Global Entropy Solutions to the Gas Flow in General Nozzle
aa r X i v : . [ m a t h . A P ] A ug GLOBAL ENTROPY SOLUTIONS TO THE GAS FLOW IN GENERALNOZZLE
WENTAO CAO, FEIMIN HUANG, AND DIFAN YUAN
Abstract.
We are concerned with the global existence of entropy solutions for thecompressible Euler equations describing the gas flow in a nozzle with general cross-sectional area, for both isentropic and isothermal fluids. New viscosities are delicatelydesigned to obtain the uniform bound of approximate solutions. The vanishing viscositymethod and compensated compactness framework are used to prove the convergence ofapproximate solutions. Moreover, the entropy solutions for both cases are uniformlybounded independent of time. No smallness condition is assumed on initial data. Thetechniques developed here can be applied to compressible Euler equations with generalsource terms. : 35L45, 35L60, 35Q35.
Key words : isentropic gas flow, isothermal gas flow, compensated compactness, uniformestimate, independent of time. 1.
Introduction
We consider one dimensional gas flow in a general nozzle for the isentropic and isother-mal flows separately. The nozzle is widely used in some types of steam turbines, rocketengine nozzles, supersonic jet engines, and jet streams in astrophysics. The motion of thenozzle flow is governed by the following system of compressible Euler equations: ρ t + m x = a ( x ) m, x ∈ R , t > ,m t + (cid:18) m ρ + p ( ρ ) (cid:19) x = a ( x ) m ρ , x ∈ R , t > , (1.1)where ρ is the density, the momentum m = ρu with u being the velocity, and p ( ρ ) is thepressure of the gas. Here the given function a ( x ) is represented by a ( x ) = − A ′ ( x ) A ( x ) with A ( x ) ∈ C ( R ) being a slowly variable cross-sectional area at x in the nozzle. For γ -lawgas, p ( ρ ) = p ρ γ with γ denoting the adiabatic exponent and p = θ γ , θ = γ − . When γ > , (1.1) is called the isentropic gas flow. When γ = 1 , (1.1) is called isothermal one.We consider the Cauchy problem for (1.1) with large initial data( ρ, m ) | t =0 = ( ρ ( x ) , m ( x )) ∈ L ∞ . (1.2)The above Cauchy problem (1.1)-(1.2) can be written in compact form as follows: (cid:26) U t + f ( U ) x = g ( x, U ) ,U | t =0 = U ( x ) , x ∈ R , (1.3) Date : August 20, 2019.2000
Mathematics Subject Classification.
Key words and phrases. isentropic flow, isothermal flow, compensated compactness, uniform estimate. where U = ( ρ, m ) ⊤ , f ( U ) = ( m, m ρ + p ( ρ )) ⊤ , and g ( x, U ) = ( − A ′ ( x ) A ( x ) m, − A ′ ( x ) A ( x ) m ρ ) ⊤ . There have been extensive studies and applications of homogeneous γ -law gas, i.e., g ( x, U ) = 0. Diperna [9] proved the global existence of entropy solutions with largeinitial data by the theory of compensated compactness and vanishing viscosity methodfor γ = 1 + n +1 , where n is a positive integer. Subsequently, Ding, Chen, and Luo [6, 7]and Chen [1] successfully extended the result to γ ∈ (1 , ] by using a Lax-Friedrichsscheme. Lions, Perthame, and Tadmor [17] and Lions, Perthame, and Souganidis [18]treated the case γ > . The existence of entropy solutions to the isothermal gas, i.e., γ = 1, was proved in Huang and Wang [14] by introducing complex entropies and utilizingthe analytic extension method.For the isentropic Euler equations with source term, Ding, Chen, and Luo [8] establisheda general framework to investigate the global existence of entropy solution through thefractional step Lax-Friedrichs scheme and compensated compactness method. Later on,there have been extensive studies on the inhomogeneous case (see [2, 3, 16, 23, 24, 30, 31]).For the nozzle flow problem, see [5, 10–12, 19–21, 33]. For converging-diverging de Lavalnozzles, as flow speed accelerates from the subsonic to the supersonic regime, the physicalproperties of nozzle and diffuser flows are altered. This kind of nozzle is particularlydesigned to converge to a minimum cross-sectional area and then expand. Liu [19] firstproved the existence of a global solution with initial data of small total variation andaway from sonic state by a Glimm scheme. Tsuge [27–29] first studied the global existenceof solutions for Laval nozzle flow and transonic flow for large initial data by introducinga modified Godunov scheme. Recently, Chen and Schrecker [4] proved the existence ofglobally defined entropy solutions in transonic nozzles in an L p compactness framework,whose uniform bound of approximate solutions may depend on time t . In our paper, weare focusing on the L ∞ compactness framework. Moreover, general cross-sectional areasof nozzles are considered, which include several important physical models, such as thede Laval nozzles with closed ends, that is, the cross-sectional areas are tending to zero as x → ∞ . In our paper, we assume the cross-sectional area function A ( x ) > C , function a ( x ) ∈ L ( R ) such that (cid:12)(cid:12)(cid:12)(cid:12) A ′ ( x ) A ( x ) (cid:12)(cid:12)(cid:12)(cid:12) = | a ( x ) | ≤ a ( x ) . (1.4)Here, A ( x ) > a ( x ) . The main purpose of the present paper is to prove the existence of a global entropy so-lution with uniform bound independent of time for large initial data in both the isentropiccase 1 < γ < γ = 1. We are interested in solutions that can reachthe vacuum ρ = 0 . Near the vacuum, the system (1.1)-(1.2) is degenerate and the velocity u cannot be defined uniquely. We define the weak entropy solution as follows. Definition 1.1.
A measurable function U ( x, t ) is called a global weak solution of theCauchy problem (1.3) if Z t> Z R U ϕ t + f ( U ) ϕ x + g ( x, U ) ϕdxdt + Z R U ( x ) ϕ ( x, dx = 0 ENERAL NOZZLE FLOW 3 holds for any test function ϕ ∈ C ( R × R + ). In addition, for the isentropic flow, if U alsosatisfies that for any weak entropy pair ( η, q ) (see Section 2), the inequality η ( U ) t + q ( U ) x − ∇ η ( U ) · g ( x, U ) ≤ U is called a weak entropy solution to (1.3). Forthe isothermal flow, U is called a weak entropy solution if U additionally satisfies (1.5) formechanical entropy pair η ∗ = m ρ + ρ ln ρ, q ∗ = m ρ + m ln ρ. Two main results of the present paper are given as follows.
