EXISTENCE OF SOLUTIONS TO SYSTEMS OF EQUATIONS MODELLING COMPRESSIBLE FLUID FLOW
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1 Electronic Journal o Dierential Equations, Vol. 15 (15, No. 16, pp ISSN: URL: or tp ejde.math.txstate.e EXISTENCE OF SOLUTIONS TO SYSTEMS OF EQUATIONS MODELLING COMPRESSIBLE FLUID FLOW DIANE L. DENNY Abstract. We study a system o nonlinear equations that models the low o a compressible, inviscid, barotropic luid. The value o the density is known at a point in the spatial domain and the initial value o the velocity is known. The purpose o this paper is to prove the existence o a unique, classical solution to this system o equations under periodic boundary conditions. 1. Introction In this article, we consider the ollowing system o equations which arises rom a model o the multi-dimensional low o a compressible, inviscid, barotropic luid: v t + v v + ρ 1 p = (1.1 1 ψ v = ψ t ψ 1 v ψ (1. p = ˆp(ρ (1.3 where v is the velocity, ρ is the density, p is the pressure, and ψ is a given positive smooth unction. The luid s thermodynamic state is determined by the density ρ, and the pressure p is determined rom the density by an equation o state p = ˆp(ρ. We assume that p is a given smooth unction o ρ, and we assume that p and dp dρ are positive unctions. It ollows rom (1.1 and (1.3 that v t + v v + ρ 1 ˆp (ρ ρ = (1.4 The density ρ is a positive unction which satisies the condition that ρ(x, t = b(t, where x is a given point in the domain Ω and where b is a given positive smooth unction o t. The initial condition or the velocity is v(x, = v (x. The purpose o this paper is to prove the existence o a unique classical solution v, ρ to the system o equations (1., (1.4, where ρ(x, t = b(t and v(x, = v (x, or t T and under periodic boundary conditions. That is, we choose or our domain the N-dimensional torus T N, where N = or N = 3. 1 Mathematics Subject Classiication. 35A5. Key words and phrases. Existence; uniqueness; compressible; luid. c 15 Texas State University - San Marcos. Submitted February 8, 15. Published August 17, 15. 1
2 D. L. DENNY EJDE-15/16 We will also present a proo that the standard conservation o mass equation ρ t + v ρ + ρ v = is approximately satisied by the solution ρ, v to equations (1., (1.4. Equations (1., (1.4 are an approximation to Euler s equations or a compressible, barotropic luid (see, e.g., []. In our previous related work, the existence o a unique classical solution to a similar system o equations modelling the low o an incompressible, barotropic luid was proven in [3], with the condition that ρ(x, t was given and where v =. 1 ψ In this paper, the luid is compressible and v = ψ t ψ 1 v ψ. The proo uses new inequalities which are proven in Section 4. The proo o the existence theorem is based on the method o successive approximations, in which an iteration scheme, based on solving a linearized version o the equations, is designed and convergence o the sequence o approximating solutions to a unique solution satisying the nonlinear equations is proven. The ramework o the proo ollows one used, or example, by A. Majda to prove the existence o a solution to a system o conservation laws [8]. Embid [7] also uses the same general ramework to prove the existence o a solution to equations or zero Mach number combustion. Under this ramework, the convergence proo is presented in two steps. In the irst step, we prove uniorm boundedness o the approximating sequence o solutions in a high Sobolev space norm. The second step is to prove contraction o the sequence in a low Sobolev space norm. Standard compactness arguments inish the proo. The paper is organized as ollows. First the main result, Theorem.1, is presented and proven in the next section. Next, we present a proo that the standard conservation o mass equation ρ t + v ρ + ρ v = is approximately satisied by the solution ρ, v to equations (1., (1.4. Finally, lemmas supporting the proo o the existence theorem are provided in Section 3 (which presents a proo o the existence o a solution to the linearized equations used in the iteration scheme and in Section 4 (which contains other lemmas used in the proo o the theorem.. Existence theorem The main tools utilized in the existence proo are a priori estimates. We will work with the Sobolev space H s (Ω (where s is an integer o real-valued unctions in L (Ω whose distribution derivatives up to order s are in L (Ω, with norm given by u s = α s Ω Dα u dx and inner proct (u, v s = α s Ω (Dα u (D α vdx. Here, we adopt the standard multi-index notation. We will also use the notation u s = r s Ω Dr u dx, where D r u is the set o all space derivatives D α u with α = r, and D r u = α =r Dα u, where r is an integer. Also, we let both u and Du denote the gradient o u. We will use standard unction spaces. L ([, T ], H s (Ω is the space o bounded measurable unctions rom [, T ] into H s (Ω, with the norm u s,t = sup t T u(t s. C([, T ], H s (Ω is the space o continuous unctions rom [, T ] into H s (Ω. The purpose o this paper is to prove the ollowing theorem: Theorem.1. Let Ω = T N, the N-dimensional torus, where N =, 3. Let ψ be a given positive smooth unction o x and t, and let ψ(x, t c or x Ω and d t T, where c is a positive constant. And let dt ψdx =. Let p be a Ω given smooth unction o ρ, and let p and dp dρ be positive unctions. Let b be a given positive smooth unction o t, and let x be a given point in Ω. Then or time interval
3 EJDE-15/16 EXISTENCE OF SOLUTIONS 3 t T, equations (1., (1.4, subject to the condition that ρ(x, t = b(t or t T and the initial condition v(x, = v (x H s (Ω where s > N +, have a unique classical solution ρ, v, where ρ is a positive unction, provided that v s, ψ t s,t, ψ tt s 1,T, and max t T b(t are suiciently small. The regularity o the solution is ρ C([, T ], C 3 (Ω L ([, T ], H s+1 (Ω v C([, T ], C (Ω L ([, T ], H s (Ω v t C([, T ], C1 (Ω L ([, T ], H s 1 (Ω Proo. We begin by making a change o variables. First, we let h = ln (ρ, and we deine the composite unction g = ˆp exp, where exp (h = e h, so that g(h = (ˆp exp(h = ˆp (e h = ˆp (e ln (ρ = ˆp (ρ. Then equation (1.4 can be equivalently written as v + v v + g(h h = (.1 t Note that g is a given positive, smooth unction o h. We deine the interval G 1 R by G 1 = ( 3 min t T ln (b(t, 1 max t T ln (b(t, where ln (b(t < or t T, since b(t < 1 by assumption. Let the positive constant c 3 = min h G 1 g(h. We will prove that h(x, t G 1 or x Ω, t T. Recall that ψ(x, t c >. We now deine = 1 c c 3 ψ, and thereore (x, tg(h(x, t = ψ(x,tg(h(x,t c c 3 1 or x Ω, t T. Under this change o variable, equation (1. can be equivalently written as 1 v = t 1 v (. Next, we let u = v. Then equations (.1, (. can be equivalently written as u t = 1 u u + 1 ( uu + u 1 g(h h (.3 t u = (.4 t We now use the Helmholtz decomposition u = P u + Qu = w + φ, where w = P u and φ = Qu and w = (see, e.g., [], [6]. Here, P and Q = I P are orthogonal projection operators. Then equations (.3, (.4 become w t = φ 1 (w + φ (w + φ t + 1 ( (w + φ(w + φ + (w + φ 1 t g(h h (.5 w = (.6 φ = (.7 t At time t =, we have u(x, = (x, v(x, H s (Ω, and w(x, is the solenoidal component o u(x,, so that w(x, = P u(x, and w(x, =. The compatibility condition or solving the elliptic equation (.7 is = Ω t dx = d dt Ω dx, where Ω = TN, the N-dimensional torus. Thereore we require that 1 ψdx =, where = d dt Ω c c 3 ψ.
