Uniqueness of the 1D Compressible to Incompressible Limit
aa r X i v : . [ m a t h . A P ] D ec Uniqueness of the 1D Compressible to Incompressible Limit
Rinaldo M. Colombo Graziano Guerra October 10, 2018
Abstract
Consider two compressible immiscible fluids in 1D in the isentropic approximation. The firstfluid is surrounded and in contact with the second one. As the Mach number of the first fluidvanishes, the coupled dynamics of the two fluids results as the compressible to incompressiblelimit and is known to satisfy an ODE–PDE system. Below, a characterization of this limit isprovided, ensuring its uniqueness.
Keywords:
Compressible to Incompressible limit, Hyperbolic Conservation Laws, Unique-ness of the Zero Mach number Limit
The literature on the compressible to incompressible limit is vast. We refer for instance to the wellknown results [12, 13, 15, 16], the more recent [3, 18], the review [17] and the references therein.In this paper, following [5], we consider two compressible immiscible fluids and study the limitas one of the two becomes incompressible. A volume of a compressible inviscid fluid, say the liquid ,is surrounded by another compressible fluid, say the gas . Using the Lagrangian formulation, inthe isentropic case, we assume that the gas obeys a fixed pressure law P g ( τ ), while for the liquidwe assume a one parameter family of pressure laws P κ ( τ ) such that P ′ κ ( τ ) → −∞ as κ →
0. Thetotal mass of the liquid is fixed so that in Lagrangian coordinates the liquid and gas phases fillthe fixed sets (see Figure 1) L = ]0 , m [ and G = R \ ]0 , m [ . For an Eulerian description, see [5]. P g ( τ ) P g ( τ ) P κ ( τ )0 m z Figure 1: In Lagrangian coordinates, the boundaries separating the two fluids are fixed.On P g ( τ ) and P κ ( τ ), we require the usual hypotheses and the incompressible limit assumption: P g , P κ ∈ C , P g ( τ ) , P κ ( τ ) > P ′ g ( τ ) , P ′ κ ( τ ) < P ′′ g ( τ ) , P ′′ κ ( τ ) > P ′ κ ( τ ) κ → −−−→ −∞ . (1.1)The standard choice P g ( τ ) = k/τ γ satisfies (1.1) for all k > γ > INDAM Unit, University of Brescia, Italy. [email protected] Department of Mathematics and Applications, University of Milano - Bicocca, Italy. [email protected] p -system [10, Formula (7.1.11)] ( ∂ t τ − ∂ z v = 0 ∂ t v + ∂ z P κ ( z, τ ) = 0 , where P κ ( z, τ ) = ( P κ ( τ ) for z ∈ L P g ( τ ) for z ∈ G , (1.2) v ( t, z ) being the fluid speed at time t and at the Lagrangian coordinate z .In Lagrangian coordinates, the conservation of mass and momentum are equivalent to theconservation of τ and v which, in turn, are equivalent along the interfaces z = 0 and z = m to theRankine–Hugoniot conditions for (1.2). Therefore, for a.e. t ≥ ( v ( t, − ) = v ( t, P g (cid:0) τ ( t, − ) (cid:1) = P κ (cid:0) τ ( t, (cid:1) , ( v ( t, m − ) = v ( t, m +) P κ (cid:0) τ ( t, m − ) (cid:1) = P g (cid:0) τ ( t, m +) (cid:1) . In other words, pressure and velocity have to be continuous across the interfaces. Hence, thepressure is a natural choice as unknown, rather than the specific volume. Following [5, 7, 9, 11],we introduce the inverse functions of the pressure laws T g ( p ) = P − g ( p ) , T κ ( p ) = P − κ ( p ) where T ′ κ ( p ) κ → −−−→ , (1.3)the last limit being a consequence of (1.1). Rewrite system (1.2) with ( p, v ) as unknowns ( ∂ t T κ ( z, p ) − ∂ z v = 0 ∂ t v + ∂ z p = 0 , where T κ ( z, p ) = ( T κ ( p ) for z ∈ LT g ( p ) for z ∈ G . (1.4)The conditions at the interfaces become continuity requirements on the unknown functions: ( v ( t, − ) = v ( t, p ( t, − ) = p ( t, ( v ( t, m − ) = v ( t, m +) p ( t, m − ) = p ( t, m +) for a.e. t ≥ . (1.5)As in [5], we fix a pressure law P and choose T = P − , so that T κ ( p ) = T (cid:16) ¯ p + κ ( p − ¯ p ) (cid:17) , lim κ → T κ ( p ) = T (¯ p ) = ¯ τ , (1.6)where ¯ τ is the constant specific volume at the incompressible limit and ¯ p = P (¯ τ ). For instance,the (modified) Tait equation of state [14, Formula (1)] fits into (1.6) with T ( p ) = p − /n with κ = n β o ¯ τ n where β o is the isothermal compressibility, n is a pressure independent parameter and β o → ( ∂ t T g ( p ) − ∂ z v = 0 ∂ t v + ∂ z p = 0 z ∈ G gas;˙ v ℓ = p ( t, − ) − p ( t,m +) m liquid; ( v ( t, − ) = v ℓ ( t ) v ( t, m +) = v ℓ ( t ) interface. (1.7)The existence of a Lipschitz continuous semigroup generated by (1.7) is proved in [1]. On theother hand, a characterization yielding the uniqueness of solutions to (1.7) is obtained in [6].In this paper we show that the incompressible limit obtained in [5] satisfies the characterizationin [6]. Hence, the solution ( p κ , v κ ) to (1.4) converges as κ →
0, the limit being the unique solutionto (1.7).The next Section is devoted to the formal statements, while Section 3 contains the technicalproofs. 2
Main Result
Throughout, we denote by LC the set of functions defined on R \ ]0 , m [ that are locally constantout of a compact set, i.e., they attain a constant value on ] −∞ , − M ] and a, possibly different,constant value on [ M, + ∞ [, for a suitable positive M .Below, solutions to (1.7) are understood in the sense of [5, Definition 3.2], see also [1, Def-inition 2.5], and are constructed in [5] as limits of solutions to (1.2). In solutions to (1.2), thepropagation speed of waves in the gas region G is uniformly bounded, independently of κ . There-fore, to prove the uniqueness of solutions to (1.7) obtained as the compressible to incompressiblelimit, it is sufficient to consider initial data (cid:0) ( τ o , v o ) , v ℓ,o (cid:1) such that ( τ o , v o ) is in LC and v ℓ,o ∈ R .Given (cid:0) ( τ, v ) , v ℓ (cid:1) ∈ BV ( R ; R ) × R such that ( τ, v ) ∈ LC , call( τ ±∞ , v ±∞ ) = lim x →±∞ ( τ, v )( x ) . Under the transformation U ( x ) = τ ( − x ) − τ −∞ v ( − x ) − v −∞ τ ( x + m ) − τ + ∞ v ( x + m ) − v + ∞ w = " v ℓ − v −∞ v ℓ − v + ∞ , (2.1)setting f ( U ) = U − P g ( U + τ −∞ ) − U P g ( U + τ + ∞ ) F ( U, w ) = 1 m " P g ( U + τ −∞ ) − P g ( U + τ + ∞ ) P g ( U + τ −∞ ) − P g ( U + τ + ∞ ) b ( U ) = " U U g ( w ) = w (2.2)the Cauchy Problem ( ∂ t τ − ∂ z v = 0 ∂ t v + ∂ z P g ( τ ) = 0 x ∈ G ˙ v ℓ = P g (cid:0) τ ( t, − ) (cid:1) − P g (cid:0) τ ( t, m +) (cid:1) m ( v ( t, − ) = v ℓ ( t ) v ( t, m +) = v ℓ ( t )( τ, v )(0 , x ) = ( τ o , v o )( x ) x ∈ G v ℓ (0) = v ℓ,o (2.3)is formally equivalent to ∂ t U ( t, x ) + ∂ x f (cid:0) U ( t, x ) (cid:1) = 0 x ∈ R + b (cid:0) U ( t, (cid:1) = g (cid:0) w ( t ) (cid:1) ˙ w ( t ) = F (cid:0) U ( t, , w ( t ) (cid:1) U (0 , x ) = U o ( x ) x ∈ R + w (0) = w o (2.4)which fits in the well posedness theory developed in [6], as proved by the following Proposition. Proposition 2.1.