Theorem 1.1. (isentropic case) Let < γ < . Assume that there is a positive constant M such that the initial data satisfies ≤ ρ ( x ) ≤ M, | m ( x ) | ≤ M ρ ( x ) , a.e., x ∈ R , and a ( x ) satisfies (1.4) with k a ( x ) k L ( R ) ≤ − θ θ . (1.6) Then, there exists a global entropy solution of (1.1) - (1.2) satisfying ≤ ρ ( x, t ) ≤ C, | m ( x, t ) | ≤ Cρ ( x, t ) , a.e., ( x, t ) ∈ R × R + , where C depends only on initial data and is independent of time t . Theorem 1.2. (isothermal case) Let γ = 1 . Assume that there is a positive constant M such that the initial data satisfy ≤ ρ ( x ) ≤ M, | m ( x ) | ≤ ρ ( x )( M + | ln ρ ( x ) | ) , a.e., x ∈ R , and a ( x ) satisfies (1.4) with k a ( x ) k L ( R ) ≤ . (1.7) Then, there exists a global entropy solution of (1.1) - (1.2) satisfying ≤ ρ ( x, t ) ≤ C, | m ( x, t ) | ≤ ρ ( x, t )( C + | ln ρ ( x, t ) | ) , a.e., x ∈ R , where C depends only on initial data and is independent of time t .Remark . Here, the conditions (1.6) (Theorem 1.1) and (1.7) (Theorem 1.2) are assumedto guarantee a uniform bound of ( ρ, m ) independent of time. This condition illustrates anew physical phenomena that is important in engineering. For example, if we consider anisothermal nozzle with a monotone cross-sectional area, a ( x ) = A ′ ( x ) A ( x ) ≥ , and denote A + and A − the far field of a variable cross-sectional area, respectively, then the ratio of theoutlet and inlet cross-sectional area can be controlled, i.e., A + A − ≤ e . Remark . The condition (1.6) in Theorem 1.1 is different from that in Tsuge [29]. Here,in our paper, we allow 1 < γ < . The main difficulty we came across is how to construct approximate solutions withuniform bound independent of time. Another difficulty is the interaction of nonlinearresonance between the characteristic modes and geometrical source terms. Our strategyis applying the maximum principle (Lemma 3.1) introduced in [13, 15], which is similar
WENTAO CAO, FEIMIN HUANG, AND DIFAN YUAN to invariant region theory [26], to a viscous equation with novel viscosity. To be morespecific, for the isentropic case, we add − εb ( x ) ρ x on the momentum equation (c.f (3.1));for the isothermal case, we raise n := A ( x ) ρ with δ and also add − εb ( x ) n x on themomentum equation (c.f (5.1)). Two modified Riemann invariants are introduced and asystem of decoupled new parabolic equations along the characteristic are derived. Owingto the hyperbolicty structure of (1.1), we can transform the integral of source termsalong characteristics with time t into the integral with space x. Finally after establishingthe estimate of H − loc compactness, we apply a compensated compactness framework in[6,7,14,18] to show the convergence of approximate solutions. To the best of our knowledge,for the isothermal flow, the uniform bound for the approximate solutions depends on time t in all the previous results. We remark that the method in our paper can be applied toobtain the existence of weak solutions of related gas dynamic models, such as Euler-Poissonfor a semiconductor model [15] or an Euler equation with geometric source terms [13], andmay also shed light on the large time behavior of entropy solutions. Besides, we avoid alaborious numerical scheme to construct approximate solutions.The present paper is organized as follows. In Section 2, we introduce some basic notionsand formulas for the isentropic Euler system. In Section 3, we prove Theorem 1.1 for theglobal existence of isentropic gas flow in general nozzle. Subsequently, in Section 4, wefurther formulate several preliminaries and formula for the isothermal Euler system. Theproof of Theorem 1.2 for global existence of isothermal gas flow in general nozzle willbe presented in Section 5. In the appendix, we provide the proof of variant version ofinvariant region theory for completeness.2. Preliminary and Formulation for Isentropic Flow
First we list some basic notation for the isentropic system (1.1). The eigenvalues are λ = mρ − θρ θ , λ = mρ + θρ θ , and the corresponding right eigenvectors are r = (cid:20) λ (cid:21) , r = (cid:20) λ (cid:21) . The Riemann invariants w, z are given by w = mρ + ρ θ , z = mρ − ρ θ , (2.1)satisfying ∇ w · r = 0 and ∇ z · r = 0. A pair of functions ( η, q ) : R + × R R is definedto be an entropy-entropy flux pair if it satisfies ∇ q ( U ) = ∇ η ( U ) ∇ " m m ρ + p ( ρ ) . When η (cid:12)(cid:12)(cid:12) mρ fixed → , as ρ → ,η ( ρ, m ) is called weak entropy. In particular, the mechanical entropy pair η ∗ ( ρ, m ) = m ρ + p ρ γ γ − , q ∗ ( ρ, m ) = m ρ + γp ρ γ − mγ − ENERAL NOZZLE FLOW 5 is a strictly convex entropy pair. As shown in [17] and [18], any weak entropy for thesystem (1.1) is given by η = ρ Z − χ ( mρ + ρ θ s )(1 − s ) λ ds, q = ρ Z − ( mρ + ρ θ θs ) χ ( mρ + ρ θ s )(1 − s ) λ ds (2.2)with λ = − γ γ − for any function χ ( · ) ∈ C ( R ).3. Proof of Theorem 1.1
Construction of approximate solutions.
We first construct approximate solu-tions to (1.1) satisfying the framework in [6, 7, 18]. Indeed, for any ε ∈ (0 ,
1) we constructapproximate solutions by adding suitable artificial viscosity as follows: ρ t + m x = a ( x ) m + ερ xx ,m t + (cid:18) m ρ + p ( ρ ) (cid:19) x = a ( x ) m ρ + εm xx − εb ( x ) ρ x (3.1)with initial data ( ρ, m ) | t =0 = ( ρ ε ( x ) , m ε ( x )) = ( ρ ( x ) + ε, m ( x )) ∗ j ε , (3.2)where b ( x ) is a function to be given later, and j ε is the standard mollifier.3.2. Global existence of approximate solutions.
For the global existence to Cauchyproblem (3.1)-(3.2), we have the following.
Theorem 3.1.
For any time
T > , there exists a unique global classical bounded solutionto the Cauchy problem (3.1) - (3.2) that has following L ∞ estimates e − C ( ε,T ) ≤ ρ ε ( x, t ) ≤ C, | m ε ( x, t ) | ≤ Cρ ε ( x, t ) . (3.3)We shall show Theorem 3.1 in two steps. In the section, we omit the upper index ε forsimplicity. Step 1. Uniform upper bound.
First, we can rewrite the first equation of (3.1) as ρ t + uρ x = ερ xx + ρ ( a ( x ) u − u x ) , and then applying the maximum principle of parabolic equation yields that ρ ≥ min ρ ( x ) e − R t k a ( x ) u − u x k L ∞ ds > , which implies w ≥ z. Second, we recall a revised version of the invariant region theory [26]introduced in [13, 15].