4 4 D. L. DENNY EJDE-15/16 We will prove the existence o a unique solution w, h, φ to (.5, (.6, (.7. It ollows that u, h is a solution to (.3, (.4, where u = w + φ. And thereore it ollows that v, ρ is a solution to (1., (1.4, where v = 1 1 u, where = c c 3 ψ, and where ρ = e h. We now proceed with the proo. We will construct the solution w, h, φ to (.5, (.6, (.7 through an iteration scheme. To deine the iteration scheme, we will let the terms o the sequence o approximate solutions consist o the unctions w k and h k. Note that φ is immediately determined by solving the elliptic equation (.7, since is a given unction. For k =, 1,,..., construct w k+1, h k+1 rom the previous iterates w k, h k by solving the linear system o equations w k+1 t = φ t 1 (wk + φ (w k+1 + φ + 1 ( (wk + φ(w k+1 + φ + (w k+1 + φ 1 t g(hk h k+1, (.8 w k+1 =, (.9 under periodic boundary conditions, and satisying h k+1 (x, t = ln(ρ(x, t = ln (b(t, and with the initial data w k+1 (x, = w(x,. Set the initial iterate w (x, t = w(x,, the initial data. And set the initial iterate h (x, t = ln (ρ(x, t = ln (b(t. Existence o a suiciently smooth solution to the system o equations (.8, (.9 or ixed k which satisies h k+1 (x, t = ln (b(t ollows rom the proo given in Lemma 3.1 in Section 3. We proceed now to prove convergence o the iterates as k to a unique classical solution o (.5, (.6, (.7. First, we will prove uniorm boundedness o the approximating sequence in a high Sobolev space norm. Then, we will prove contraction o the approximating sequence in a low Sobolev space norm. A standard compactness argument completes the proo (see Embid [6], Majda [8]. Proposition.. Assume that the hypotheses o Theorem.1 are satisied. Let ɛ = max t T b(t and let max{ v s, t s,t, tt s 1,T } max b(t = ɛ, t T where ɛ < 1, and suppose that ɛ is suiciently small. Let R be a given positive constant such that R 1 max t T ln (b(t. Then there exist positive constants L 1, L, L 3, L 4, L 5 such that the ollowing hold or k = 1,, 3..., (a w k s,t ɛl 1, (b h k s,t ɛl, (c h k s+1,t L 3, (d w k / t s 1,T L 4, (e h k h L,T R ( g(h k (x, t s,t L 5 where s > N +, N =, 3, and where ɛl 1 1, ɛl 1. And there exist constants c 1, c, c 3, c 4 such that c 1 h k (x, t c < and < c 3 g(h k (x, t c 4 or x Ω, t T, k = 1,, 3....
5 EJDE-15/16 EXISTENCE OF SOLUTIONS 5 Proo. The proo is by inction on k. We show only the inctive step. We will derive estimates or h k+1, h k+1, and w k+1, and then use these estimates to prescribe L 1, L, L 3, L 4, L 5 a priori, independent o k, so that i h k, h k, w k satisy the estimates in (a-(, then h k+1, h k+1, w k+1 also satisy the same estimates. In the estimates that ollow, we use C to denote a generic constant whose value may change rom one instance to the next, but is independent o ɛ, L 1, L, L 3, L 4, L 5. We requently use the well-known Sobolev space inequality uv r C u r v r or r (rom Lemma 4.1 in Section 4 while making the needed estimates. And in the estimates, s > N +, s = [ N ] + 1 = or N =, 3, and s 4, and s 1 = max{s 1, s } = s 1. Also, we will requently use the estimate φ r C t r 1 or r 1 rom Lemma 4.7 in Section 4, where φ = t. Estimate or h k+1 s: Applying the divergence operator to equation (.8 yields (g(h k h k+1 = φ t ( ( 1 (wk + φ T : ( (w k+1 + φ 1 (wk + φ φ + ( 1 ( (wk + φ(w k+1 + φ + ((w k+1 + φ 1 t (.1 Applying estimate (4. rom Lemma 4.7 in Section 4 to equation (.1, where we use the inequality (x, tg(h k (x, t 1 by the inction hypothesis, yields h k+1 s C D(g(h k j s 1 ( φ t s 1 j= + ( ( 1 (wk + φ T : ( (w k+1 + φ s 1 j= D(g(h k j s 1 ( 1 (wk + φ φ s ( (wk + φ(w k+1 + φ s D(g(h k j s 1 (w k+1 + φ 1 t s C j= j= g(h k j s ( φ t s 1 + ( 1 (wk + φ s 1 (w k+1 + φ s 1 j= g(h k j s ( 1 s 1 w k + φ s 1 φ s s s w k + φ s w k+1 + φ s g(h k j s w k+1 + φ s 1 t s j=
6 6 D. L. DENNY EJDE-15/16 C C C j= s g(h k j s ( φ t s s( w k s + φ s( w k+1 s + φ s j= j= j= j= s g(h k j s 1 s 1( w k s 1 + φ s 1 φ s s g(h k j s 1 s s( w k s + φ s( w k+1 s + φ s s g(h k j s ( w k+1 s + φ s 1 s t s s (L 5 j ( tt s s(ɛl 1 + t s 1( w k+1 s + t s 1 j= s (L 5 j ( 1 s 1(ɛL 1 + t s t s s s(ɛl 1 + t s 1( w k+1 s + t s 1 j= j= s (L 5 j ( w k+1 s + t s 1 1 s t s s,t (L 5 j( ɛ + 1 s,t (ɛl 1 + ɛ( w k+1 s + ɛ + 1 s 1,T (ɛl 1 + ɛ(ɛ j= s,t (L 5 j( 1 4 s,t s,t (ɛl 1 + ɛ( w k+1 s + ɛ + ( w k+1 s + ɛ 1 s,t (ɛ C 1 (ɛ + w k+1 s (.11 where g(h k s L 5 by the inction hypothesis. And L 5 = C g s,g 1, where we deine g s,g1 = max{ dj g dh (h j : h G 1, j s}. And we used the estimate w k s ɛl 1 by the inction hypothesis, where ɛl 1 1. We used the act that φ = t, and we used the estimate φ r C t r 1 or r 1 rom Lemma 4.7 in Section 4. And we used the assumptions that tt s 1 ɛ and t s ɛ. The constant C 1 depends on s,t, 1 s,t, s,t, and g s,g1. Estimate or w k+1 s: Recall that the initial condition w k+1 (x, = w(x, = P u(x, = P ((x, v, where P is an orthogonal projection operator. Then by applying estimate (4.1 rom Lemma 4.8 to equation (.8 we obtain the estimate w k+1 s e βt P ((x, v s + e βt C + e βt C ( φ t s + 1 (wk + φ ( φ sdτ ( 1 ( (wk + φ φ s + φ t s + g(h k h k+1 sdτ
7 EJDE-15/16 EXISTENCE OF SOLUTIONS 7 e βt (x, v s + e βt C + e βt C ( φ t s + 1 s w k + φ s φ s+1dτ ( 1 s s w k + φ s φ s + φ s 1 s t s + s g(h k s h k+1 sdτ e βt C (x, s v s + e βt C + e βt C ( φ t s + 1 s( w k s + φ s φ s+1dτ ( 1 s s( w k s + φ s φ s + φ s 1 s t s + s g(h k s h k+1 sdτ e βt C s,t v s + e βt C + e βt C ( tt s s(ɛl 1 + t s 1 t sdτ ( 1 s s(ɛl 1 + t s 1 t s 1 + t s 1 1 s t s + s g(h k sc 1 (ɛ + w k+1 sdτ ( e βt Cɛ s,t + e βt CT ɛ + 1 s,t (ɛl 1 + ɛɛ s,t s,t (ɛl 1 + ɛɛ + e βt CT ɛ 1 s,t + e βt CT s,t L 5 C 1 ɛ + e βt C s,t L 5 C 1 w k+1 sdτ e βt C ɛ(1 + T + e βt C w k+1 sdτ (.1 where we used the estimates w k s ɛl 1 1 and g(h k s L 5, by the inction hypothesis, where L 5 = C g s,g 1. And we used estimate (.11 or h k+1 s. We also used the estimates φ r C t r 1 and φ t r C tt r 1 or r 1 rom Lemma 4.7, where φ = t. And we used the assumptions that t s,t ɛ, tt s 1,T ɛ, and v s ɛ. Here C depends on s,t, 1 s,t, s,t, and g s,g1. From Lemma 4.8 in Section 4 we obtain the estimate: β C(1 + 1 s 1+1,T ( w k s1+1,t + φ s1+1,t 1 s 1+1,T s1+1,t ( w k s1+1,t + φ s1+1,t 1 C ( s,t (ɛ 1/ L 1/ 1 + t s 1,T t s 1+1,T (.13 1 s,t s,t (ɛ 1/ L 1/ 1 + t s 1,T 1 s,t t s,t C 3 where s 1 = s 1 and s 4. We used the estimate w k s ɛ 1/ L 1/ 1 1, by the inction hypothesis. And we used the assumption that t s,t ɛ < 1. And C 3 depends on 1 s,t, s,t.