Let P g satisfy (1.1) . Fix τ −∞ , τ + ∞ ∈ ˚ R + . Then, system (2.4) generates asemigroup S : R + × D → D uniquely characterized by the properties (i)–(iv) in [6, Theorem 4].Moreover, for a suitable positive δ , D ⊇ (cid:26) ( U, w ) ∈ ( L ∩ BV )( R + ; R ) × R : TV( U ) + (cid:13)(cid:13)(cid:13) b (cid:0) U (0+) (cid:1) − g ( w ) (cid:13)(cid:13)(cid:13) < δ (cid:27) . (2.5)3he above proposition leads to the main result of this paper. Theorem 2.2.
Let t → (cid:0) ( τ, v ) , v ℓ (cid:1) ( t ) be a solution to (1.7) obtained as limit for κ → of solutionsto (1.2) , with an initial datum in LC and satisfying for all t ∈ R + TV (cid:0) ( τ, v )( t ); G (cid:1) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)" v ( t, − ) − v ℓ ( t ) v ( t, m +) − v ℓ ( t ) < δ (2.6) with δ as in (2.5) . Correspondingly, define t → ( U, w )( t ) as in (2.1) . Then,1. for all t ∈ R + , the map t → ( U, w )( t ) coincides with an orbit of the semigroup S defined inProposition 2.1.2. The semigroup S is defined globally in time for all initial data with sufficiently small totalvariation. In the above statement, as well as below, we use the obvious notationTV (cid:0) ( τ, v ); G (cid:1) = TV (cid:0) ( τ, v ); ] −∞ , (cid:1) + TV (cid:0) ( τ, v ); [ m, + ∞ [ (cid:1) . Proof of Proposition 2.1.
On the basis of (2.2) and with the help of (1.1), we verify that (2.4)satisfies the assumptions of [6, Theorem 4]. With reference to the notation therein, set n = 4, l = 2, m = 2. Now, observe that (H1) holds. Clearly, f is of class C by (1.1). The stricthyperbolicity of (1.7) can easily be recovered through a rescaling of the space variable, since thedifferent p –systems in (1.7) interact only through the boundary, see [8, Lemma 4.1]. Besides, withstandard notation, we have: λ ( U ) = − q − P ′ g ( U + τ −∞ ) λ ( U ) = − q − P ′ g ( U + τ + ∞ ) λ ( U ) = q − P ′ g ( U + τ −∞ ) λ ( U ) = q − P ′ g ( U + τ + ∞ ) r = λ ( U )00 r = − λ ( U ) r = λ ( U )00 r = − λ ( U ) . Concerning (H2) , b is clearly of class C and b (0) = 0. Moreover,det h Db ( U ) (cid:2) r ( U ) r ( U ) (cid:3)i = det " λ ( U ) 00 − λ ( U ) = − λ ( U ) λ ( U )and the latter expression above is non zero by (1.1).Assumptions (H3) and (H4) are immediate by (2.2) and (1.1).An application of [6, Theorem 4] yields the existence of a Lipschitz continuous local semigroup S defined on a domain D enjoying [6, Properties (i)–(iv) in Theorem 4]. Note that (2.5) holdsby [6, Formula (4) and Theorem 4]. (cid:3) Proof of Theorem 2.2.