Lemma 3.1. (Maximum principle) Let p ( x, t ) , q ( x, t ) , ( x, t ) ∈ R × [0 , T ] be any boundedclassical solutions of the quasilinear parabolic system (cid:26) p t + µ p x = εp xx + a p + a q + R ,q t + µ q x = εq xx + a p + a q + R (3.4) with initial data p ( x, ≤ , q ( x, ≥ , where µ i = µ i ( x, t, p ( x, t ) , q ( x, t )) , a ij = a ij ( x, t, p ( x, t ) , q ( x, t )) , and the source terms R i = R i ( x, t, p ( x, t ) , q ( x, t ) , p x ( x, t ) , q x ( x, t )) , i, j = 1 , , ∀ ( x, t ) ∈ R × [0 , T ] WENTAO CAO, FEIMIN HUANG, AND DIFAN YUAN µ i , a ij are bounded with respect to ( x, t, p, q ) ∈ R × [0 , T ] × K, where K is an arbitrarycompact subset in R , a , a , R , R are continuously differentiable with respect to p, q. Assume the following conditions hold: (C1):
When p = 0 and q ≥ , there is a ≤ when q = 0 and p ≤ , there is a ≤ . (C2): When p = 0 and q ≥ , there is R = R ( x, t, , q, ζ, η ) ≤ when q = 0 and p ≤ , there is R = R ( x, t, p, , ζ, η ) ≥ . Then for any ( x, t ) ∈ R × [0 , T ] , p ( x, t ) ≤ , q ( x, t ) ≥ . Remark . The modified version of invariant region theory (Lemma 3.1) is valid notonly for the Cauchy problem with source terms, but also for the initial boundary valueproblem with Dirichlet and Neumann boundary conditions.We shall apply maximum principle Lemma 3.1 to get the uniform bound of ρ, m . Bythe formulas of Riemann invariants (2.1), the viscous perturbation system (3.1) can betransformed as w t + λ w x = εw xx + 2 ε ( w x − b ) ρ x ρ − εθ ( θ + 1) ρ θ − ρ x + θ w − z a ( x ) ,z t + λ z x = εz xx + 2 ε ( z x − b ) ρ x ρ + εθ ( θ + 1) ρ θ − ρ x − θ w − z a ( x ) . (3.5)Set the control functions ( φ, ψ ) as φ = C + ε k b ′ ( x ) k L ∞ t + Z x −∞ b ( y ) dy,ψ = C + ε k b ′ ( x ) k L ∞ t + Z ∞ x b ( y ) dy. Then a simple calculation shows that φ t = ε k b ′ ( x ) k L ∞ , φ x = b ( x ) , φ xx = b ′ ( x ); ψ t = ε k b ′ ( x ) k L ∞ , ψ x = − b ( x ) , ψ xx = − b ′ ( x ) . Define the modified Riemann invariants ( ¯ w, ¯ z ) as¯ w = w − φ, ¯ z = z + ψ. (3.6)Inserting (3.6) into (3.5) yields the decoupled equations for ¯ w and ¯ z : ¯ w t + λ ¯ w x = ε ¯ w xx + εφ xx − φ t − λ φ x + 2 ε ρ x ρ ¯ w x − εθ ( θ + 1) ρ θ − ρ x + θ ( ¯ w + φ ) − (¯ z − ψ ) a ( x ) , ¯ z t + λ ¯ z x = ε ¯ z xx − εψ xx + ψ t + λ ψ x + 2 ε ρ x ρ ¯ z x + εθ ( θ + 1) ρ θ − ρ x − θ ( ¯ w + φ ) − (¯ z − ψ ) a ( x ) . (3.7) ENERAL NOZZLE FLOW 7
Noting that λ = w + z − θ w − z ,λ = w + z θ w − z , the system (3.7) becomes ¯ w t + ( λ − ε ρ x ρ ) ¯ w x = ε ¯ w xx + a ¯ w + a ¯ z + R , ¯ z t + ( λ − ε ρ x ρ )¯ z x = ε ¯ z xx + a ¯ w + a ¯ z + R , (3.8)where a = − (cid:18) θ φ x − θ ¯ w + 2 φ a ( x ) (cid:19) , a = − (cid:18) − θ φ x + θ ¯ z − ψ a ( x ) (cid:19) ,a = (cid:18) − θ ψ x − θ ¯ w + 2 φ a ( x ) (cid:19) , a = (cid:18) θ ψ x + θ ¯ z − ψ a ( x ) (cid:19) , and R = εφ xx − φ t − θ φφ x + 1 − θ ψφ x − εθ ( θ + 1) ρ θ − ρ x + θ φ − ψ a ( x ) ,R = − εψ xx + ψ t + 1 − θ φψ x − θ ψψ x + εθ ( θ + 1) ρ θ − ρ x − θ φ − ψ a ( x ) . To apply Lemma 3.1, we need to verify (C1) and (C2) . For (C1) , when ¯ w = 0 , ¯ z ≥ , wehave 0 ≤ ¯ z = z + ψ ≤ w + ψ = φ + ψ, and then a = − − θ (cid:18) b ( x ) + θ − θ ) (¯ z − ψ ) a ( x ) (cid:19) ≤ − − θ (cid:18) b ( x ) − θ − θ ) ( φ − ψ ) | a ( x ) | (cid:19) if a ( x ) < , − − θ (cid:18) b ( x ) − θ − θ ) 2 ψ | a ( x ) | (cid:19) if a ( x ) ≥ . Hence, we take b ( x ) = M a ( x ) with M ≥ θ − θ ) max( φ − ψ, ψ ) , and using (1.4), one has a ≤ . Moreover, when ¯ w ≤ , ¯ z = 0 , we have0 ≥ ¯ w = w − φ ≥ z − φ = − ψ − φ, WENTAO CAO, FEIMIN HUANG, AND DIFAN YUAN and then a = − − θ (cid:18) b ( x ) + θ − θ ) ( ¯ w + 2 φ ) a ( x ) (cid:19) ≤ − − θ (cid:18) b ( x ) − θ − θ ) 2 φ | a ( x ) | (cid:19) if