suppose that ψ ∈ H loc (Ω) is a weak solution to problem (2.2) . Then there is aconstant α ∈ (0 , such that for any subregion Ω ⋐ (Γ ∪ Ω) , there holds sup x ∈ Ω |∇ ψ | + sup x ,x ∈ Ω |∇ ψ ( x ) − ∇ ψ ( x ) || x − x | α ≤ C, (2.22) where C depends on Ω , Ω , γ, ǫ, ρ , κ , κ . Proof : Denote ¯ ψ k = ∂ k ψ for k = 1 ,
2. It is easy to verify that ∂ i ( a ij ∂ j ¯ ψ k ) = 0 , (2.23)where a ij = F ′ ( |∇ ψ | ) δ ij + 2 F ′′ ( |∇ ψ | ) ∂ i ψ∂ j ψ . [ a ij ] has uniformly positive eigenvalues.For the interior estimate, let U be a bounded interior subregion of Ω. It follows fromTheorem 8.24 in [15] that k ¯ ψ k k C α ( U ) ≤ C k ¯ ψ k k L (Ω) . According to (2.9), we know that k ¯ ψ k k L (Ω) is uniformly bounded. From the standard elliptic estimate, we obtain (2.22).Next for the boundary estimate near Γ, one can apply Theorem 8.29 in [15] and followthe arguments above to show (2.22) holds.Then a standard bootstrap argument implies that ψ actually belongs to C ∞ (Ω) and ψ isa classical solution.Now, we use L ∞ bound of gradient to show that |∇ ( ψ − ψ ) | tend to zero as | x | → ∞ with decay rate | x | − in Ω. Lemma 2.4.
There is an estimate of the continuity of ∇ ψ at infinity: |∇ ψ − ∇ ψ | ≤ C (1 + | x | ) . (2.24)10 roof : For a fixed R ≫ N = {| x | ≥ R } ⊂ Ω. Set ¯ ψ = ∂ ψ = − ρu and¯ ψ = ∂ ψ = ρu . Then we have the following elliptic system − a ( x ) ∂ x ¯ ψ = a ( x ) ∂ x ¯ ψ + 2 a ( x ) ∂ x ¯ ψ , ∂ x ¯ ψ = ∂ x ¯ ψ , (2.25)where a ( x ) = F ′ ( |∇ ψ | ) + 2 F ′′ ( |∇ ψ | ) ¯ ψ ,a ( x ) = 2 F ′ ( |∇ ψ | ) + 2 F ′′ ( |∇ ψ | ) ¯ ψ ¯ ψ ,a ( x ) = F ′ ( |∇ ψ | ) + 2 F ′′ ( |∇ ψ | ) ¯ ψ . Applying Theorem 3 in [12], we know that ¯ ψ and ¯ ψ can be represented in the form (cid:18) ¯ ψ ¯ ψ (cid:19) = (cid:18) ¯ ψ ¯ ψ (cid:19) + 1 c x + b x x + a x (cid:18) β x + β x γ x + γ x (cid:19) + (cid:18) O ( | x | − − α ) O ( | x | − − α ) (cid:19) , as | x | → ∞ , (2.26)where 0 < α < lim | x |→∞ ¯ ψ = ¯ ψ , lim | x |→∞ ¯ ψ = ¯ ψ ,c = F ′ (( ¯ ψ + ¯ ψ ) ) + 2 F ′′ (( ¯ ψ + ¯ ψ ) ) ¯ ψ ,b = 2 F ′′ (( ¯ ψ + ¯ ψ ) ) ¯ ψ ¯ ψ ,a = F ′ (( ¯ ψ + ¯ ψ ) ) + 2 F ′′ (( ¯ ψ + ¯ ψ ) ) ¯ ψ ,c ( γ + β ) + 2 b β = 0 , c γ = a β . Since ∇ ψ − ∇ ψ ∈ L (Ω), then it follows from (2.26) that ¯ ψ = ¯ ψ = 0. Thus a = c = F ′ (0), b = 0, γ + β = 0 and γ = β . Hence (2.26) can be rewritten as ∇ ψ = (cid:18) ¯ ψ ¯ ψ (cid:19) = 1 F ′ (0)( x + x ) (cid:26) β (cid:18) x x (cid:19) + β (cid:18) x − x (cid:19)(cid:27) + (cid:18) O ( | x | − − α ) O ( | x | − − α ) (cid:19) , as | x | → ∞ . (2.27)Then ∇ ψ − ∇ ψ ∈ L (Ω) implies that ∇ ψ = 1 F ′ (0)( x + x ) (cid:26) β (cid:18) x x (cid:19) + β (cid:18) x − x (cid:19)(cid:27) , as | x | → ∞ . (2.28)Since ψ = φ + ψ , thus ∇ φ = (cid:18) O ( | x | − − α ) O ( | x | − − α ) (cid:19) , as | x | → ∞ . (2.29)Furthermore, it follows from (2.25) that we derive X i,j =1 a ij ( x ) ∂ x i x j ¯ ψ + X i =1 c i ( x ) ∂ x i ¯ ψ = 0 , (2.30)where c ( x ) = ∂ x a ( x ) − a ( x ) a ( x ) ∂ x a ( x ) ,c ( x ) = 2 ∂ x a ( x ) − a ( x ) a ∂ x a ( x ) .
