Joint work with Prof. Akitaka Matsumura, Osaka universit

Hashimoto, Itsuko

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Kyoto University Research Information Repository

RIGHT:URL:CITATION:AUTHOR(S):ISSUE DATE:TITLE:Existence of radially symmetric stationarysolutions for the compressible Navier-Stokesequation (Mathematical Analysis of ViscousIncompressible Fluid)Hashimoto, ItsukoHashimoto, Itsuko. Existence of radially symmetric stationary solutions for the compressible Navier-Stokes equation(Mathematical Analysis of Viscous Incompressible Fluid). 数理解析研究所講究録 2021, 2205: 108-1192021-12http://hdl.handle.net/2433/267830108Existence of radially symmetric stationary solutions for the corn pressi ble N avier-Stokes equation I tsuko Hashimoto Kanazawa University /OGAMI, College of Science and Engineering (joint work with Prof. Akitaka Matsumura, Osaka university) 1 Introduction and main theorem This is a short survey of our recent results announced in [5] on the existence of radially symmetric stationary solutions for exterior problems in Rn(n 2: 2) to the compressible Navier-Stokes equation: { Pt + div(pU) = 0, (pU)t + div(pU ® U) + 'vp = v L. U + (v + .\)'v(divU), (1.l) t > 0, x E n, where n = {x E :!Rn (n 2: 2); lxl > r 0 } (r0 is a positive con-stant), p = p(t, x) > 0 is the mass density, U = (u1(t, x), · · · , un(t, x)) is the fluid velocity, and p = p(p) is the pressure given by a smooth function of p satisfying p'(p) > 0 (p > 0). Furthermore, v and ,\ are the shear and bulk viscosity coefficients respectively, which are assumed to be constants satisfy-ing v > 0, 2v + n,\ > 0. We focus our attention on the radially symmetric solutions, which have the form p(t, x) = p(t, r), X U(t, x) = - u(t, r), r = lxl, r (1.2) where u(t, r) is a scalar function. By plugging (1.2) to (1.1), we can rewrite (1.1) as in the form { (rn-lP)t + (rn-lpu)r = 0, pu2 (rn-lu) (pu)t + (pu2 + p(p))r + (n - 1)--;- = µ( rn-1 r)r, t > 0, r > ro, (1.3) 109whereµ= 2v + >.. Now, we consider the initial boundary value problems to (1.3) under the initial condition (p, u)(0, r) = (Po, uo)(r), r > ro, the far field condition lim (p, u)(t, r) = (P+, u+), t > 0, r--+oo (1.4) (1.5) and also the following two types of boundary conditions depending on the sign of the velocity on the boundary { (p,u)(t,~o) = (p_,u_), u(t, r0 ) - u_, t > 0, t>0, (u_>0), (u_ :::; 0), (1.6) where P± > 0, U± are given constants. The case u_ > 0 is known as "inflow problem", the case u_ = 0 as "impermeable wall problem", and the case u_ > 0 as "outflow problem". For these initial boundary value problems, when the space dimension is one ( n = 1), there have been many results on the existence of time-global solutions and their asymptotic behaviors toward various nonlinear waves depending on the far field and boundary conditions, for example, toward stationary waves, rarefaction waves, viscous shock waves, and even their composite waves (cf. [3],[6],[7],,, etc.). On the other hand, when the problems are multi-dimensional (n ~ 2), there seem no results except the case U± = 0 studied by Jiang [2] and Nakamura-Nishibata-Yanagi [8]. They study more general compressible N avier-Stokes equation, describing the motion of viscous polytropic idea gas, Jiang [2] first showed the global asymptotic stability of the