a ( x ) < , − − θ (cid:18) b ( x ) − θ − θ ) ( ψ − φ ) | a ( x ) | (cid:19) if a ( x ) ≥ , ≤ , provided M ≥ θ − θ ) max( φ − ψ, φ ) . Thus we require M ≥ θ − θ ) max (cid:18)Z x −∞ b ( y ) dy − Z ∞ x b ( y ) dy, C + 2 ε k b ′ ( x ) k L ∞ t + 2 Z ∞−∞ b ( y ) dy (cid:19) . Taking ε sufficient small such that ε k b ′ ( x ) k L ∞ T ≤ , we have M ≥ θ − θ ( C + 1 + M k a ( x ) k L ) , that is, k a ( x ) k L ≤ − θθ − C + 1 M . (3.9)Hence (C1) is also satisfied by ( ¯ w, ¯ z ) . As for (C2) , one can derive R ≤ εb ′ ( x ) − ε k b ′ ( x ) k L ∞ + b ( x ) (cid:18) − θ φ + 1 − θ ψ (cid:19) + θ ( φ + ψ )( φ − ψ )4 a ( x ) ≤ b ( x ) (cid:18) − θC − εθ k b ′ ( x ) k L ∞ t − θ Z x −∞ b ( y ) dy + 1 − θ Z ∞ x b ( y ) dy (cid:19) + θ C + 2 ε k b ′ ( x ) k L ∞ t + k b k L ) (cid:18)Z x −∞ b ( y ) dy − Z ∞ x b ( y ) dy (cid:19) a ( x ) ≤ − (cid:20) M ( θC + θε k b ′ ( x ) k L ∞ t − − θ k b k L ) − θ C + 2 ε k b ′ ( x ) k L ∞ t + k b k L ) k b k L (cid:21) a ( x ) ≤ − M a ( x ) (cid:20) θC −
12 ( θC + (1 − θ ) M + θ M k a k L ) k a k L (cid:21) ≤ . The last inequality holds on the condition that k a ( x ) k L ≤ θC θC + M and k a ( x ) k L ≤ . (3.10) ENERAL NOZZLE FLOW 9
Then we also have R ≥ εb ′ ( x ) + ε k b ′ ( x ) k L ∞ + b ( x ) (cid:18) − − θ φ + 1 + θ ψ (cid:19) − θ ( φ + ψ )( φ − ψ )4 a ( x ) ≥ b ( x ) (cid:18) θC + εθ k b ′ ( x ) k L ∞ t − − θ Z x −∞ b ( y ) dy + 1 + θ Z ∞ x b ( y ) dy (cid:19) − θ C + 2 ε k b ′ ( x ) k L ∞ t + k b k L ) (cid:18)Z x −∞ b ( y ) dy − Z ∞ x b ( y ) dy (cid:19) a ( x ) ≥ (cid:20) M ( θC + θε k b ′ ( x ) k L ∞ t − − θ k b k L ) − θ C + 2 ε k b ′ ( x ) k L ∞ t + k b k L ) k b k L (cid:21) a ( x ) ≥ M a ( x ) (cid:20) θC −
12 ( θC + (1 − θ ) M + θ M k a k L ) k a k L (cid:21) ≥ . Hence (C2) is verified for ( ¯ w, ¯ z ). From (3.9) and (3.10), a must satisfy k a k L ≤ min (cid:26) , θC θC + M , − θθ − C + 1 M (cid:27) . (3.11)Now we turn to choose M and C . Considering the initial values of approximate solutions,we shall choose C large enough first such that C ≥ max { sup w ( x, , − inf z ( x, } , and then we have w ( x, ≤ φ ( x, , z ( x, ≥ − ψ ( x, . One choice of M is M = C θ + θ − θ , and then 2 θC θC + M = 1 − θ θ and 1 − θθ − C M − M ≥ − θ θ if M is large enough. Thus our condition (1.6) on a satisfies (3.11), which is the keyreason for (1.6). Therefore, an application of Lemma 3.1 yields¯ w ( x, t ) ≤ , ¯ z ( x, t ) ≥ , which implies w ( x, t ) ≤ φ ( x, t ) ≤ C + k b k L + 1 = C,z ( x, t ) ≥ − ψ ( x, t ) ≥ − C − k b k L − − C, where we can see that C is independent of time. Hence we obtain0 ≤ ρ ( x, t ) ≤ C, | m ( x, t ) | ≤ Cρ ( x, t ) . (3.12) Step 2. Lower bound of density.
By (3.12), we know that the velocity u = mρ isuniformly bounded, i.e., | u | ≤ C . Then the lower bound of density can be derived by the method of [13]. Set v = ln ρ , and then we get a scalar equation for vv t + v x u + u x = εv xx + εv x + a ( x ) u (3.13)from which we have v = Z R G ( x − y, t ) v ( y ) dy + Z t Z R ( εv x − v x u − u x + a ( x ) u ) G ( x − y, t − s ) dyds, where G is the heat kernel satisfying Z R G ( x − y, t ) dy = 1 , Z R | G y ( x − y, t ) | dy ≤ C √ εt . Then it follows that v = Z R G ( x − y, t ) v ( y ) dy + Z t Z R ( εv y − v y u − u y + au ) G ( x − y, t − s ) dyds ≥ Z R G ( x − y, t ) v ( y ) dy + Z t Z R uG y ( x − y, t − s ) + ( au − u ε ) G ( x − y, t − s ) dyds ≥ ln ε − Ctε − C √ t √ ε := − C ( ε, t ) . Thus ρ ≥ e − C ( ε,t ) . (3.14)From (3.12) and (3.14), we get (3.3). The lower bound of density guarantees that thereis no singularity in (3.1). Then we can apply classical theory of quasilinear parabolicsystems to complete the proof of Theorem 3.1.3.3. Convergence of approximate solutions.