11e set ¯ φ = ∂ φ and ¯ ψ = ∂ ψ . Then ¯ φ satisfies X i,j =1 a ij ( x ) ∂ x i x j ¯ φ + X i =1 c i ( x ) ∂ x i ¯ φ + X i,j =1 a ij ( x ) ∂ x i x j ¯ ψ + X i =1 c i ( x ) ∂ x i ¯ ψ = 0 (2.31)Introduce the Kelvin’s transform y = R | x | x =: Φ ( x ) , y = R | x | x =: Φ ( x ) , (2.32)and write y = Φ ( x ) , x = Ψ ( y ) , where Ψ = Φ − .Set φ ♯ ( y ) = ¯ φ ( Ψ ( y )) and ψ ♯ ( y ) = ¯ ψ ( Ψ ( y )). Then it follows from (2.29) and (2.32) that φ ♯ ( y ) = O ( | y | α ). Thus φ ♯ (0) = 0 , ∇ φ ♯ (0) = 0 . (2.33)Meanwhile, under this transformation, (2.31) can be written as X i,j =1 a ♯ij ( y ) ∂ y i y j φ ♯ + X i =1 c ♯j ( y ) ∂ y j φ ♯ = f ♯ ( y ) , (2.34)where a ♯ ( y ) = a ( Ψ ( y ))(Φ x ) + 2 a ( Ψ ( y ))Φ x Φ x + a ( Ψ ( y ))(Φ x ) ,a ♯ ( y ) = a ♯ ( y ) = a ( Ψ ( y ))Φ x Φ x + a ( Ψ ( y ))(Φ x Φ x + Φ x Φ x ) + a ( Ψ ( y ))Φ x Φ x ,a ♯ ( y ) = a ( Ψ ( y ))(Φ x ) + 2 a ( Ψ ( y ))Φ x Φ x + a ( Ψ ( y ))(Φ x ) ,c ♯ ( y ) = a ( Ψ ( y ))Φ x x + 2 a ( Ψ ( y ))Φ x x + a ( Ψ ( y ))Φ x x + X i =1 (cid:18) ∂ y i a ( Ψ ( y )) − a ( Ψ ( y )) a ( Ψ ( y )) ∂ y i a ( Ψ ( y )) (cid:19) Φ x Φ ix + 2 X i =1 (cid:18) ∂ y i a ( Ψ ( y )) − a ( Ψ ( y )) a ( Ψ ( y )) ∂ y i a ( Ψ ( y )) (cid:19) Φ ix Φ x ,c ♯ ( y ) = a ( Ψ ( y ))Φ x x + 2 a ( Ψ ( y ))Φ x x + a ( Ψ ( y ))Φ x x + X i =1 (cid:18) ∂ y i a ( Ψ ( y )) − a ( Ψ ( y )) a ( Ψ ( y )) ∂ y i a ( Ψ ( y )) (cid:19) Φ x Φ ix + 2 X i =1 (cid:18) ∂ y i a ( Ψ ( y )) − a ( Ψ ( y )) a ( Ψ ( y )) ∂ y i a ( Ψ ( y )) (cid:19) Φ ix Φ x ,f ♯ ( y ) = − X i,j =1 a ♯ij ( y ) ∂ y i y j ψ ♯ − X i =1 c ♯j ( y ) ∂ y j ψ ♯ . Applying Schauder estimates in [15], we deduce that | φ ♯ | ,α ; B R/ ≤ C ( | φ ♯ | B R + | f ♯ | ,α ; B R ) . (2.35)12ince ∂ i ( a ij ∂ j ¯ φ )+ ∂ i ( a ij ∂ j ¯ ψ ) = 0, we can apply Theorem 8.15 in [15] to derive that | ¯ φ | N ≤ C . By substituting | φ ♯ | B R = | ¯ φ | N ≤ C into (2.35), we infer that | φ ♯ | ,α ; B R/ ≤ C. (2.36)Combing (2.33), we obtain | φ ♯ | ≤ C | y | , in B R/ . (2.37)Hence | ¯ φ | ≤ C | x | , in {| x | ≥ R } . (2.38)Thus (2.24) can be obtained from (2.38). For the modified equation (2.2), we have the following theorem:
Theorem 2.5.