constant states, and later Nakamura-Nishibata-Yanagi [8] extended the results to the case with external potential forces. As the first one step to study the multi-dimensional problems in more general cases u_ -/=- 0 or u+ -/=- 0, we showed in [5] the existence of the stationary solution in a suitably small neighborhood of the far field state, by using the similar arguments as in Germain-Iwabuchi [1]. In this survey we present the overview of our paper [5]. The stationary problem corresponding to the problem (1.3)-(1.6) is writ-ten as r > r0 , (1. 7) lim (p, u)(r) = (P+, u+), r--+oo (p, u)(r0 ) = (P-, u_) (u_ > 0), u(r0 ) = u_ (u_ :::; 0). 110From the first equation in ( 1. 7), we easily see it holds r ?: r0 , (1.8) for some constant E, and it also holds from the boundary conditions that n-l E = r0 p_u_ (u_ > 0), n-l ( ) E = r0 p r0 u_ (u_::; 0), (1.9) where note that in the case u_ < 0, E includes the unknown p(r0 ) which should be determined later. The formula (1.8) implies that if n ?: 2, u+ = lim u(r) = lim E1 = 0. r---too r---too rn- P+ (1.10) Hence, we need to assume u+ = 0 for the existence of multi-dimensional stationary solutions. Now we are ready to state the main results. Theorem 1.1. Let n ?: 2 and u+ = 0. Then, for any P+ > 0, there exist positive constants Eo and C satisfying the following: (I) Let u_ > 0. If lu- I+ IP- - P+ I ::; Eo, there exists a unique smooth solution (p, u) of the problem (1. 7) satisfying lp(r) - P+I ::; cr-(n-ll(lu-1 2 + IP- - P+I), (1.11) Furthermore, for any positive constant h, there exists a positive constant Ch such that it holds sup lp(r) - P+I::; Chlu-l 2 - (1.12) r2ro+h (II) Let u_ ::; 0. If lu- I ::; Eo, there exists a unique smooth solution (p, u) of the problem (1. 7) satisfying lp(r) - P+I::; cr-2(n-l)lu-l 2 , c-1r-(n-l)lu-l::; lu(r)I::; cr-(n-l)lu-1, r ?: ro. (1.13) 1112 Preliminary In this section, we reformulate the problem (1.7). First, we assume u_ =/- 0 in what follows. We also assume r0 = 1 without of loss of generality. Next, introduce the specific volume v by v = 1/ p (accordingly, denote V± by 1/ P±)-Then, by (1.8) and (1.9), the velocity u is given in terms of v as E u(r) = - 1 v(r), r ~ 1, rn- (2.1) where E = u_/v_ (u_ > 0), and E = u_/v(l) (u_ :s; 0). Substituting (2.1) into the second equation of (1.7), and introducing a new unknown function rJ, as the deviation of v from the far field state v+, by rJ(r) = v(r) - v+, r ~ 1, (2.2) we obtain the following differential equation in terms of rJ(r) in the following reformulated problem: n-1 rJr = _r - (fi( V+ + rJ) - P( V+)) Eµ + EV+ _1_ + __!!J_ _ E( n - 1 )rn-l 100 rJ( s) ds 2µ rn-1 µrn-1 µ r s2n-1 ' r > 1, { lim rJ(r) = 0, r--+oo rJ(l) = rJ- := v_ - v+ (u_ > 0), no boundary condition (u_ < 0), (2.3) where E = u_/v_ (u_ > 0), and E = u_/(v+ + rJ(l)) (u_ < 0). We note p(v) := p(l/v), and it holds p'(v) < 0 (v > 0) by the assumption on p(v). Once the desired solution rJ of (2.3) is obtained, the velocity u is imme-diately obtained by (2.1) as ( ) - u_(v++rJ(r)) ( ) ( )- u_(v++rJ(r)) ( ) ( ) u r - 1 , u_ > 0 , u r - ( ( )) 1 , u_ < 0 . 