In this section, we will provide the proofof Theorem 1.1. Since we are focusing on the uniform bound of ρ and m, in this sectionwe assume 1 < γ ≤ < γ ≤
3, one can follow the similarargument in [18] or [32] to obtain the same conclusions.Denote Π T = R × [0 , T ] for any T ∈ (0 , ∞ ) . Step 1. H − loc compactness of the entropy pair. We consider η ( ρ ε , m ε ) t + q ( ρ ε , m ε ) x , where ( η, q ) is any weak entropy-entropy flux pair given in (2.2). We will apply the Muratlemma to achieve the goal. Lemma 3.2. (Murat [25]) Let Ω ∈ R n be an open set, then ( compact set of W − ,qloc (Ω)) ∩ ( bounded set of W − ,rloc (Ω)) ⊂ ( compact set of H − loc (Ω)) , where < q ≤ < r. Let K ⊂ Π T be any compact set, and choose ϕ ∈ C ∞ c (Π T ) such that ϕ | K = 1 and0 ≤ ϕ ≤ . Multiplying (3.1) by ϕ ∇ η ∗ with η ∗ the mechanical entropy, we obtain ε Z Z Π T ϕ ( ρ x , m x ) ∇ η ∗ ( ρ x , m x ) ⊤ dxdt = Z Z Π T ( a ( x ) m ρ − εb ( x ) ρ x ) η ∗ m ϕ + a ( x ) mη ∗ ρ ϕ + η ∗ ϕ t + q ∗ ϕ x + εη ∗ ϕ xx dxdt. (3.15) ENERAL NOZZLE FLOW 11
A direct calculation tells us that( ρ x , m x ) ∇ η ∗ ( ρ x , m x ) ⊤ = p γρ γ − ρ x + ρu x . Noting that | ( a ( x ) m ρ − εb ( x ) ρ x ) η ∗ m | ≤ εp γ ρ γ − ρ x + εCb m ρ − γ + a m ρ , we get ε Z Z Π T ϕ ( ρ x , m x ) ∇ η ∗ ( ρ x , m x ) ⊤ dxdt ≤ Z Z Π T ( Cεb m ρ − γ + a m ρ ) ϕ + η ∗ ϕ t + q ∗ ϕ x + εη ∗ ϕ xx + ( m ρ + γγ − mρ γ − p ) a ϕdxdt ≤ C ( ϕ ) . Hence ε ( ρ x , m x ) ∇ η ∗ ( ρ x , m x ) ⊤ ∈ L loc (Π T ) , (3.16)i.e., ερ γ − ρ x + ερu x ∈ L loc (Π T ) . (3.17)For any weak entropy-entropy flux pairs given in (2.2), as in (3.15), we have η t + q x = εη xx − ε ( ρ x , m x ) ∇ η ( ρ x , m x ) ⊤ + ( η ρ a ( x ) m + η m a ( x ) m ρ ) − εη m ρ x b ( x )=: X i =1 I i . (3.18)Using (3.17), it is straightforward to check that I is compact in H − loc (Π T ) . Note that forany weak entropy, the Hessian matrix ∇ η is controlled by ∇ η ∗ ( [18]), that is,( ρ x , m x ) ∇ η ( ρ x , m x ) ⊤ ≤ ( ρ x , m x ) ∇ η ∗ ( ρ x , m x ) ⊤ , (3.19)and thus I is bounded in L loc (Π T ) and thus compact in W − ,αloc (Π T ) for some 1 < α < I , we have | I | = | η ρ a ( x ) m + η m a ( x ) m ρ | ≤ Ca , which implies that I is bounded in L loc (Π T ) . For the last term I , we get | I | ≤ Cερ γ/ − | ρ x | . It follows from (3.16) that I is compact in H − loc (Π T ) . Therefore, η t + q x is compact in W − ,αloc (Π T ) with some 1 < α < . On the other hand, since ρ and m are uniformly bounded, we have η t + q x is bounded in W − , ∞ loc (Π T ) . We conclude that η t + q x is compact in H − loc (Π T ) (3.20)for all weak entropy-entropy flux pairs with the help of the Murat lemma 3.2. Step 2. Strong convergence and consistency.
By (3.20) and the compactnessframework established in [6, 7, 9, 18], we can prove that there exists a subsequence of( ρ ε , m ε ) (still denoted by ( ρ ε , m ε )) such that( ρ ε , m ε ) → ( ρ, m ) in L ploc (Π T ) , p ≥ , (3.21)from which it is easy to show that ( ρ, m ) is a weak solution to the Cauchy problem (1.1)-(1.2). We omit the proof for brevity. Step 3. Entropy inequality.
We shall also prove that ( ρ, m ) satisfies the entropyinequality in the sense of distributions for all weak convex entropies. Let ( η, q ) be anyentropy-entropy flux pair with η being convex. Multiplying (3.1) by ϕ ∇ η with 0 ≤ ϕ ∈ C ∞ c (Π T ), we get Z Z Π T η t ϕ + q x ϕdxdt = Z Z Π T εη xx ϕ − εϕ ( ρ x , m x ) ∇ η ( ρ x , m x ) ⊤ + η ρ a ( x ) mϕ + η m ( a ( x ) m ρ − εb ( x ) ρ x ) ϕdxdt. As in Step 1, we have (cid:12)(cid:12)(cid:12)(cid:12)Z Z Π T εη xx ϕdxdt (cid:12)(cid:12)(cid:12)(cid:12) → ε → . Moreover, (cid:12)(cid:12)(cid:12)(cid:12)Z Z Π T ερ x b ( x ) η m ϕdxdt (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:20)Z Z Π T Cερ − γ b ϕdxdt (cid:21) (cid:20)Z Z Π T ϕερ x ρ γ − dxdt (cid:21) ≤ Cε → ε → . Noting that εϕ ( ρ x , m x ) ∇ η ( ρ x , m x ) ⊤ ≥ , we conclude that Z Z Π T ηϕ t + qϕ x dxdt + ( η m m ρ + mη ρ ) a ( x ) ϕdxdt ≥ ε → , that is, ( ρ, m ) is indeed an entropy solution to the Cauchy problem (1.1)-(1.2). Therefore,the proof of Theorem 1.1 is completed.4. Preliminary and Formulation for Isothermal Flow
In this section, we provide some preliminaries and formulation for the isothermal case.Here, we adopt a similar notion as in Section 2 with no confusion. Letting n = A ( x ) ρ, J = A ( x ) m, ENERAL NOZZLE FLOW 13 and using γ = 1 , we can rewrite (1.1) as n t + J x = 0 ,J t + (cid:18) J n + n (cid:19) x = − a ( x ) n, x ∈ R (4.1)with a ( x ) = − A ′ ( x ) A ( x ) , J = nu . Then seeking weak entropy solutions of (1.1)-(1.2) isequivalent to solving (4.1) with the following initial data:( n, J ) | t =0 = ( n ( x ) , J ( x )) = ( A ( x ) ρ ( x ) , A ( x ) m ( x )) ∈ L ∞ ( R ) . (4.2)The eigenvalues of (4.1) are λ = Jn − , λ = Jn + 1 , and the corresponding right eigenvectors are r = (cid:20) λ (cid:21) , r = (cid:20) λ (cid:21) . The Riemann invariants ( w, z ) are given by w = Jn + ln n, z = Jn − ln n. The mechanical energy η ∗ ( n, J ) and mechanical energy flux q ∗ ( n, J ) have the followingformula η ∗ ( n, J ) = J n + n ln n, q ∗ ( n, J ) = J n + J ln n. Proof of Theorem 1.2
We first recall the compactness framework in Huang and Wang [14].
Theorem 5.1.
Let ( n ε , J ε ) be a sequence of bounded approximate solutions of (4.1) - (4.2) satisfying < δ ≤ n ε ≤ C, | J ε | ≤ n ε ( C + | ln n ε | ) with C being independent of ε, T, δ = o ( ε ) . Assume that ∂ t η ( n ε , J ε ) + ∂ x q ( n ε , J ε ) is compact in H − loc (Π T ) , where ( η, q ) is defined as η = n − ξ e ξ − ξ Jn , q = (cid:18) Jn + ξ (cid:19) η for any fixed ξ ∈ ( − , . Then there exists a subsequence of ( n ε , J ε ) , still denoted by ( n ε , J ε ) , such that ( n ε ( x, t ) , J ε ( x, t )) → ( n ( x, t ) , J ( x, t )) in L ploc ( R × R + ) , p ≥ , for some function ( n ( x, t ) , J ( x, t )) satisfying ≤ n ≤ C, | J | ≤ n ( C + | ln n | ) , where C is a positive constant independent on T. Construction of approximate solutions.