Given t → (cid:0) ( τ, v ) , v ℓ (cid:1) ( t ), define t → ( U, w )( t ) by means of (2.1). SinceTV (cid:0) U ( t ) (cid:1) + (cid:13)(cid:13)(cid:13) b (cid:0) U ( t, (cid:1) − g (cid:0) w ( t ) (cid:1)(cid:13)(cid:13)(cid:13) = TV (cid:0) ( τ, v )( t ); G (cid:1) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)" v ( t, − ) − v ℓ ( t ) v ( t, m +) − v ℓ ( t ) thanks to (2.5) we obtain that for all t ∈ R + , ( U, w )( t ) ∈ D , D being the domain defined inProposition 2.1. 4or ε > κ >
0, call ( p κ,ε , v κ,ε ) the wave front tracking approximate solutions to (1.4),see also [5, Formula (2.5)] as defined in [5, Section 4], converging to (cid:0) ( P g ( τ ) , v ) , v ℓ (cid:1) first as ε → κ →
0. To simplify the notation, here we omit the introduction of sequences andsubsequences.In the limit ε →
0, by [5, Proof of Theorem 3.3] we have thatlim ε → ( p κ,ε , v κ,ε )( t ) = ( p κ , v κ )( t ) for all t ≥ L ( R ; R ) , where ( p κ , v κ ) solves [5, Formula (2.5)] in the sense of [5, Definition 3.1].In the limit κ →
0, we have thatlim κ → ( p κ , v κ )( t, · ) = ( p, v )( t, · ) for all t ≥ L ( G ; R )lim κ → v κ ( t, · ) = v ℓ ( t ) for all t ≥ L ( L ; R ) v ℓ being independent of z . Introduce v κ,ε ( t ) = 1 m Z m v κ,ε ( t, z ) d zu κ,ε ( t ) = ( p κ,ε , v κ,ε )( s ) (cid:12)(cid:12)(cid:12) G , v κ,ε ( t ) (3.1)and note that the above L convergence implies that u κ,ε ( t ) → u κ ( t ) = ( p κ , v κ )( t ) (cid:12)(cid:12)(cid:12) G , m Z m v κ ( t, z ) d z as ε → t ≥ ,u κ ( t ) → u ( t ) = (cid:0) ( p, v ) , v ℓ (cid:1) ( t ) as κ → t ≥ . (3.2)Following (2.1) and (3.1), introduce the variables U κ,ε ( t, x ) = T g (cid:0) p κ,ε ( t, − x ) (cid:1) − τ −∞ v κ,ε ( t, − x ) − v −∞ T g (cid:0) p κ,ε ( t, x + m ) (cid:1) − τ + ∞ v κ,ε ( t, x + m ) − v + ∞ w κ,ε ( t ) = " v κ,ε ( t ) − v −∞ v κ,ε ( t ) − v + ∞ (3.3)and the distance d (cid:16) ( U, w ) , ( ˜ U , ˜ w ) (cid:17) = (cid:13)(cid:13)(cid:13) ˜ U − U (cid:13)(cid:13)(cid:13) L ( R + ; R ) + k ˜ w − w k . By the convergences (3.2), the definition (3.3) and the continuity of S t d (cid:16) ( U, w )( t ) , S t (cid:0) ( U, w )(0) (cid:1)(cid:17) ≤ lim κ → lim ε → d (cid:16) ( U κ,ε , w κ,ε )( t ) , S t (cid:0) ( U κ,ε , w κ,ε )(0) (cid:1)(cid:17) . (3.4)By [2, Theorem 2.9], denoting by L a Lipschitz constant of S t , d (cid:16) ( U κ,ε , w κ,ε )( t ) , S t (cid:0) ( U κ,ε , w κ,ε )(0) (cid:1)(cid:17) ≤ L Z t lim inf h → h d (cid:16) ( U κ,ε , w κ,ε )( s + h ) , S h (cid:0) ( U κ,ε , w κ,ε )( s ) (cid:1)(cid:17) d s = L Z t lim inf h → h d (cid:16) ( U κ,ε , w κ,ε )( s + h ) , F ( h ) (cid:0) ( U κ,ε , w κ,ε )( s ) (cid:1)(cid:17) d s (3.5)where F is the local flow defined in [6, Formula (5)]. By construction, the last term in the integrandabove is d (cid:16) ( U