For every ψ , (2.2) has a unique classical spiral solution ψ such that ψ = φ + ϕ + ψ with φ ∈ V . Furthermore, for each fixed κ , ∇ ψ depends on κ continuously. Inparticular, max Ω |∇ ψ | is a continuous function of | κ | . For each fixed κ , we have the similarconclusion. Proof : The existence follows from Theorem 2.2 and the regularity estimate followsLemma 2.3 and 2.4. To prove the uniqueness, we assume that classical solutions ψ i = φ i + ϕ + ψ , i = 1 , φ i ∈ V would be both critical points of I [ φ ; ϕ , ψ ]. Let I ′ φ be the Fr´echet derivative. Then we have0 = ( I ′ φ ( φ ; ϕ , ψ ) − I ′ φ ( φ ; ϕ , ψ ) , φ − φ )= Z Ω [ ∂ p F ( ∇ φ + ϕ + ψ ) − ∂ p F ( ∇ φ + ∇ ϕ + ∇ ψ )]( ∇ φ − ∇ φ )d x = Z Ω Z ∂ p i p j F ( t ∇ φ + (1 − t ) ∇ φ + ∇ ϕ + ∇ ψ ) ∂ i ( φ − φ ) ∂ j ( φ − φ )d t d x ≥ λ k φ − φ k V . Hence φ = φ .Now we examine the continuity of I [ φ ; ϕ , ψ ]. Let ψ i = φ i + ϕ + ψ , i = 1 ,
2, one has F ( ∇ φ + ∇ ϕ + ∇ ψ ) − F ( ∇ ψ ) − ∂ p F ( ∇ ψ )( ∇ φ + ∇ ϕ ) − [ F ( ∇ φ + ∇ ϕ + ∇ ψ ) − F ( ∇ ψ ) − ∂ p F ( ∇ ψ )( ∇ φ + ∇ ϕ )]= Z ∂ p F ( t ∇ φ + (1 − t ) ∇ φ + ∇ ϕ + ∇ ψ )d t ( ∇ φ − ∇ φ ) − ∂ p F ( ∇ ψ )( ∇ φ − ∇ φ )= Z Z ∂ p i p j F ( st ∇ φ + s (1 − t ) ∇ φ + s ∇ ϕ + ∇ ψ )( t∂ i φ + (1 − t ) ∂ i φ + ∇ ϕ )d s d t × ( ∂ j φ − ∂ j φ ) . | I [ φ ; ϕ , ψ ] − I [ φ ; ϕ , ψ ] | ≤ C (1 + k φ k V + k φ k V )( k φ − φ k V ) . (2.39)Since ψ = ψ + ψ , ϕ = − ζψ , so the continuity of I [ φ ; ϕ , ψ ] on ψ and ψ followsfrom the equality below I [ φ ; ϕ , ψ ] = Z t∂ p i p j F ( ∇ ψ + (1 − t )( ∇ φ + ∇ ϕ ) ∂ i ( φ + ϕ ) ∂ j ( φ + ϕ )d t = Z t∂ p i p j F ( ∇ ψ + ∇ ψ + (1 − t )( ∇ φ − ∇ ( ζψ ))) ∂ i ( φ − ζψ ) ∂ j ( φ − ζψ )d t. (2.40)Now, for each fixed κ , we will prove the continuous of the solution on κ . Let κ m be aconvergent sequence such that κ m → κ ∗ as m → ∞ . Denote ψ m = φ m + ϕ m + ψ + ψ m = φ m + ψ + (1 − ζ ) ψ m , where ψ m = − κ m R r ρ b ( s ) s d s . First we show that φ m ⇀ φ ∗ in V byusing that φ m is a minimizing sequence of I [ φ ; − ζψ ∗ , ψ + ψ ∗ ]. Since κ m → κ ∗ as m → ∞ ,so (1 − ζ ) ψ m → (1 − ζ ) ψ ∗ , where ψ ∗ = − κ ∗ R r ρ b ( s ) s d s . All estimates in Lemma 2.3 and2.4 as well as (2.9) can be taken uniformly. In particular, R Ω |∇ φ m | d x and max Ω |∇ φ m | areuniformly bounded.For any given δ > m , it follows from (2.39) and (2.40) that wehave ( | I [ φ m ; − ζψ m , ψ + ψ m ] − I [ φ m ; − ζψ ∗ , ψ + ψ ∗ ] | < δ, | I [ φ ∗ ; − ζψ m , ψ + ψ m ] − I [ φ ∗ ; − ζψ ∗ , ψ + ψ ∗ ] | < δ. Combing with the minimality