8 8 D. L. DENNY EJDE-15/16 By applying Gronwall s inequality to (.1, we obtain or w k+1 the estimate w k+1 s ɛc 4 (1 + T (1 4 T e C4T (.14 where C 4 = e C3T C. We now deine L 1 = C 4 (1 + T (1 4 T e C4T. It ollows that w k+1 s ɛl 1, where we choose ɛ suiciently small so that ɛl 1 1. Substituting inequality (.14 into (.11 yields h k+1 s C 1 (ɛ + w k+1 s ɛc 1 (1 + L 1 (.15 We now deine L = C 1 (1+L 1. It ollows that h k+1 s ɛl, where we choose ɛ suiciently small so that ɛl 1. And L 1, L depend on s,t, 1 s,t, s,t, and g s,g1. This completes the proo o parts (a and (b o Proposition.. Estimate or h k+1 s+1: By Lemma 4.3 in Section 4, and using inequality (.15, we obtain h k+1 s+1 C( h k+1 (x, t s+1 + (h k+1 (x, t s + h k+1 s C( Ω max t T hk+1 (x, t + ɛl C( Ω max t T ln (b(t + 1 = L 3 (.16 since h k+1 (x, t = ln (b(t and ɛl 1. This completes the proo o part (c o Proposition.. Estimate or wt k+1 s 1: We immediately obtain rom equation (.8 the ollowing: wt k+1 s 1 C φ t s 1 1 s 1( w k s 1 + φ s 1( w k+1 s + φ s 1 s 1 s 1( w k s 1 + φ s 1( w k+1 s 1 + φ s 1 ( w k+1 s 1 + φ s 1 1 t s 1 s 1 g(h k s 1 h k+1 s 1 C tt s,t 1 s 1,T (ɛl 1 + t s,t (ɛl 1 + t s 1,T 1 4 s 1,T s 1,T (ɛl 1 + t s,t (ɛl 1 + t s,t (ɛl 1 + t s,t 1 s 1,T t s 1,T s 1,T L 5 (ɛl (.17 where we used the estimates w k+1 s ɛl 1 1 and h k+1 s 1 ɛl 1 rom (.14, (.15. And we used the estimates w k s ɛl 1 1, g(h k s L 5, by the inction hypothesis, where L 5 = C g. s,g 1 And we used the estimate φ r C t r 1 or r 1 by Lemma 4.7, where φ = t. And we used the assumptions that t s,t ɛ < 1 and tt s 1,T ɛ < 1. And C 5 depends on 1 s 1,T, s 1,T, s 1,T, and g s,g1. We now deine L 4 = C 5. This completes the proo o part(d o Proposition..
9 EJDE-15/16 EXISTENCE OF SOLUTIONS 9 Estimate or h k+1 h L : By Lemma 4.3, we obtain the inequality h k+1 h L C (h k+1 h 1 = C h k+1 1 Cɛ 1/ L 1/ R (.18 where we used estimate (.15, and where we choose ɛ suiciently small so that Cɛ 1/ L 1/ R. Here we used the acts that h k+1 (x, t = h (x, t and that h (x, t = ln (b(t. This completes the proo o part (e. Since R 1 max t T ln (b(t, h k+1 h 1 max t T ln (b(t. And since h (x, t = ln (b(t, where max t T b(t = ɛ and ɛ < 1, so that ln (b(t <, we obtain the inequalities h k+1 (x, t h (x, t + 1 max ln (b(t = ln (b(t 1 t T max ln (b(t t T 1 max ln (b(t = 1 (.19 t T ln (ɛ h k+1 (x, t h (x, t 1 max ln (b(t = ln (b(t + 1 t T max ln (b(t t T 3 (. min (ln (b(t t T We have deined the interval G 1 R by G 1 = (c 1, c = ( 3 min ln (b(t, 1 t T max ln (b(t t T = ( 3 min ln (b(t, 1 ln (ɛ, t T so that h k+1 (x, t G 1 or x Ω, t T. Since g is a smooth, positive unction o h, there exist constants c 3 = min h G 1 g(h, c 4 = max h G 1 g(h, so that < c 3 g(h c 4 or all h G 1. And so < c 3 g(h k+1 (x, t c 4 or x Ω, t T. Since g(h = (ˆp exp(h = ˆp (e h, where ˆp is a smooth unction, it ollows that dg dh (h = ˆp (e h e h max h G 1 ˆp (e h e 1 ln ɛ Cɛ 1/ (.1 d g dh (h = ˆp (e h e h + ˆp (e h (e h max h G 1 ˆp (e h e 1 ln ɛ + max h G 1 ˆp (e h ( e 1 ln ɛ Cɛ 1/ or h G 1, where we assume that max h G 1 ˆp (e h C and that ɛ 1/ max h G 1 ˆp (e h C, where C does not depend on ɛ. From (.1, (., it ollows that where we deine r,g1 (. dg dh 1,G1 Cɛ 1/ (.3 = max{ dj dh j (h : h G 1, j r}.