κ,ε , w κ,ε )( s + h ) , F ( h ) (cid:0) ( U κ,ε , w κ,ε )( s ) (cid:1)(cid:17) (cid:13)(cid:13)(cid:13) U κ,ε ( s + h ) − ¯ S h (cid:0) ¯ U κ,ε ( s ) (cid:1)(cid:13)(cid:13)(cid:13) L ( R + ; R ) + (cid:13)(cid:13)(cid:13)(cid:13) w κ,ε ( s + h ) − h w κ,ε ( s ) + hF (cid:0) U σ , w κ,ε ( s ) (cid:1)i(cid:13)(cid:13)(cid:13)(cid:13) (3.6)where F is as in (2.2), ¯ S is the Standard Riemann Semigroup [2, Chapter 9] generated by ∂ t U + ∂ x f ( U ) = 0, with f as in (2.2),¯ U κ,ε ( s, x ) = ( U κ,ε ( s, x ) x ≥ U σ x < U σ is the unique state satisfying b ( U σ ) = g (cid:0) w κ,ε ( s ) (cid:1) that can be connected to U κ,ε ( t, b and g as in (2.2).Introduce (¯ p, ¯ v )( z ) = ( p κ,ε , v κ,ε )( s, z ) z < p σ , v k,ε )( s ) z ∈ (cid:2) , m/ (cid:2) ( p σm , v k,ε )( s ) z ∈ [ m/ , m ]( p κ,ε , v κ,ε )( s, z ) z > m where v k,ε is defined in (3.1) and the pressure p σ , respectively p σm , is such that the RiemannProblem ∂ t T ( p ) − ∂ z v = 0 ∂ t v + ∂ z p = 0( p, v )(0 , x ) = ( ( p κ,ε , v κ,ε )( s, − ) x < p σ , v κ,ε )( s ) x > , resp. ∂ t T ( p ) − ∂ z v = 0 ∂ t v + ∂ z p = 0( p, v )(0 , x ) = ( ( p σm , v κ,ε )( s ) x < p κ,ε , v κ,ε )( s, m +) x > , is solved by waves with negative, respectively positive, speed. Note that by [4, Lemma 4.1] (cid:12)(cid:12) p κ,ε ( s, − p σ (cid:12)(cid:12) = O (1) (cid:12)(cid:12) v κ,ε ( s, − v κ,ε ( s ) (cid:12)(cid:12) , (cid:12)(cid:12) p κ,ε ( s, m ) − p σm (cid:12)(cid:12) = O (1) (cid:12)(cid:12) v κ,ε ( s, m ) − v κ,ε ( s ) (cid:12)(cid:12) , (3.7)recall that z → p κ,ε ( s, z ) and z → v κ,ε ( s, z ) are locally constant in neighborhoods of z = 0 and z = m , see [5, Formula (4.12)]. Call Σ the Standard Riemann Semigroup [2, Chapter 9] generatedby the p -system ( ∂ t T ( p ) − ∂ z v = 0 ∂ t v + ∂ z p = 0 for z varying on all the real line. Observe that the firstaddend in (3.6) reads (cid:13)(cid:13)(cid:13) U κ,ε ( s + h ) − ¯ S h (cid:0) ¯ U κ,ε ( s ) (cid:1)(cid:13)(cid:13)(cid:13) L ( R + ; R ) ≤ Z G (cid:13)(cid:13)(cid:13) ( p κ,ε , v κ,ε )( s + h, z ) − (cid:0) Σ h (¯ p, ¯ v ) (cid:1) ( z ) (cid:13)(cid:13)(cid:13) d z . (3.8)Assume that at time s no interaction takes place and choose h sufficiently small so that in thetime interval [ s, s + h ] no interaction takes place and no wave hits any of the lines z = ± ε , z = 0, z = m ± ε and z = m .We now continue to estimate the right hand side in (3.8) limited to ] −∞ , z , z , . . . be the points of jump of the map z → ( p κ,ε , v κ,ε )( s, z ). Denote by b λ an upper bound for thecharacteristic speeds in the gas phase. Then, we have Z −∞ (cid:13)(cid:13)(cid:13) ( p κ,ε , v κ,ε )( s + h, z ) − (cid:0) Σ h (¯ p, ¯ v ) (cid:1) ( z ) (cid:13)(cid:13)(cid:13) d z = X z i < − ε Z z i + b λhz i − b λh (cid:13)(cid:13)(cid:13)(cid:13) ( p κ,ε , v κ,ε )( s + h, z ) − (cid:16) Σ h (cid:0) ( p κ,ε , v κ,ε )( s ) (cid:1)(cid:17) ( z ) (cid:13)(cid:13)(cid:13)(cid:13) d z (3.9)+ X z i ∈ ] − ε , [ Z z i + b λhz i − b λh (cid:13)(cid:13)(cid:13)(cid:13) ( p κ,ε , v κ,ε )( s + h, z ) − (cid:16) Σ h (cid:0) ( p κ,ε , v κ,ε )( s ) (cid:1)(cid:17) ( z ) (cid:13)(cid:13)(cid:13)(cid:13) d z (3.10)6 Z − b λh (cid:13)(cid:13)(cid:13) ( p κ,ε , v κ,ε )( s + h, z ) − Σ h (cid:0) (¯ p, ¯ v )( s ) (cid:1)(cid:13)(cid:13)(cid:13) d z . (3.11)A standard procedure yields the estimate of (3.9) by means of [2, (ii) in Lemma 9.1], so that[(3.9)] = O (1) ε h TV( p κ,ε ( s ); ] − ∞ , − ε [ ) . Similarly, since all waves in the strip (cid:3) − ε , (cid:2) have speed ±
1, by [2, (i) in Lemma 9.1] we have[(3.10)] = O (1) h TV( p κ,ε ( s ); ] − ε ,
0[ ) . Consider now (3.11). We use [4, Point 2) in Theorem 2.2] to estimate the difference between( p κ,ε , v κ,ε ) and Σ h (¯ p, ¯ v ) that are solutions, respectively, to the two initial–boundary value problems ∂ t T ( p ) − ∂ z v = 0 ∂ t v + ∂ z p = 0( p, v )(0 , z ) = ( p κ,ε , v κ,ε )( s, v ( t,
0) = v κ,ε ( s,
0) and ∂ t T ( p ) − ∂ z v = 0 ∂ t v + ∂ z p = 0( p, v )(0 , z ) = ( p κ,ε , v κ,ε )( s, v ( t,
0) = v κ,ε ( s )with the mean value v κ,ε as defined in (3.1). Then, we apply [5, Proposition 4.9] to obtain[(3.11)] ≤ O (1) b λ h (cid:12)(cid:12) v κ,ε ( s, − v κ,ε ( s ) (cid:12)(cid:12) by [4, Point 2) in Theorem 2.2] ≤ O (1) b λ h TV( v κ,ε ( s ); L ) by (3.1) ≤ O (1) h κ by [5, Proposition 4.9].Entirely analogous estimates can be applied to bound the similar terms on [ m, + ∞ [. We thuscontinue (3.8) as follows: (cid:13)(cid:13)(cid:13) U κ,ε ( s + h ) − ¯ S h (cid:0) ¯ U κ,ε ( s ) (cid:1)(cid:13)(cid:13)(cid:13) L ( R + ; R ) ≤ O (1) h ε TV( p κ,ε ( s ); ] − ∞ , − ε [ ∪ ] m + ε , + ∞ [ )+ O (1) h TV( p κ,ε ( s ); ] − ε , ∪ ] m, m + ε [ )+ O (1) h κ . We pass now to the second addend in (3.6), using (3.7) and [5, Proposition 4.9], (cid:13)(cid:13)(cid:13)(cid:13) w κ,ε ( s + h ) − h w κ,ε ( s ) + hF (cid:0) U σ , w κ,ε ( s ) (cid:1)i(cid:13)(cid:13)(cid:13)(cid:13) (3.12)= (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) v κ,ε ( s + h ) − v κ,ε ( s ) − hF (cid:16) U σ , w k,ε ( s ) (cid:17) v κ,ε ( s + h ) − v κ,ε ( s ) − hF (cid:16) U σ , w k,ε ( s ) (cid:17) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = √ (cid:12)(cid:12)(cid:12)(cid:12) v κ,ε ( s + h ) − v κ,ε ( s ) − m h ( p σ − p σm ) (cid:12)(cid:12)(cid:12)(cid:12) = √ m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z L v κ,ε ( s + h, z ) d z − Z L v κ,ε ( s, z ) d z − Z s + hs (cid:0) p κ,ε ( σ, − p κ,ε ( σ, m ) (cid:1) d σ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + √ hm (cid:12)(cid:12)(cid:12) ( p σ − p