2.4 V_rn- V+ + rJ 1 rn-The theorem for the reformulated problem (2.3) which we need to prove is Theorem 2.1. Let n ~ 2. Then, for any V+ > 0, there exist positive con-stants Eo and C satisfying the following: (I) Let u_ > 0. If lu- I + lrJ-1 :s; Eo, there exists a unique smooth solution rJ of the problem (2.3) satisfying r ~ l. (2.5) 112Furthermore, for any positive constant h, there exists a positive constant Ch satisfying sup l77(r)I ~ Chlu-1 2 - (2.6) r:2'.ro+h (II) Let u_ < 0. If lu- I ~ Eo, there exists a unique smooth solution rJ of the problem (2.3) satisfying r?. l. (2.7) We can easily see that the main Theorem 1.1 is a direct consequence by Theorem 2.1, the formula (2.4). 3 Proof of theorem 3.1 Inflow problem In this subsection, we consider the case u_ > 0, that is, inflow problem, and show the result (I) in the Theorem 1.2. In this case, recalling E = u_/v_ > 0, we further rewrite the equation of T/ in (2.3) as in the form (3.1) where F[ ]( ) ·- EV+ 1 ErJ(r) E(n - l)rn-I ioo rJ(s) d rn-IN( ( )) rJ r .----+-------- -- s+-- rJr, 2µ rn-1 µrn-1 µ r s2n-1 µE N(TJ) := p(v+ + TJ) - p(v+) - p'(v+)TJ. Solving the linear differential equation (3.1) in terms of rJ with the initial data rJ(l) = T/- and the inhomogeneous term F, we have, by the Duhamel's principle, where K, := -p'(v+)/(µn) > 0. Thus, to prove the existence of the solution of (2.3) with the decay rate estimate (2.5), we look for a solution of (3.2) in the Banach space X, with its norm II · llx, defined by X = {TJ E C([l, oo)); sup lrn-lrJ(r)I < oo}, IITJllx = sup lrn-lrJ(r)I- (3.3) r:2':1 r:2':1 113To do that, we construct the approximate sequence { 'T/(m) }m~o by To show {'T/(m)}m~o is a Cauchy sequence in X for suitably small lu-1 + l'Tl-1, we prepare the following lemma. Lemma 3.1. (I) If E :::; nn_'\, then it holds that r?.. l. (3.5) (II) If E :::; 4(~~1), then there exists a positive constant C which is independent of E such that r ?._ l, f EX. (3.6) The proof is given by elementary calculations. Refer to [5] for the details. Now, by using Lemma 3.1, we show the uniform boundedness of 'T/(m) (m?.. 0) in X for suitably small lu-1 + l'Tl-1• That is, we show that for any fixed V+, there exist positive constants Eo and C which are independent of u_ and 'T/- such that if lu- I + l'Tl-1 :::; Eo, then there exists a positive constant M satisfying ll'T/(m) llx :::; M:::; C(lu- 12 + l'Tl-1), m?.. 0. (3.7) Here and in what follows, we use the letter C and Eo to denote generic positive constants which are independent of u_ and 'T/-, but may depend on V+ and other fixed constants like µ, n, ... , etc. For the proof, in particular, to use Lemma 3.1, we first assume V+ l'Tl-1 = Iv- - v+I :::; 2' nfW+ u_:::; 8(n-1)' (3.8) which assures u_ nK E=-<---. v_ - 4(n - 1) (3.9) Let us show (3.7) by mathematical induction: Case m = 0. Due to (3.9), we have from Lemma 3.1 that 114Hence, we ask the constant M satisfy (3.11) so that it holds llr/0) llx :::; M. Case m = k + l (k ~ 0). Suppose (3.7) with m = k hold. Here we ask the constant M satisfy another assumption which, in particular, implies M V+ <-- 2 ' (3.12) (3.13) Then, by using Lemma 3.1, (3.12),(3.13), and also the Taylor's theorem, we estimate ryk+l as follows: (3.14) (3.15) (3.16) I Cc2(n -1) I 2(n-1) 100 sn-lT/(k)(s) d I Cc2M Cl 12M 3 :::; ----sup r 3n_2 s :::; - 3-:::; u_ ; µ r2".1 r S µ (3.17) 115n-1 /4 :S CEsuplrn-l(_r -N(r/k)(r)))I r21 µE :SQ· sup lfi"(t)I · sup(r2(n-l)l1/h)(r)l2) :S CM2 • 2µ v+/2g9v+/2 r21 Substituting (3.15)-(3.18) into (3.14), we obtain ll77(k+l)llx :S 177-1 + Clu-1 2 + CM2 . Therefore, we further assume 1 CM<-- 2' (3.18) (3.19) (3.20) so that (3.19) gives the desired estimate ll77(k+l) llx :S M. By elementary calculations, it is easy to see there exists a positive constant Eo such that if 177-1 + lu-1 :S Eo, all the assumptions (3.8),(3.11),(3.12), and (3.20) hold, and in particular, M can be chosen by Thus, the uniform boundedness of 77(m) (m ~ 0) in X with the estimate (3.7) is proved. Once (3. 