of φ m for I [ φ ; − ζψ m , ψ + ψ m ], we derive that I [ φ m ; − ζψ ∗ , ψ + ψ ∗ ] ≤ I [ φ m ; − ζψ m , ψ + ψ m ] + δ ≤ I [ φ ∗ ; − ζψ ∗ , ψ + ψ ∗ ] + 2 δ. Therefore, φ m is a minimizing sequence of I [ φ ; − ζψ ∗ , ψ + ψ ∗ ]. By the proof of existence ofminimizer, we know that φ m ⇀ φ ∗ in V . The uniformly convergence of ∇ ψ m to ∇ ψ ∗ followsfrom uniform estimates in Lemma 2.3, 2.4 and a contradiction argument. Then we concludethat max Ω |∇ ψ m | → max Ω |∇ ψ ∗ | . Hence max Ω |∇ ψ | is a continuous function of | κ | .Finally, for each fixed κ , we will prove the continuous of the solution on κ . Let κ n be aconvergent sequence such that κ n → κ ⋆ as n → ∞ . Denote ψ n = φ n + ϕ + ψ n + ψ . Similarto the above proof, we have max Ω |∇ ψ n | → max Ω |∇ ψ ⋆ | . Hence max Ω |∇ ψ | is a continuousfunction of | κ | . In this subsection, we remove the truncation and complete the proof of Theorem 1.1 andTheorem 1.3. We only need prove Theorem 1.1, the proof of Theorem 1.3 is similar.Up to now, we have shown for fixed parameter ε , (2.2) has a unique classical solution ψ . Let { ε i } ∞ i =1 be a strictly decreasing sequence such that ε i → i → ∞ . For fixed κ and i , there exists a maximum interval [0 , κ i ) such that for | κ | ∈ [0 , κ i ), |∇ ψ | < √ − ε i ,Then ψ is the solution to the origin equation (1.18). From the uniqueness of (2.2), we cansee κ i ≤ κ j for i ≤ j . So { κ i } ∞ i =1 is an increasing sequence with the upper bound 1, whichimplies the convergence of the sequence. Therefore, we haveˆ κ = lim i →∞ κ i . (2.41)14f κ i < ˆ κ for any i , then for any | κ | ∈ [0 , ˆ κ ), there exists an index i such that | κ | ∈ [0 , κ i ), so the truncation can be removed such that ψ is the classical solution of (1.18) andsup x ∈ Ω |∇ ψ | < ψ i = φ i + ϕ + ψ , i = 1 , φ ∈ V and max Ω |∇ ψ i | <
1. A small ε canbe picked such that max Ω |∇ ψ | < √ − ε . Both solutions will be solutions to the modifiedequation (2.2) with that ε . It follows from the uniqueness of (2.2) that ψ = ψ . Thereforethe proof of Theorem 1.1 is completed.Moreover, we have max Ω |∇ ψ | → | κ | → ˆ κ . It is expected that subsonic spiral flowswill tend to some subsonic-sonic spiral flows as | κ | → ˆ κ . we will study this limiting behaviorby compensated compactness framework. In this section, similar to the subsonic case, we only need prove Theorem 1.2. Firstly, letus recall the compensated compactness framework for steady irrotational flows in [18].
Theorem 3.1.