σm ) − (cid:0) p κ,ε ( s, − p κ,ε ( s, m ) (cid:1)(cid:12)(cid:12)(cid:12) ≤ √ m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z s + hs dd σ Z L v κ,ε ( σ, z ) d z d σ − Z s + hs (cid:0) p κ,ε ( σ, − p κ,ε ( σ, m ) (cid:1) d σ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + O (1) h κ ≤ √ m (cid:12)(cid:12)(cid:12)(cid:12)Z s + hs h X z i ∈ ]0 ,m [ (cid:0) v κ,ε ( σ, z i − ) − v κ,ε ( σ, z i +) (cid:1) ˙ z i X z i ∈ ]0 ,m [ (cid:0) p κ,ε ( σ, z i +) − p κ,ε ( σ, z i − ) (cid:1)i d σ (cid:12)(cid:12)(cid:12)(cid:12) + O (1) h κ ≤ √ m Z s + hs X z i ∈ ]0 ,m [ (cid:12)(cid:12)(cid:12)(cid:0) v κ,ε ( σ, z i − ) − v κ,ε ( σ, z i +) (cid:1) ˙ z i + (cid:0) p κ,ε ( σ, z i +) − p κ,ε ( σ, z i − ) (cid:1)(cid:12)(cid:12)(cid:12) d σ + O (1) h κ We estimate the integral term in the latter term above in different ways, depending on the locationof z i : Z s + hs X z i ∈ ]0 ,ε [ ∪ ] m − ε ,m [ (cid:12)(cid:12)(cid:12)(cid:0) v κ,ε ( σ, z i − ) − v κ,ε ( σ, z i +) (cid:1) ˙ z i + (cid:0) p κ,ε ( σ, z i +) − p κ,ε ( σ, z i − ) (cid:1)(cid:12)(cid:12)(cid:12) d σ ≤ Z s + hs X z i ∈ ]0 ,ε [ ∪ ] m − ε ,m [ (cid:12)(cid:12) v κ,ε ( σ, z i − ) − v κ,ε ( σ, z i +) (cid:12)(cid:12) + (cid:12)(cid:12) p κ,ε ( σ, z i +) − p κ,ε ( σ, z i − ) (cid:12)(cid:12) d σ = O (1) h TV (cid:16) p κ,ε ( s ); ]0 , ε [ ∪ ] m − ε , m [ (cid:17) since in ]0 , ε [ ∪ ] m − ε , m [ we have ˙ z i = 1. To bound the remaining terms in (3.12), let z i ∈ (cid:3) ε , m − ε (cid:2) , use [5, Section 4, Lemma 4.1 and Formula (4.3)] and assume that the jump at z i issolved by a 2-rarefaction: (cid:12)(cid:12)(cid:12) − (cid:0) v κ,ε ( σ, z i +) − v κ,ε ( σ, z i − ) (cid:1) ˙ z i + (cid:0) p κ,ε ( σ, z i +) − p κ,ε ( σ, z i − ) (cid:1)(cid:12)(cid:12)(cid:12) ≤ ε (cid:12)(cid:12) v κ,ε ( σ, z i +) − v κ,ε ( σ, z i − ) (cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − κ vuut − T ′ (cid:16) Π κ (cid:0) p κ,ε ( σ, z i − ) (cid:1)(cid:17) κ (cid:0) p κ,ε ( σ, z i +) − p κ,ε ( σ, z i − ) (cid:1) × F (cid:16) Π κ (cid:0) p κ,ε ( σ, z i − ) (cid:1) , Π κ (cid:0) p κ,ε ( σ, z i +) (cid:1)(cid:17) + (cid:0) p κ,ε ( σ, z i +) − p κ,ε ( σ, z i − ) (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ε (cid:12)(cid:12) v κ,ε ( σ, z i +) − v κ,ε ( σ, z i − ) (cid:12)(cid:12) + (cid:12)(cid:12) p κ,ε ( σ, z i +) − p κ,ε ( σ, z i − ) (cid:12)(cid:12) F (cid:16) Π κ (cid:0) p κ,ε ( σ, z i − ) (cid:1) , Π κ (cid:0) p κ,ε ( σ, z i − ) (cid:1)(cid:17) × (cid:12)(cid:12)(cid:12)(cid:12) − F (cid:16) Π κ (cid:0) p κ,ε ( σ, z i − ) (cid:1) , Π κ (cid:0) p κ,ε ( σ, z i +) (cid:1)(cid:17) + F (cid:16) Π κ (cid:0) p κ,ε ( σ, z i − ) (cid:1) , Π κ (cid:0) p κ,ε ( σ, z i − ) (cid:1)(cid:17)(cid:12)(cid:12)(cid:12)(cid:12) = ε (cid:12)(cid:12) v κ,ε ( σ, z i +) − v κ,ε ( σ, z