10 1 D. L. DENNY EJDE-15/16 Estimate or g(h k+1 s: By Lemma 4.4 and by (.15 and by the inequality g(h k+1 (x, t c 4 which was just proven, it ollows that g(h k+1 s g(h k+1 D(g(h k+1 s 1 c 4 Ω dg dh s 1,G 1 Dh k+1 j s 1 c 4 Ω dg dh 1 j s s 1,G 1 1 j s (ɛl j (.4 C g s,g 1 = L 5 where we used that ɛl 1, and we used the act that c 4 = (max h G 1 g(h = g,g 1. This completes the proo o Proposition.. Next, we give the proo o contraction in a low Sobolev space norm. Proposition.3. Assume that the hypotheses o Theorem.1 hold. Then (a k= ( wk+1 w k 1,T + ɛ (hk+1 h k 1,T <, (b k= hk+1 h k,t <, (c k= wk+1 t wt k,t <. Proo. Subtracting equation (.8 or w k, ρ k rom equation (.8 or w k+1, ρ k+1 yields t (wk+1 w k = 1 (wk + φ (w k+1 w k 1 (wk w k 1 (w k + φ + 1 ( (wk + φ(w k+1 w k + 1 ( (wk w k 1 (w k + φ + (w k+1 w k 1 t g(hk (h k+1 h k (g(h k g(h k 1 h k (.5 where (w k+1 w k = and where w k+1 (x, w k (x, =. In the estimates that ollow, we will requently use the well-known Sobolev space inequalities uv 1 C u 3 v 1, and uv r C u r v r or r, and u L C u s, where s = [ N ] + 1 = or N =, 3 (rom Lemma 4.1 in Section 4. Estimate or (h k+1 h k 1: Applying the divergence operator to (.5 yields (g(h k (h k+1 h k = ( ( 1 (wk + φ T : ( (w k+1 w k ( 1 (wk w k 1 (w k ( 1 + φ + ( (wk + φ(w k+1 w k + ( 1 ( (wk w k 1 (w k + φ ((g(h k g(h k 1 h k + ( (w k+1 w k 1 t (.6
11 EJDE-15/16 EXISTENCE OF SOLUTIONS 11 Applying estimate (4. rom Lemma 4.7 to equation (.6, and using the act that (x, tg(h k (x, t 1, and using Lemma 4.1, yields (h k+1 h k 1 C D(g(h k j ( ( 1 (wk + φ T : ( (w k+1 w k j= j= ( D(g(h k j 1 (wk w k 1 (w k + φ ( (wk + φ(w k+1 w k 1 ( D(g(h k j 1 ( (wk w k 1 (w k + φ 1 C j= + (w k+1 w k 1 j= t 1 j= D(g(h k j (g(hk g(h k 1 h k 1 ( g(h k j 3 ( 1 (wk + φ L (wk+1 w k + 1 (wk w k 1 1 (w k + φ 3 ( g(h k j 3 1 ( (wk + φ 3 w k+1 w k 1 j= + 1 ( (wk w k 1 1 w k + φ 3 ( g(h k j 3 w k+1 w k 1 1 t 3 + g(h k g(h k 1 1 h k 3 C j= j= ( 3 g(hk j 3 ( 1 (wk + φ w k+1 w k w k w k 1 1 w k + φ 4 ( 3 g(hk j w k + φ 3 w k+1 w k 1 j= w k w k 1 1 w k + φ 3 ( 3 g(hk j 3 w k+1 w k t 3 + g(h k j= g(h k h k 3
12 1 D. L. DENNY EJDE-15/16 C j= 3 (L 5 j( 1 3( w k 3 + φ 3 w k+1 w k w k w k 1 1( w k 4 + φ 4 3 (L 5 j( 1 3( w k 3 + φ 3 w k+1 w k 1 j= w k w k 1 1( w k 3 + φ 3 3 (L 5 j( w k+1 w k t 3 C j= + g(h k g(h k h k 3 j= 3,T (L 5 j( 1 3,T (ɛl 1 + t,t w k+1 w k ,T w k w k 1 1(ɛL 1 + t 3,T 3,T (L 5 j 1 4 3,T 3,T (ɛl 1 + t,t w k+1 w k 1 j= 3,T (L 5 j 1 4 3,T 3,T w k w k 1 1(ɛL 1 + t,t j= j= 3,T (L 5 j( w k+1 w k 1 1 3,T t 3,T + g(h k g(h k 1 1 3,T (ɛl C 6 w k+1 w k 1 6 w k w k g(h k g(h k 1 1 (.7 where we used the estimates w k 4 ɛl 1 1, h k 3 ɛl 1, g(h k 3 L 5, where L 5 = C g s,g 1, rom Proposition.. We used the assumption that t 3,T ɛ < 1. And we used the estimate φ r C t r 1 or r 1 rom Lemma 4.7. And C 6 depends on 1 3,T, 3,T, 3,T, g s,g1. By Lemma 4.5 and Lemma 4.3, we obtain the estimate g(h k g(h k 1 1 C dg (1 + h k dh 1,G 1 + h k 1 h k h k 1 C dg dh 1,G 1 (1 + ɛl (h k h k 1 (.8 1 Cɛ (h k h k 1 1 where we used the estimate dg Cɛ by inequality (.3, and we used the dh 1,G 1 estimates h k ɛl 1, h k 1 ɛl 1 rom Proposition.. We used Lemma 4.3 rom Section 4, as well as the act that h k (x, t = h k 1 (x, t, to obtain the estimate h k h k 1 C (h k h k 1 1.
13 EJDE-15/16 EXISTENCE OF SOLUTIONS 13 Substituting (.8 into (.7 yields (h k+1 h k 1 C 7 w k+1 w k 1 7 w k w k ɛc 7 (h k h k 1 1 (.9 where C 7 = C 6 (1. Estimate or w k+1 w k 1: Applying estimate (4.1 rom Lemma 4.8 to equation (.5, and using Lemma 4.1, yields w k+1 w k 1 Ce βt 1 (wk w k 1 (w k + φ 1dτ e βt ( 1 ( (wk w k 1 (w k + φ 1 + g(h k (h k+1 h k 1dτ e βt (g(h k g(h k 1 h k 1dτ ( Ce βt 1 (wk w k 1 1 (w k + φ (wk w k 1 1 w k + φ 3 dτ e βt ( g(h k 3 (h k+1 h k 1 + g(h k g(h k 1 1 h k 3dτ Ce βt ( 1 3 w k w k 1 1( w k 4 + φ w k w k 1 1( w k 3 + φ 3dτ e βt ( 3 g(h k 3 (h k+1 h k 1 + g(h k g(h k h k 3dτ Ce βt ( 1 3 w k w k 1 1(ɛL 1 + t w k w k 1 1(ɛL 1 + t dτ ( e βt ( 3(L 5 C 7 w k+1 w k 1 7 w k w k ɛc 7 (h k h k 1 1 dτ e βt ɛ (h k h k 1 1 3(ɛL dτ C 8 ( w k w k ɛ (h k h k 1 1dτ 8 w k+1 w k 1dτ where β C ( ,T ( w k 3,T + φ 3,T + 1 3,T 3,T ( w k 3,T (.3