Let u ǫ ( x , x ) = ( u ǫ , u ǫ )( x , x ) be sequence of functions satisfying the fol-lowing set of conditions ( A ) : (A . q ǫ ( x , x ) = | u ǫ ( x , x ) | ≤ a.e. in Ω . (A .
2) curl u ǫ and div ( g ( q ǫ ) ) u ǫ ) are confined in a compact set in H − loc (Ω) .Then there exists a subsequence (still labeled) u ǫ that converges a.e. as ǫ → to ˆu satisfying ˆ q ( x , x ) = | ˆu ( x , x ) | ≤ a.e. ( x , x ) ∈ Ω . Let ( ρ ǫ , u ǫ , u ǫ ) denote the solutions obtained in Theorem 1.1 to problem ( κ , κ ǫ ). Thenwe have (cid:26) ∂ x (( g ( q ǫ ) ) u ǫ ) + ∂ x (( g ( q ǫ ) ) u ǫ ) = 0 ,∂ x u ǫ − ∂ x u ǫ = 0 . (3.1)Thus conditions ( A
1) and ( A
2) in Theorem 3.1 are all satisfied. Theorem 3.1 implies thatthe solution sequence has a subsequence (still denoted by) ( ρ ǫ , u ǫ , u ǫ ) that converges a.e. toa vector function (ˆ ρ, ˆ u , ˆ u ). Then the boundary condition are satisfied for ˆ ρ ˆ u in the sense ofChen-Frid [5]. Since (1.1) and (1.1) hold for the sequence of subsonic solutions ( ρ ǫ , u ǫ , u ǫ ),so it is easy to see that (ˆ ρ, ˆ u , ˆ u ) also satisfies (1.1) and (1.1) in the sense of distribution.Thus the proof of Theorem 1.2 is completed. Acknowledgement.
Weng is partially supported by National Natural Science Founda-tion of China 11701431, 11971307, 12071359.
References [1] Bers, L.: Existence and uniqueness of a subsonic flow past a given profile. Comm. PureAppl. Math., 7 (1954) 441-504.[2] Chen, C., Du, L., Xie, C., Xin, Z.: Two dimensional subsonic Euler flows past a wall ora symmetric body, Arch. Ration. Mech. Anal. 221(2), 559-602 (2016).153] Cui, D.-C., Li, J.: On the existence and stability of 2-D perturbed steady subsonic circu-latory flows. Sci. China Math. 54 (2011), no. 7, 1421-1436.[4] Chen, G.-Q., Dafermos, C. M., Slemrod, M., Wang, D.-H.: On two-dimensional sonic-subsonic flow, Commun. Math. Phys., 271 (2007), 635-647.[5] Chen, G.-Q., Frid, H.: Divergence-measure fields and hyperbolic conservation laws, Arch.Rational. Mech. Anal., 147 (1999), 89-118.[6] Chen, G.-Q., Huang, F.-M., Wang, T.-Y.: Subsonic-sonic limit of approximate solutionsto multidimensional steady Euler equations. Arch. Ration. Mech. Anal. 219 (2016), no.2, 719-740.[7] Courant, R., Friedrichs, K.O.: Supersonic Flow and Shock Waves, Interscience PublishersInc.: New York, 1948.[8] Duan, B., Weng, S.: Global smooth axisymmetric subsonic flows with nonzero swirl in aninfinitely long axisymmetric nozzle. Z. Angew. Math. Phys. 69 (2018), no. 5, Paper No.135, 17 pp.[9] Dong, G.-C., Ou, B.: Subsonic flows around a body in space, Comm. Partial DifferentialEquations, 18 (1993) 355-379.[10] Du, L., Xin, Z., Yan, W.: Subsonic flows in a multi-dimensional nozzle, Arch. Ration.Mech. Anal. 201 (2011) 965-1012.[11] Du, L., Xie, C., Xin, Z.: Steady subsonic ideal flows through an infinitely long nozzlewith large vorticity, Commun. Math. Phys. 328 (2014) 327-354.[12] Finn, R., Gilbarg, D.: Asymptotic behavior and uniqueness of plane subsonic flows,Comm. Pure Appl. Math., 10 (1957), 23-63.[13] Finn, R., Gilbarg, D.: Three-dimensional subsonic flows and asymptotic estimates forelliptic partial differential equations, Acta Math., 98 (1957) 265-296.[14] Frankl, F., Keldysh, M.: Die¨ a ussere neumann’she aufgabe f¨ uu