i − ) (cid:12)(cid:12) + O (1) (cid:12)(cid:12) p κ,ε ( σ, z i +) − p κ,ε ( σ, z i − ) (cid:12)(cid:12) (cid:12)(cid:12)(cid:12) Π κ (cid:0) p κ,ε ( σ, z i − ) (cid:1) − Π κ (cid:0) p κ,ε ( σ, z i − ) (cid:1)(cid:12)(cid:12)(cid:12) = ε (cid:12)(cid:12) v κ,ε ( σ, z i +) − v κ,ε ( σ, z i − ) (cid:12)(cid:12) + O (1) κ (cid:12)(cid:12) p κ,ε ( σ, z i +) − p κ,ε ( σ, z i − ) (cid:12)(cid:12) = ε (cid:12)(cid:12) v κ,ε ( σ, z i +) − v κ,ε ( σ, z i − ) (cid:12)(cid:12) + O (1) κ ε (cid:12)(cid:12) p κ,ε ( σ, z i +) − p κ,ε ( σ, z i − ) (cid:12)(cid:12) . When dealing with a 2-shock we obtain the simpler estimate (cid:12)(cid:12)(cid:12) − (cid:0) v κ,ε ( σ, z i +) − v κ,ε ( σ, z i − ) (cid:1) ˙ z i + (cid:0) p κ,ε ( σ, z i +) − p κ,ε ( σ, z i − ) (cid:1)(cid:12)(cid:12)(cid:12) ≤ ε (cid:12)(cid:12) v κ,ε ( σ, z i +) − v κ,ε ( σ, z i − ) (cid:12)(cid:12) while the cases of waves of the first family are entirely analogous.Summarizing:[(3.12)] ≤ O (1) h TV (cid:16) p κ,ε ( s ); ]0 , ε [ ∪ ] m − ε , m [ (cid:17) O (1) h ε (cid:18) TV (cid:16) v κ,ε ( s ); ] ε , m − ε [ (cid:17) + TV (cid:16) p κ,ε ( s ); ] ε , m − ε [ (cid:17)(cid:19) + O (1) h κ By [5, Formula (4.32) in Proposition 4.9] we finally obtain, d (cid:16) ( U κ,ε , w κ,ε )( s + h ) , F ( h ) (cid:0) ( U κ,ε , w κ,ε )( s ) (cid:1)(cid:17) ≤ O (1) h ε TV( p κ,ε ( s ); ] − ∞ , − ε [ ∪ ] m + ε , + ∞ [ )+ O (1) h TV( p κ,ε ( s ); ] − ε , ∪ ] m, m + ε [ )+ O (1) h κ + O (1) h TV (cid:16) p κ,ε ( s ); ]0 , ε [ ∪ ] m − ε , m [ (cid:17) + O (1) h ε (cid:18) TV (cid:16) v κ,ε ( s ); ] ε , m − ε [ (cid:17) + TV (cid:16) p κ,ε ( s ); ] ε , m − ε [ (cid:17)(cid:19) = O (1) h (cid:18) ε + κ + TV (cid:16) p κ,ε ( s ); [ − ε , ε [ ∪ ] m − ε , m + ε [ (cid:17)(cid:19) whence, by (3.5) d (cid:16) ( U κ,ε , w κ,ε )( t ) , S t (cid:0) ( U κ,ε , w κ,ε )(0) (cid:1)(cid:17) ≤ O (1) Z t (cid:18) ε + κ + TV (cid:16) p κ,ε ( s ); [ − ε , ε [ ∪ ] m − ε , m + ε [ (cid:17)(cid:19) d s Changing the order of integration and using [5, Formula (4.33) in Proposition 4.9], we get Z t TV (cid:16) p κ,ε ( s ); [ − ε , ε [ ∪ ] m − ε , m + ε [ (cid:17) d s = Z [ − ε ,ε [ ∪ ] m − ε ,m + ε [ TV (cid:0) p κ,ε ( · , z ); [0 , t ] (cid:1) d z = O (1) ε κ so that d (cid:16) ( U κ,ε , w κ,ε )( t ) , S t (cid:0) ( U κ,ε , w κ,ε )(0) (cid:1)(cid:17) = O (1) ( ε + κ ) t + ε κ ! Using (3.4), the proof of 1. is completed.Hence, the trajectory of the semigroup S with initial datum ( U, w )(0) is defined for all t ∈ R + . (cid:3) Acknowledgment:
The present work was supported by the PRIN 2012 project
Nonlinear Hyper-bolic Partial Differential Equations, Dispersive and Transport Equations: Theoretical and Applica-tive Aspects and by the GNAMPA 2014 project
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