14 14 D. L. DENNY EJDE-15/16 + φ 3,T + 1 t 3,T C 3 rom estimate (.13. Here we used estimate (.9 or (h k+1 h k 1, and we used estimate (.8 or g(h k g(h k 1 1. And we used the estimates w k 4 ɛl 1 1, h k 3 ɛl 1, and g(h k 3 L 5, where L 5 = C g s,g 1, rom Proposition.. And we used the assumption that t 3 ɛ < 1. And C 8 depends on 1 3,T, 3,T, 3,T, and g s,g1. Applying Gronwall s inequality to (.3 yields w k+1 w k 1 C 9 ( w k w k ɛ (h k h k 1 1dτ (.31 where C 9 = C 8 (1 8 T e C8T. Substituting (.31 into (.9 yields (h k+1 h k 1 C 1 ( w k w k ɛ (h k h k ( w k w k ɛ (h k h k 1 1dτ (.3 where C 1 = C 7 (1 9. Multiplying (.3 by ɛ and then adding the resulting inequality to (.31, yields w k+1 w k 1 + ɛ (h k+1 h k 1 ɛc 11 ( w k w k ɛ (h k h k ( w k w k ɛ (h k h k 1 1dτ (.33 where C 11 = C 9 1. Applying Lemma 4.6 to (.33, where we deine Z k (t = w k w k 1 1+ɛ (h k h k 1 1, yields the inequality w k+1 w k 1,T + ɛ (h k+1 h k 1,T (ɛc 11 k ( e T ɛ w 1 w 1,T + ɛ (h 1 h (.34 1,T where we choose ɛ suiciently small so that ɛc 11 < 1. Then it ollows rom (.34 that ( w k+1 w k 1,T + ɛ (h k+1 h k 1,T <. (.35 k= This completes the proo o part (a o Proposition.3. By Lemma 4.3 and the act that h k+1 (x, t = h k (x, t, we have the inequality h k+1 h k C (h k+1 h k 1. Then rom (.35, it ollows that h k+1 h k,t C (h k+1 h k 1,T <. (.36 k= This completes the proo o part (b o Proposition.3. Estimate or wt k+1 wt k : From (.5 we obtain the inequality t (wk+1 w k k= C 1 (wk + φ (w k+1 w k 1 (wk w k 1 (w k + φ
15 EJDE-15/16 EXISTENCE OF SOLUTIONS 15 1 ( (wk + φ(w k+1 w k 1 ( (wk w k 1 (w k + φ (w k+1 w k 1 t g(h k (h k+1 h k (g(h k g(h k 1 h k C 1 ( w k L L + φ L (wk+1 w k 1 w k w k 1 L ( Dw k L + D( φ L 1 L L ( wk L + φ L wk+1 w k 1 L L wk w k 1 ( w k L + φ L w k+1 w k 1 t L L g(hk L (hk+1 h k L g(hk g(h k 1 h k L C 1 (ɛl 1 + t 1 w k+1 w k 1 1 wk w k 1 1(ɛL 1 + t 1 4 (ɛl 1 + t 1 w k+1 w k w k w k 1 1(ɛL 1 + t 1 w k+1 w k 1 1 t (L 5 (h k+1 h k 1 g(h k g(h k 1 (ɛl C 1 ( w k+1 w k 1,T + w k w k 1 1,T + (h k+1 h k 1,T + ɛ h k h k 1,T (.37 where we used the estimates g(h k L C g(hk CL 5, where L 5 = C g s,g 1, and w k L C wk CɛL 1, Dw k L C wk 3 CɛL 1, and h k L C h k CɛL rom Lemma 4.1 and Proposition., where ɛl 1 1 and ɛl 1. And we used the estimate g(h k g(h k 1 C dg dh h k h k 1,G 1 Cɛ h k h k 1 by Lemma 4.5 and estimate (.3, where we use the act that dg dh,g 1 dg dh Cɛ. And we used the estimates φ 1,G 1 L C φ C t 1 and D( φ L C φ 3 C t rom Lemma 4.7. And we used the assumption that t ɛ < 1. And C 1 depends on 1,T,,T,,T, and g s,g1. Then by (.35, (.36, (.37, it ollows that t (wk+1 w k,t <. (.38 k= This completes the proo o Proposition.3.
16 16 D. L. DENNY EJDE-15/16 Using Proposition. and Proposition.3, we now complete the proo o Theorem.1. From Proposition.3, w k+1 w k 1,T and hk+1 h k,t as k. Thereore, we conclude that there exist w C([, T ], H 1 (Ω and h C([, T ], H (Ω such that w k w 1,T, and h k h,t, as k. Using the interpolation inequalities g s C g γ 1 g 1 γ s and g s +1 C g γ g 1 γ s+1, where γ = s s s 1, and 1 < s < s, and s 4 (see Lemma 4.1, and using Proposition., we can conclude that w k w s,t and h k h s +1,T as k or 1 < s < s. For s > N +, Sobolev s lemma implies that wk w in C([, T ], C (Ω and h k h in C([, T ], C 3 (Ω. From the linear system o equations (.8, (.9 it ollows that wt k w t s 1,T, as k, so that wt k w t in C([, T ], C 1 (Ω, and w, h is a classical solution o the system o equations (.5, (.6. The additional acts that w L ([, T ], H s (Ω, and h L ([, T ], H s+1 (Ω can be deced using boundedness in high norm and a standard compactness argument (see, or example, Embid [6], Majda [8]. The uniqueness o the solution ollows rom the proo o Proposition.3 by a standard argument using estimates similar to the estimates used in the proo o Proposition.3. And thereore v, ρ is a unique classical solution o the system o equations (1., (1.4, where v = w+ φ, = 1 c c 3 ψ, and ρ = e h. This completes the proo o the theorem. A inal remark We now present a proo that the standard conservation o mass equation ρ t + v ρ + ρ v = is approximately satisied by the solution ρ, v to the system o equations (1., (1.4, provided that the hypotheses o Theorem.1 are satisied and that max t T b (t b(t C, where C is a generic constant which does not depend on ɛ. We use the ollowing inequality: ρ t + v ρ + ρ v L C ρ t L + C v L ρ L + C ρ L v (.39 L C ρ t L 13 ρ L 13 ρ L where we used the estimate v L (w + φ C v = C Similarly, C 1 ( w + φ C 1 (ɛl 1 + t 1 C 1,T. v L C v C v 3 (w + φ = C 3 C 1 3( w 3 + φ 3 C 1 3(ɛL 1 + t C 1 3,T. We used the estimate w s ɛl 1 1, and the assumption that t ɛ < 1. Also we use C 13 = C 1 3,T.