Next we construct approximate solu-tions satisfying the conditions in Theorem 5.1. Raising density, which is motivated by [22],we add artificial viscosity as follows: n t + ( J − δ Jn ) x = εn xx ,J t + (cid:18) J n − δ J n + Z nδ t − δt dt (cid:19) x = εJ xx − a ( x )( n − δ ) + 2 b ( x ) δ Jn − εb ( x ) n x (5.1)with initial data ( n, J ) | t =0 = ( n ε ( x ) , J ε ( x )) = ( n ( x ) + δ, J ( x )) ∗ j ε , (5.2)where b is a function to be determined later, δ = o ( ε ) , and j ε is the standard mollifier and0 < ε < . By a direct computation, the eigenvalues are λ δ = Jn − n − δn , λ δ = Jn + n − δn , (5.3)and the Riemann invariants are w = Jn + ln n, z = Jn − ln n. Global existence of approximate solutions.
In this section, we show the globalexistence of classical solutions to the Cauchy problem of quasilinear parabolic system(5.1)-(5.2) and obtain the following theorem.
Theorem 5.2.
There exists a unique global classical bounded solution ( n ε , J ε ) to theCauchy problem (5.1) - (5.2) satisfying δ ≤ n ε ≤ C, | J ε | ≤ n ε ( C + | ln n ε | ) . (5.4)We divide the proof of Theorem 5.2 into three steps. In this section, we omit the upindex ε. Step 1. Local existence and lower bound of density.
The local existence of thesolution for (5.1)-(5.2) can be proved by using the heat kernel and the same way in [9].For the lower bound of density, we denote v = n − δ, and then v satisfies v t + ( uv ) x = εv xx , v | t =0 = v ( x ) (5.5)with u = Jn . From the definition of n , we have v ≥
0. Rewrite (5.5) as v t + uv x = εv xx − u x v, and then it is easy to obtain from the maximum principle of the parabolic equation that v ( x, t ) ≥ min v ( x ) e −k u x k L ∞ t ≥ , and hence we gain n ≥ δ. Step 2. Uniform upper bound.
We apply Lemma 3.1 to obtain the uniform L ∞ estimates. As before, to estimate the uniform bound of the approximate solution, we shall ENERAL NOZZLE FLOW 15 investigate a parabolic system derived by Riemann invariants. We transform (5.1) intothe following form: w t + λ δ w x = εw xx + 2 ε ( w x − b ( x )) n x n − ε n x n − a ( x ) n − δn + 2 b ( x ) δ Jn ,z t + λ δ z x = εz xx + 2 ε ( z x − b ( x )) n x n + ε n x n − a ( x ) n − δn + 2 b ( x ) δ Jn . Set the control functions ( φ, ψ ) as follows: φ = M + 2 Z x −∞ b ( y ) dy + 2 ε k b ′ ( x ) k L ∞ t,ψ = M + 2 Z ∞ x b ( y ) dy + 2 ε k b ′ ( x ) k L ∞ t. We remark that φ, ψ in this Section is different from those in Section 3 for simplicity.Then we obtain φ t = 2 ε k b ′ ( x ) k L ∞ , φ x = 2 b ( x ) , φ xx = 2 b ′ ( x ); ψ t = 2 ε k b ′ ( x ) k L ∞ , ψ x = − b ( x ) , ψ xx = − b ′ ( x ) . Let ¯ w = w − φ, ¯ z = z + ψ. A simple calculation yields ¯ w t + (cid:16) λ δ − ε n x n (cid:17) ¯ w x = ε ¯ w xx + 2 εb ′ ( x ) − ε k b ′ ( x ) k L ∞ − ε n x n − (cid:18) Jn + n − δn (cid:19) b ( x ) − n − δn a ( x ) + 2 b ( x ) δ Jn , ¯ z t + (cid:16) λ δ − ε n x n (cid:17) ¯ z x = ε ¯ z xx + 2 εb ′ ( x ) + 2 ε k b ′ ( x ) k L ∞ + ε n x n − (cid:18) Jn − n − δn (cid:19) b ( x ) − n − δn a ( x ) + 2 b ( x ) δ Jn . (5.6)Note that Jn = w + z w + φ + ¯ z − ψ , and then the system (5.6) becomes ¯ w t + (cid:16) λ δ − ε n x n (cid:17) ¯ w x = ε ¯ w xx + a ¯ w + a ¯ z + R , ¯ z t + (cid:16) λ δ − ε n x n (cid:17) ¯ z x = ε ¯ z xx + a ¯ w + a ¯ z + R with a = − b ( x ) n − δn , a = − b ( x ) n − δn ≤ ,a = − b ( x ) n − δn ≤ , a = − b ( x ) n − δn and R =2 εb ′ ( x ) − ε k b ′ ( x ) k L ∞ − ε n x n + ( − a − b ) n − δn + b ( x ) n − δn (cid:18) Z ∞ x b ( y ) dy − Z x −∞ b ( y ) dy − (cid:19) ,R =2 εb ′ ( x ) + 2 ε k b ′ ( x ) k L ∞ + ε n x n + ( − a + b ) n − δn + b ( x ) n − δn (cid:18) Z ∞ x b ( y ) dy − Z x −∞ b ( y ) dy + 1 (cid:19) , where we have used n ≥ δ . Sincesup (cid:26) x ∈ R (cid:12)(cid:12)(cid:12) Z ∞ x b ( y ) dy − Z x −∞ b ( y ) dy (cid:27) = 2 k b k L , we can take b ( x ) ∈ C ( R ) such that k b ( x ) k L ≤ , | a ( x ) | ≤ b ( x ) , and then we have R ≤ , R ≥ . In fact, from our assumption on a ( x ), we take b ( x ) = a ( x ) , which is our key reason for the condition (1.7). By our conditions on initial data,we can take M large enough such that¯ w ( x, ≤ , ¯ z ( x, ≥ . Then, Lemma 3.1 yields ¯ w ( x, t ) ≤ , ¯ z ( x, t ) ≥ , which implies that w ( x, t ) ≤ φ ( x, t ) ≤ M + 2 k b k L + 2 ε k b ′ k L ∞ t ≤ C,z ( x, t ) ≥ − ψ ( x, t ) ≥ − M − k b k L − ε k b ′ k L ∞ t ≥ − C, where for any fixed time T, we choose ε small such that ε k b ′ k L ∞ t ≤ ε k b ′ k L ∞ T ≤ . Hence we obtain (5.4).From Steps 1 and 2, using the classical theory of quasilinear parabolic systems, we cancomplete the proof of Theorem 5.2.5.3.
Convergence of approximate solutions.