17 EJDE-15/16 EXISTENCE OF SOLUTIONS 17 We next obtain estimates or ρ L, ρ L, and ρ t L (.39. to use in inequality Estimate or ρ L : By inequality (.19, and using the act that ρ = eln (ρ = e h, we have: ρ L = eh L (e 1 ln (ɛ = ɛ (.4 Estimate or ρ L : Using the act that ρ = ρ (ln (ρ = ρ h, we obtain the estimate ρ L = ρ h L ρ L h L ɛ(cɛl ɛc (.41 where we used (.4 and we used the estimate h L C h C(ɛL, where ɛl 1. Estimate or ρ t L : Using the act that ρ t = ρ(ln (ρ t = ρh t, and using (.4, we obtain ρ t L = ρh t L ρ L h t L ɛ h t L ɛc h t (.4 By Lemma 4.3 in Section 4 and by the act that h t (x, t = ρt(x,t ρ(x,t obtain the estimate h t C( h t (x, t + (h t (x, t 1 + h t 1 C( Ω max t T h t(x, t + h t 1 C( Ω max t T b (t b(t + h t 1 C( Ω + h t 1 where we used the assumption that max t T b (t b(t C. Estimate or h t 1: Applying t to equation (.5 yields w t + g(h h t = t ( 1 (w + φ (w + φ + t ( 1 ( (w + φ(w + φ + t ((w + φ 1 t tg(h h dg h dh t h φ t Next, applying the divergence operator to (.44 yields = b (t b(t, we (.43 (.44 (g(h h t ( ( ( 1 = (w + φ (w + φ ( 1 t (w t + φ t (w + φ ( 1 ( (w + φ (w t + φ t + ( t ( 1 (w + φ (w + φ + ( 1 ( (w t + φ t (w + φ + ( 1 ( (w + φ(w t + φ t + ((w t + φ t t + ((w + φ ( t ( t g(h h t ( dg dh h t h ( dg dh (h t h dg dh ( h t h dg dh h t h φ t (.45
18 18 D. L. DENNY EJDE-15/16 We will use the well-known inequality uv 1 uv C u v (see Lemma 4.1. Then applying estimate (4. rom Lemma 4.7 to (.45, and using the act that (x, tg(h(x, t 1, yields h t 1 C j= D(g(h j ( ( t + 1 (w t + φ t (w + φ 1 j= ( 1 (w + φ (w + φ 1 D(g(h j ( 1 (w + φ (w t + φ t 1 ( + t ( 1 (w + φ (w + φ 1 D(g(h j ( 1 ( (w t + φ t (w + φ 1 j= + 1 ( (w + φ(w t + φ t 1 j= D(g(h j ( (w t + φ t t 1 + (w + φ t ( t 1 + t g(h h 1 + ( dg dh h t h D(g(h j dg ( ( dh (h t h + dg dh ( h t h j= + dg dh h t h + φ t C 3 g(h j 3 ( ( 1 t w + φ (w + φ j= + 1 w t + φ t (w + φ j= 3 g(h j 3 ( 1 w + φ (w t + φ t + ( 1 t w + φ w + φ 3 g(h j 3 ( 1 w t + φ t w + φ j= + 1 w + φ w t + φ t
19 EJDE-15/16 EXISTENCE OF SOLUTIONS 19 j= 3 g(h j 3 ( w t + φ t t + w + φ ( t t + t g(h h dg 3 g(h j 3 ( L dh L h t L h j= + ( dg dh L h t L h L dg 3 g(h j 3 ( L dh L h t h L j= + dg L dh L h t L h + φ t C j= 3 (L 5 j ( ( 1 t (ɛl 1 + t 1(ɛL 1 + t + 1 (L 4 + tt 1(ɛL 1 + t j= 3 (L 5 j ( 1 (ɛl 1 + t 1(L 4 + tt + ( 1 t (ɛl 1 + t 1(ɛL 1 + t 1 3 (L 5 j ( 1 (L 4 + tt 1(ɛL 1 + t 1 j= + 1 (ɛl 1 + t 1(L 4 + tt 1 j= 3 (L 5 j ((L 4 + tt 1 t + (ɛl 1 + t 1 t ( t + t L 5 (ɛl + (ɛ h t (ɛl 3 (L 5 j( (ɛ(ɛl h t (ɛl + (ɛ h t 1(ɛL j= + (ɛ h t (ɛl + ttt C 14 (1 + ɛ h t (.46 Here we used the estimates (w t + φ t C w t + φ t 3 C( w t 3 + φ t 3 C(L 4 + tt, and (w + φ C w + φ 3 C( w 3 + φ 3 C(ɛL 1 + t, where ɛl 1 1. We used the act that φ = t, and we used the estimate φ t 3 C tt rom Lemma 4.7 in Section 4. And we used the estimate h C h 1 C(ɛL, where ɛl 1. And we used the inequalities t 3 ɛ < 1, tt ɛ < 1, g(h 3 L 5. We used the estimate dg dh L dg dh,g 1 dg 1,G 1 Cɛ by inequality (.3. And we used Lemma 4.4 dh
20 D. L. DENNY EJDE-15/16 to obtain the estimate ( dg dh d g dh,g 1 h dg dh 1,G 1 h (Cɛ(ɛL. Here C 14 depends on s,t, 1 s,t, s,t, g s,g1, and ttt,t. By substituting estimate (.46 into (.43, it ollows that h t C( Ω + h t 1 C( Ω 14 (1 + ɛ h t (.47 Choosing ɛ suiciently small so that CC 14 ɛ 1, and re-arranging terms in (.47, yields h t C( Ω 14 (.48 By substituting estimate (.48 into estimate (.4, it ollows that ρ t L ɛc h t ɛc( Ω 14 = ɛc 15 (.49 where C 15 depends on s,t, 1 s,t, s,t, g s,g1, and ttt,t. From substituting estimates (.4, (.41, (.49 into (.39, we obtain ρ t + v ρ + ρ v L C ρ t L 13 ρ L 13 ρ L ɛc 16 (.5 where C 16 depends on s,t, 1 s,t, s,t, g s,g1, and ttt,t. It ollows rom (.5 that the standard conservation o mass equation ρ t + v ρ + ρ v = is approximately satisied by the solution ρ, v to the system o equations (1., ( Existence or the linear system o equations In this section we present the proo o the existence o a solution to the linear equations (.8, (.9 used in the iteration scheme in Section. Lemmas supporting the proo appear in Section 4. Lemma 3.1. Let w H s (Ω where w =, and let b 1 C([, T ], H (Ω L ([, T ], H s (Ω, b C([, T ], H (Ω L ([, T ], H s (Ω, b 3 C([, T ], H (Ω L ([, T ], H s (Ω, g C([, T ], H (Ω L ([, T ], H s (Ω, where s > N +, or N =, 3, and where b 3 (x, t 1 or x Ω = T N, t T. Let b be a given positive smooth unction o t or t T. Let x Ω be a given point. Then there exists a unique classical solution w, h or x Ω and t T o and w t + b 1 w + b w + b 3 h = g, (3.1 w =, (3. h(x, t = ln (b(t, w(x, = w (x, w =, (3.3 w C([, T ], C (Ω L ([, T ], H s (Ω, h C([, T ], C 3 (Ω L ([, T ], H s+1 (Ω. Proo. First, we change the equations to an equivalent system by employing the projections P and Q = I P, where P is the orthogonal projection o L (Ω onto the solenoidal vector ield and Q is the orthogonal projection o L (Ω onto the gradient vector ield. Applying the operator P to (3.1, and using the act that P w = w, we obtain the equation w t + b 1 w + b w = J := Q(b 1 P w + Q(b P w P (b 3 h + P g (3.4
21 EJDE-15/16 EXISTENCE OF SOLUTIONS 1 Applying the operator Q to (3.1, and using the act that P w = w, we obtain the equation Q (b 1 P w + b P w + b 3 h g =. (3.5 With the given initial data w, where w =, the system o equations (3.1, (3. and the system o equations (3.4, (3.5 are equivalent. (The proo is standard; see, or example, Embid [6]. From the deinition o Q, it ollows that equation (3.5 is equivalent to Re-arranging terms yields (b 1 P w + b P w + b 3 h g =. (3.6 (b 3 h = (b 1 P w (b P w + g = ( b 1 T : (P w b (P w + g (3.7 Next, we construct the solution w, h o the system o equations (3.4, (3.7 using the method o successive approximation as ollows: Set the initial iterate w (x, t = w (x, the initial data, and set the initial iterate h (x, t = ln (b(t. For k =, 1,,... deine w k+1, h k+1, as the solution o the equations w k+1 + b 1 w k+1 + b w k+1 = J k, (3.8 t (b 3 h k+1 = ( b 1 T : (P w k b (P w k + g, (3.9 where J k = Q(b 1 P w k + Q(b P w k P (b 3 h k+1 + P g, and w k+1 (x, = w (x. The irst step is to solve (3.9 or h k+1. By the inction hypothesis, ( b 1 T : (P w k C([, T ], H (Ω L ([, T ], H s 1 (Ω and b (P w k belongs to C([, T ], H (Ω L ([, T ], H s 1 (Ω. Also g C([, T ], H (Ω L ([, T ], H s 1 (Ω and b 3 C([, T ], H (Ω L ([, T ], H s (Ω, and b 3 1. Thereore, by Lemma 4.7 there exists a unique zeromean solution h k+1 C([, T ], H (Ω L ([, T ], H s+1 (Ω to equation (3.9. It ollows that h k+1 (x, t = h k+1 (x, t + ln (b(t h k+1 (x, t is a unique solution to equation (3.9 which satisies the condition h k+1 (x, t = ln (b(t. The next step is to solve (3.8 or w k+1. By Lemma 4. and by the inction hypothesis, Q(b 1 P w k C([, T ], H (Ω L ([, T ], H s (Ω. And by the inction hypothesis, Q(b P w k C([, T ], H (Ω L ([, T ], H s (Ω. And rom the previous step, P (b 3 h k+1 C([, T ], H (Ω L ([, T ], H s (Ω. And P g C([, T ], H (Ω L ([, T ], H s (Ω. Thereore, by Lemma 4.8 there exists a unique solution w k+1 C([, T ], H (Ω L ([, T ], H s (Ω to equation (3.8. Next, we obtain estimates or (h k+1 h k s,t, hk+1 h k s+1,t, and wk+1 w k s,t. From equations (3.8, (3.9 we obtain the ollowing system o equations or h k+1 h k and w k+1 w k : (w k+1 w k + b 1 (w k+1 w k + b (w k+1 w k = J k J k 1 (3.1 t (b 3 (h k+1 h k = ( b 1 T : (P (w k w k 1 b P (w k w k 1, (3.11 where J k J k 1 = Q(b 1 P (w k w k 1 + Q(b P (w k w k 1 P (b 3 (h k+1 h k. Initially we have (w k+1 w k (x, =. And h k+1 (x, t = h k (x, t = ln (b(t.
22 D. L. DENNY EJDE-15/16 Next, applying Lemma 4.7 to equation (3.11, and using the act that b 3 (x, t 1, yields (h k+1 h k s C j= Db 3 j s 1 ( ( b 1 T : (P (w k w k 1 s 1 + b P (w k w k 1 s 1 C Db 3 j s 1 ( Db 1 s 1 + Db s 1 w k w k 1 s j= C 1 w k w k 1 s (3.1 where s > N + and s 1 = s 1. And C 1 depends on Db 1 s 1,T, Db s 1,T, Db 3 s 1,T. By Lemma 4.3, and by the act that h k+1 (x, t = h k (x, t, we have the inequality From (3.1, (3.13, it ollows that h k+1 h k s+1 C (h k+1 h k s (3.13 h k+1 h k s+1 C (h k+1 h k s C w k w k 1 s (3.14 where C depends on Db 1 s 1,T, Db s 1,T, Db 3 s 1,T. Applying Lemma 4.8 to equation (3.1, and using estimate (3.14 or (h k+1 h k s, yields w k+1 w k s Ce βt Q(b 1 P (w k w k 1 s + Q(b P (w k w k 1 s + P (b 3 (h k+1 h k sdτ Ce βt b 1 s w k w k 1 s + b s w k w k 1 s + b 3 s (h k+1 h k sdτ C 3 w k w k 1 sdτ (3.15 where we used Lemma 4. to obtain the estimate Q(b 1 P (w k w k 1 s C b 1 s w k w k 1 s. Here β = C(1 + b 1 s1+1,t + b s1+1,t, and C 3 depends on b 1 s,t, b s,t, b 3 s,t. Repeated application o (3.15 yields w k+1 w k (C3T k s,t k! w 1 w s,t, rom which it ollows that k= wk+1 w k s,t <, and thereore k= hk+1 h k s+1,t < by (3.14. Thereore, there exists w C([, T ], H (Ω L ([, T ], H s (Ω such that w k w as k strongly in C([, T ], H (Ω L ([, T ], H s (Ω, and there exists h C([, T ], H (Ω L ([, T ], H s+1 (Ω such that h k h as k strongly in C([, T ], H (Ω L ([, T ], H s+1 (Ω. And the act that w and h is a solution to (3.1, (3., (3.3 ollows by a standard argument (see, e.g., Majda [8], Embid [6].
23 EJDE-15/16 EXISTENCE OF SOLUTIONS 3 4. Lemmas supporting the proo Lemma 4.1 (Standard Calculus Inequalities. (a I H s1 (Ω, g H s (Ω and s 3 = min{s 1, s, s 1 + s s }, where s = [ N ] + 1, then g Hs3 (Ω, and g s3 C s1 g s. We note that s = or N = or N = 3. Here, the constant C depends on s 1, s, Ω. (b I H s (Ω L (Ω, where s = [ N ] + 1, then L C s. (c I H r (Ω and r = θr 1 + (1 θr, with θ 1 and r 1 < r, then r C θ r 1 1 θ r. Here C is a constant which depends on r 1, r, Ω. The above inequalities are well known. They appear, or example, in Embid [6]. Lemma 4.. I w, v H r (Ω, r > N + 1, Ω = TN, then Q(v P w H r (Ω and Q(v P w r C v r w r. Here P, Q are the projection operators rom the Helmholtz Decomposition u = P u+qu, where P u = and Qu is a gradient vector ield. A proo o the above lemma appears in Embid [7]. Lemma 4.3. Let, g be H r (Ω unctions on a bounded domain Ω, where r. And let (x = g(x at a point x Ω. Then g and satisy the estimates g C ( g 1, (4.1 g r C ( g r 1, (4. g L C ( g 1 (4.3 r C g r g r 1 r 1 (4.4 where C is a constant which depends on Ω and on r. Proo. A proo o inequality (4.1 appears in Denny [4]. From (4.1 we obtain the inequality g r g ( g r 1 C ( g 1 ( g r 1 C ( g r 1 or r. This completes the proo o (4.. From (4. with r =, and rom Lemma 4.1, we obtain g L C g C ( g 1 This completes the proo o (4.3. From (4. we obtain the inequality r = g + g r C g r g r C ( g r 1 g r C r 1 g r 1 g r or r. This completes the proo o the lemma. Lemma 4.4. Let be a smooth unction on an interval G R, and let u be a continuous unction such that u(x G 1 or x Ω, where G 1 G and u H r+1 (Ω, where r. Then D((u d,g 1 Du, (4.5 D((u 1 C d ( Du 1,G Du Du, (4.6
24 4 D. L. DENNY EJDE-15/16 D((u r C d r,g 1 1 j r+1 where r and C depends on r, Ω. And we deine g r,g1 G 1, j r}. Proo. We immediately obtain the estimate D((u = d Du d,g 1 Du Next, we obtain the inequality D((u 1 = d Du 1 Du j r, (4.7 = max{ dj g j (u : u = d Du + D ( d Du d Du D ( d L Du d,g 1 D u = d Du d Du L Du d D u,g 1 d,g 1 Du d,g 1 Du L Du d,g 1 D u C d 1,G 1 ( Du 1 + Du L Du C d 1,G 1 ( Du 1 + Du Du where we used Lemma 4.1 to obtain the estimate Du L C Du. I r, then we obtain the inequality D((u r = d Du r C d r Du r C( d + D ( d r 1 Du r (4.8 (4.9 By (4.9 and by repeating the above argument applied to the terms D ( d j j r j, or j = 1,,..., r, that appear on the right-hand side o the inequality, we obtain D((u r C( d + D ( d r 1 Du r = C( d + d Du r 1 Du r C( d + d r 1 Du r 1 Du r C( d + d r 1 Du r Du r C( d + ( d + D ( d r Du r Du r C 1 j r 1 dj j Du j r D ( d r 1 r 1 1 Du r (r 1 where C is a generic constant which changes rom one instance to the next. (4.1