As stated in Section 2, (1.1)-(1.2) isequivalent to (4.1)-(4.2). Thus we only need to show that a subsequence of ( n ε , J ε ) inSection 5.2 converges to the solutions of (4.1)-(4.2) by verifying the conditions in Theorem5.1. We also divide the proof into three steps. Step 1. H − loc compactness of the entropy pair. We will verify the H − loc compactnessof the entropy pair η ( n ε , J ε ) t + q ( n ε , J ε ) x for some weak entropy ( η, q ) of (4.1) with η = n − ξ e ξ − ξ Jn , q = (cid:18) Jn + ξ (cid:19) η ENERAL NOZZLE FLOW 17 for any fixed ξ ∈ ( − , . It is easy to calculate that η n = 11 − ξ (cid:18) − ξ Jn (cid:19) ηn , η J = ξ − ξ ηn ,η nn = ξ (1 − ξ ) (cid:18) − ξ Jn + J n (cid:19) n ξ − ξ − e ξ − ξ Jn ,η nJ = ξ (1 − ξ ) (cid:18) ξ − Jn (cid:19) n ξ − ξ − e ξ − ξ Jn ,η JJ = ξ (1 − ξ ) n ξ − ξ − e ξ − ξ Jn . Hence η nn η JJ − η nJ = ξ (1 − ξ ) n ξ − ξ − e ξ − ξ Jn > . It indicates that η is strictly convex for any ξ ∈ ( − , . Then( n x , J x ) ∇ η ( n x , J x ) ⊤ = ξ (1 − ξ ) n − ξ − e ξ − ξ Jn " n x + (cid:18) Jn n x − J x (cid:19) − ξn x (cid:18) Jn n x − J x (cid:19) ≥ ξ (1 − ξ ) ηn " (1 − | ξ | ) n x + (1 − | ξ | ) (cid:18) Jn n x − J x (cid:19) . Let K ⊂ Π T be any compact set, and choose ϕ ∈ C ∞ c (Π T ) such that ϕ | K = 1 , and0 ≤ ϕ ≤ . After multiplying (5.1) by ϕ ∇ η, and integrating over Π T , we obtain ε Z Z Π T ϕ ( n x , J x ) ∇ η ( n x , J x ) ⊤ dxdt = Z Z Π T [ − εn x b − a ( n − δ ) + 2 bδ Jn + δ n x n + δ J n ) x ] η J ϕ + ηϕ t + εηϕ xx dxdt. Due to (cid:12)(cid:12)(cid:12)(cid:12) Jn (cid:12)(cid:12)(cid:12)(cid:12) ≤ C + | ln n | , and η ≤ n − ξ e | ξ | − ξ ( C − ln n ) ≤ Cn −| ξ | − ξ , it is easy to get | − a ( n − δ ) + 2 bδ Jn η J | ≤ Cb. (5.7)Besides, | εn x bη J | ≤ εb | n x | | ξ | − ξ ηn ≤ εξ (1 − | ξ | )4(1 − ξ ) ηn n x n + Cεηb . (5.8)Moreover, we have (cid:12)(cid:12)(cid:12) δ n x n η J (cid:12)(cid:12)(cid:12) ≤ | n x | n | ξ | − ξ ηn ≤ εξ (1 − | ξ | )4(1 − ξ ) ηn n x n + C δ ε ηn , (cid:12)(cid:12)(cid:12)(cid:12) δ (cid:18) J n (cid:19) x η J (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) δ Jn | ξ | − ξ ηn (cid:18) Jn (cid:19) x (cid:12)(cid:12)(cid:12)(cid:12) ≤ εξ (1 − | ξ | )4(1 − ξ ) η (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) Jn (cid:19) x (cid:12)(cid:12)(cid:12)(cid:12) + C δ ε ηn J n . (5.9) Taking δ = ε such that δ /ε ≤ δ / ≤ n / , and choosing small | ξ | 6 = 0 , from the twofacts ηn = n − ξ − e ξ − ξ Jn ≤ Cn − ξ − − | ξ | − ξ = Cn − | ξ | +11+ | ξ | ,ηn J n = n − ξ − e ξ − ξ Jn J n ≤ Cn − ξ − − | ξ | − ξ (1 + | ln n | ) ≤ Cn − | ξ | +11+ | ξ | we get ε Z Z Π T ϕ ( n x , J x ) ∇ η ( n x , J x ) ⊤ dxdt ≤ C ( ϕ ) (5.10)with constant C ( ϕ ) depending on the H (Π T ) norm of ϕ . Hence for small | ξ | 6 = 0 ,ε ηn n x + ε ηn (cid:18) Jn n x − J x (cid:19) = ε ηn n x + εη (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) Jn (cid:19) x (cid:12)(cid:12)(cid:12)(cid:12) ∈ L loc (Π T ) . (5.11)Now we investigate the dissipation of the entropy as follows: η t + q x = εη xx − ε ( n x , J x ) ∇ η ( n x , J x ) ⊤ + [ − a ( n − δ ) + 2 bδ Jn ] η J − εn x bη J + (cid:18) δ ( Jn ) x η n + [ δ n x n + δ J n ) x ] η J (cid:19) := X k =1 I k . Combining (5.7), (5.8), (5.9), (5.10), we obtain that I + I + I + I is bounded in L loc (Π T ) , and then compact in W − ,αloc (Π T ) with some 1 < α < I , from (5.11), for any ϕ ∈ H (Π T ) , (cid:12)(cid:12)(cid:12)(cid:12)Z Z Π T εη xx ϕdxdt (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z Z Π T ε ( η n n x + η J J x ) ϕ x dxdt (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z Z Π T εη | ϕ x | − ξ (cid:12)(cid:12)(cid:12)(cid:12) n x n − ξn (cid:18) Jn n x + J x (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dxdt ≤√ ε (cid:18)Z Z Π T ηϕ x n (1 − ξ ) dxdt (cid:19) (cid:20) (cid:18)Z Z Π T εηn x n dxdt (cid:19) + Z Z Π T εηn (cid:18) Jn n x − J x (cid:19) dxdt ! (cid:21) , and thus we have that I is compact in H − loc (Π T ) . Finally, we get η t + q x is compact in W − ,αloc (Π T ) with 1 < α < . Moreover, q = (cid:18) Jn + ξ (cid:19) η ≤ ( C − ln n + | ξ | ) η ≤ C + | ln n | n − ξ e ξ − ξ Jn ≤ C, and then η t + q x bounded in W − , ∞ loc (Π T ) . ENERAL NOZZLE FLOW 19
Therefore, taking | ξ | small, we conclude that η t + q x is compact in H − loc (Π T ) for small | ξ | ≤ , by Lemma 3.2. Step 2. Convergence and consistency.