25 EJDE-15/16 EXISTENCE OF SOLUTIONS 5 Substituting estimate (4.8 or the term D ( d r 1 r 1 1 into (4.1 yields D((u r C C 1 j r 1 1 j r 1 C d C d dj j Du j r D ( d r 1 r 1 1 Du (r 1 r dj j Du j r dr r 1,G 1 ( Du 1 + Du Du Du r (r 1 r,g 1 1 j r 1 r,g 1 1 j r+1 Du j r Du j r This completes the proo o the lemma. d r,g 1 ( Du r r + Du (r+1 r Lemma 4.5. Let be a smooth unction on an interval G R, and let u 1, u be continuous unctions such that u 1 (x G 1, u (x G 1 or x Ω, where G 1 G, and u 1 H r (Ω, u H r (Ω, where r. Then (u 1 (u d u,g 1 u 1, (4.11 (u 1 (u 1 C d 1,G 1 (1 + Du 1 + Du u 1 u, (4.1 where C depends on Ω. Here we deine g r,g1 r}. Proo. We begin by writing the calculus identity where I (u 1, u is deined as ollows: = max{ dj g j (u : u G 1, j (u 1 (u = I (u 1, u (u 1 u (4.13 I (u 1, u = 1 d (τu 1 + (1 τu dτ (4.14 and where the ollowing estimates hold or I (u 1, u : I (u 1, u L d (4.15,G 1 D(I (u 1, u C d ( Du 1,G Du (4.16 A proo o inequalities (4.15-(4.16 appears in Embid [7]. From (4.13, (4.15, it ollows that (u 1 (u = I (u 1, u (u 1 u I (u 1, u L u 1 u d u,g 1 u 1 From (4.13, (4.15, (4.16 and the estimate or (u 1 (u, it ollows that (u 1 (u 1 = (u 1 (u + D((u 1 (u (u 1 (u D(I (u 1, u (u 1 u I (u 1, u D(u 1 u
26 6 D. L. DENNY EJDE-15/16 (u 1 (u D(I (u 1, u u 1 u L I (u 1, u L D(u 1 u d,g 1 u 1 u d 1,G 1 ( Du 1 + Du u 1 u d u,g 1 u 1 1 C d 1,G 1 (1 + Du 1 + Du u 1 u where u 1 u L C u 1 u by Lemma 4.1. This completes the proo o the lemma. Lemma 4.6. Let Z k (t be a positive, integrable unction or t T and or k = 1,, 3,..., and let Z k+1 (t L ( ɛz k (t + Zk (τ 1 dτ 1, where ɛ and L are positive constants which do not depend on k. Then Z k+1 (t (ɛl k e t/ɛ sup t T Z 1 (t Proo. We repeatedly apply the inequality given or Z k+1 as ollows: Z k+1 (t ɛlz k (t + ( ɛl ɛlz k 1 (t + + τ1 LZ k (τ 1 dτ 1 LZ k 1 (τ dτ dτ 1 = ɛ L Z k 1 (t + ɛ ɛ L ( ɛlz k (t + + ɛ + τ1 LZ k 1 (τ 1 dτ 1 + L ( ɛlz k (τ 1 + L ( ɛlz k (τ + L Z k 1 (τ 1 dτ 1 + LZ k (τ 1 dτ 1 τ1 τ = ɛ 3 L 3 Z k (t + 3ɛ L 3 Z k (τ 1 dτ 1 + 3ɛ τ1 L k ɛ k Z 1 (t + L k L 3 Z k (τ dτ dτ 1 + k j=1 ( k j ɛ k j ( L ɛlz k 1 (τ 1 τ1 LZ k (τ dτ dτ 1 LZ k (τ 3 dτ 3 dτ dτ 1 τ1 τ τ1 τ... L Z k 1 (τ dτ dτ 1 L 3 Z k (τ 3 dτ 3 dτ dτ 1 τj 1 Z 1 (τ j dτ j... dτ 3 dτ dτ 1 (4.17
27 EJDE-15/16 EXISTENCE OF SOLUTIONS 7 Then we have the inequality τ1 τ... sup Z 1 (t t T = sup t T τj 1 τ1 τ Z 1 (t ( t j j! Z 1 (τ j dτ j... dτ 3 dτ dτ 1 τj 1... dτ j... dτ 3 dτ dτ 1 (4.18 Substituting estimate (4.18 into inequality (4.17, and using the inequality ( k j = k or each j k (see Abramowitz and Stegun [1], yields k ( k j= j This completes the proo. Z k+1 L k k j= ( k ɛ k j j = (ɛl k sup t T (ɛl k (ɛl k sup t T sup t T Z 1 (t Z 1 (t sup t T Z 1 (t k j= k j= Z 1 (t ( t j j! ( k (( t j ɛ j j! k( ( t ɛ j j! ( ( t j= = (ɛl k e t/ɛ sup Z 1 (t t T ɛ j j! The next lemma on the existence o a solution to an elliptic equation is standard. Lemma 4.7. Let a C (Ω H s (Ω, C 1 (Ω H s 1 (Ω, and g C (Ω H s (Ω, where s > N +, Ω = TN, N =, 3. Let a(x 1 or x Ω. And let Ω ( + gdx =. Then there exists a solution h C3 (Ω H s+1 (Ω o (a h = + g (4.19 which is unique up to a constant. And the zero-mean unction h = h 1 Ω Ω hdx is also a solution. And the ollowing inequality holds or r 1 r h r C Da j r 1 ( r 1 + g r (4. j= where r 1 = max{r 1, s } and s = [ ] N + 1 =, or N =, 3, and where C depends on r. A proo o the above lemma appears in Denny [5]. The next lemma on the existence o a solution to a linear system o equations is standard. Lemma 4.8. Given w H s (Ω, b 1 C([, T ], H (Ω L ([, T ], H s (Ω, b C([, T ], H (Ω L ([, T ], H s (Ω, g C([, T ], H (Ω L ([, T ], H s (Ω,
28 8 D. L. DENNY EJDE-15/16 where s > N +, Ω = TN, N =, 3, t T, there exists a unique classical solution w C([, T ], C (Ω L ([, T ], H s (Ω o And or r 1, w satisies w t + b 1 w + b w = g w(x, = w (x w r e βt ( w r g rdτ (4.1 where β = C(1+ b 1 r1+1,t + b r1+1,t and C depends on r. Here, r 1 = max{r 1, s }, where s = [ N ] + 1 = or N =, 3. The proo o the above lemma is standard; see, or example, Embid [6]. Reerences [1] I. Abramowitz, I. Stegun, ed.; Handbook o Mathematical Functions, Dover Publications, Inc., New York, [] J. Chorin, P. Marsden; A Mathematical Introction to Fluid Mechanics, Springer, New York, [3] Diane L. Denny; Existence and uniqueness o global solutions to a model or the low o an incompressible, barotropic luid with capillary eects, Electronic Journal o Dierential Equations 7, no. 39, (7, 1 3. [4] D. Denny; Existence o a unique solution to a quasilinear elliptic equation, Journal o Mathematical Analysis and Applications 38 (11, [5] D. Denny; A unique solution to a nonlinear elliptic equation, Journal o Mathematical Analysis and Applications 365 (1, [6] P. Embid; On the Reactive and Non-diusive Equations or Zero Mach Number Combustion, Comm. in Partial Dierential Equations 14, nos. 8 and 9, (1989, [7] P. Embid; Well-posedness o the Nonlinear Equations or Zero Mach Number Combustion, Comm. in Partial Dierential Equations 1, no. 11, (1987, [8] A. Majda; Compressible Fluid Flow and Systems o Conservation Laws in Several Space Variables, Springer-Verlag: New York, Diane L. Denny Department o Mathematics and Statistics, Texas A&M University - Corpus Christi, Corpus Christi, TX 7841, USA address: diane.denny@tamucc.e
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