Since our approximate solutions satisfy allthe conditions in Theorem 5.1, applying Theorem 5.1 yields( n ε , J ε ) → ( n, J ) in L ploc (Π T ) , p ≥ . This implies that ( n, J ) is a weak solution to the Cauchy problem (4.1)-(4.2). Similar tothe previous argument, we can show that ( n, J ) satisfies the energy inequality. Thus ( n, J )is an entropy solution. The proof of Theorem 1.2 is completed.6.
Appendix
Here we provide the proof of Lemma 3.1 for completeness.
Proof.
Let ¯ M = k p k L ∞ ( R × [0 ,T ]) + k q k L ∞ ( R × [0 ,T ]) . We define two new variables ¯ p = p − ξ, ¯ q = q + ξ, where ξ = ξ ( x, t ) = 2 ¯ M cosh x cosh N e Λ t , N > , and Λ > i, j ) = (1 ,
2) or (2 , , we write a ij ( x, t, p, q ) = a ij ( x, t, ¯ p, ¯ q )+ (cid:18)Z ∂a ij ∂p ( x, t, ¯ p + τ ξ, ¯ q − τ ξ ) dτ − Z ∂a ij ∂q ( x, t, ¯ p + τ ξ, ¯ q − τ ξ ) dτ (cid:19) ξ and R i ( x, t, p, q, ζ, η ) = R i ( x, t, ¯ p, ¯ q, ζ, η )+ (cid:18)Z ∂R i ∂p ( x, t, ¯ p + τ ξ, ¯ q − τ ξ, ζ, η ) dτ − Z ∂R i ∂q ( x, t, ¯ p + τ ξ, ¯ q − τ ξ ) dτ (cid:19) ξ. Denote a ij = a ij ( x, t, ¯ p, ¯ q ) , R i = R i ( x, t, ¯ p, ¯ q ) ,b ij = Z ∂a ij ∂p ( x, t, ¯ p + τ ξ, ¯ q − τ ξ ) dτ − Z ∂a ij ∂q ( x, t, ¯ p + τ ξ, ¯ q − τ ξ ) dτ,c i = Z ∂R i ∂p ( x, t, ¯ p + τ ξ, ¯ q − τ ξ, ζ, η ) dτ − Z ∂R i ∂q ( x, t, ¯ p + τ ξ, ¯ q − τ ξ, ζ, η ) dτ. Then we get a system for (¯ p, ¯ q ) , ¯ p t + µ ¯ p x = ε ¯ p xx + a ¯ p + a ¯ q + b ¯ qξ + ¯ R + ¯ c ξ − µ ¯ M sinh x cosh N e Λ t + ξ ( − Λ + ε + a − a ) , ¯ q t + µ ¯ q x = ε ¯ q xx + a ¯ p + a ¯ q + b ¯ pξ + ¯ R + ¯ c ξ + 2 µ ¯ M sinh x cosh N e Λ t + ξ (Λ − ε + a − a ) . Note that for any Λ >
0, ¯ p ( x, t ) < q ( x, t ) > | x | ≥ N . Next we show Claim :There exists Λ = Λ( ¯ M ) such that¯ p ( x, t ) ≤ q ( x, t ) ≥ x ∈ ( − N, N ) , ≤ t ≤ s ∗ = 1Λ . To this end, let A = { t ∈ [0 , s ∗ ] | there exist x ∈ [ − N, N ] such that ¯ p ( x, t ) > q ( x, t ) < } . We shall prove the set A is empty by contradiction. In fact, if A is not empty, let t ∗ =inf A >
0, and then there exists | x ∗ | ≤ N such that ¯ p ( x ∗ , t ∗ ) = 0 or ¯ q ( x ∗ , t ∗ ) = 0 . Withoutloss of generality, we assume ¯ p ( x ∗ , t ∗ ) = 0 . Then, ¯ p ( x, ≤ , ¯ q ( x, ≥ , | x | ≤ N. For0 ≤ t < t ∗ , ¯ p ( ± N, t ) < , ¯ q ( ± N, t ) > , | x | ≤ N, and thus ¯ p ( x, t ) takes the maximum value over [ − N, N ] × [0 , t ∗ ] at the point ( x ∗ , t ∗ ). Wehave ¯ p x ( x ∗ , t ∗ ) = 0 , ¯ p xx ( x ∗ , t ∗ ) ≤ , ¯ p t ( x ∗ , t ∗ ) ≥ , ¯ q ( x ∗ , t ∗ ) ≥ . Note that at the point ( x ∗ , t ∗ ) , a ≤ , ¯ R ≤ , Λ t ∗ ≤ Λ s ∗ = 1 . Moreover, for any τ ∈ [0 , , | ¯ p + τ ξ | ≤ | p | + 2 ξ ≤ ¯ M + 4 ¯ M e ≤ C ( ¯ M ) , | ¯ q − τ ξ | ≤ | q | + 2 ξ ≤ ¯ M + 4 ¯ M e ≤ C ( ¯ M ) . Therefore, | b | ≤ C ( ¯ M ) , | b | ≤ C ( ¯ M ) , | c | ≤ C ( ¯ M ) , | c | ≤ C ( ¯ M ) . A direct com-putation yields that at the point ( x ∗ , t ∗ ),¯ p t + µ ¯ p x ≤ a ¯ q + b ¯ qξ + ¯ R + ¯ c ξ + ξ ( − Λ + ε + a − a + | µ | ) ≤ ξ (cid:0) − Λ + ε + a − a + | µ | + C ( ¯ M ) ¯ M + 2 C ( ¯ M ) ¯ M e + 2 C ( ¯ M ) M e (cid:1) . Then, choosingΛ =: 2 ε + X i =1 k µ i k L ∞ + X i,j =1 k a ij k L ∞ + C ( ¯ M ) ¯ M (2 e + 1) + 2 C ( ¯ M ) M e, we get ¯ p t + µ ¯ p x < . It contradicts with¯ p t + µ ¯ p x ≥ , at ( x ∗ , t ∗ ) . Hence A is empty and Claim holds. Letting N tend to infinity, we obtain that p ( x, t ) ≤ , and q ( x, t ) ≥ ∀ x ∈ R , ≤ t ≤ s ∗ . From the above analysis, we have proved that theset Ω = { t ∈ [0 , T ] | p ( x, s ) ≤ , q ( x, s ) ≥ ∀ x ∈ R , ≤ s ≤ t } is an open set. It is obvious that Ω is a closed subset of [0 , T ] . Therefore, Ω = [0 , T ]. Wethus complete Lemma 3.1. (cid:3)
ENERAL NOZZLE FLOW 21
Acknowledgments
Wentao Cao’s research is supported by ERC Grant Agreement No.724298. FeiminHuang is partially supported by National Center for Mathematics and Inter-disciplinarySciences, AMSS, CAS, and NSFC Grant No.11371349 and 11688101. Difan Yuan is sup-ported by China Scholarship Council No.201704910503. The authors would like to thankProfessor Naoki Tsuge for valuable comments and suggestions.
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