7) is proved, the proof to show { 77(m) }m2:o is a Cauchy sequence in X is very standard. In fact, by using Lemma 3.1, we may estimate ll77(m+l) - 77(m)llx = sup lrn-l J,r e-~(rn-sn)(F[77(m)] - F[77(m-l)])(s) dsl r21 1 :S CEIIF[77(m)] - F[77(m-1)] llx, m ~ 1, (3.22) in the same way as in (3.15)-(3.18), and taking Eo suitably small again if needed, we can show which proves that {77(m)}m2:o is a Cauchy sequence in X. Thus, as the limit, the solution 77 of (2.3) with the desired estimate (2.5) is obtained. The arguments on the regularity and uniqueness of the solution are also very standard, so we omit them. Finally, we show the estimate (2.6), that is, the existence of the boundary layer for the density. We turn back to the equation in (2.3), and again rewrite it as a(77)rn-l 77r - ---77 = q[77], r > 1, E (3.24) 116where (3.25) Here we note that we already know the existence of the solution r, of (3.24) with the estimate (2.5). Solving the equation (3.24) in terms of r, with the initial data 77(1) = T/-, we have, by the Duhamel's principle, (3.26) By the estimate (2.5) and the assumption p(v) < 0 (v > 0), it is easy to see that there exist positive constants <5 and C satisfying -a(77(r)) 2:: <5, lq[r,](r)I S Cc, r 2:: 1. (3.27) Therefore, for any positive constant h, if r 2:: 1 + h, it follows from (3.26) and (3.27) that l77(r)I Se-¾ J; sn-ldsl'Tl-1 + lr e-¾ J; sn-1dslq[77](T)I dT S e-¾(r-l)IT/-I + Cc lr e-¾(r-T) dT S e-¾hl'Tl-1 + ~ c2 S Chlu-1 2 , (3.28) which proves the desired estimate (2.6). Thus, the proof for the case u_ > 0 in Theorem 2.1 is completed. 3.2 Outflow problem In this section, we consider the case u_ < 0, that is, outflow problem, and show the result (II) in the Theorem 1.2. In this case, recalling c = u_ / ( V+ + 77(1)) < 0, we again rewrite the equation of 77 in (2.3) as in the form V+f5'(v+) n-l - G[ ] T/r - r 77 - 77 ' µu_ r > 1, (3.29) 117where G[ ]( ) V+P'(v+)77(r)77(l) n-l u_v+ l 77 r ·= ------r + -------. µu_ 2µ(v++77(l))rn-l u_rJ(r) (n - l)u_rn-l 100 77(s) d +------------ ------ s µ(v++ 77 (l))rn-1 µ r (v++ 77 (l))s2n-1 + (v+ + 77(l))rn-l N(77(r)). µu_ Noting u_ < 0, and solving the equation (3.29) in terms of 77 with the inho-mogeneous term G and the far field condition 77( oo) = 0, we obtain r 2". 1, (3.30) where we recall K, = -p' ( V+) / (µn) > 0. This time, to prove the existence of the solution of (3.30) with the decay rate estimate (2.7), we look for a solution of (3.30) in the Banach space Y, with its norm II· IIY, defined by Y = {77 EC([l,oo)); suplr2(n-1)77(r)I < oo}, r::>1 ll77IIY = sup lr2(n-l)77(r)l-(3.31) r::>1 By the same way as in the last section, we construct the approximate se-quence { rJ(m) }m>O by To show the uniform boundedness, we prepare the following lemma. Lemma 3.2. For g EX, it holds that r 2(n-l)l100 e-~v~l(sn-rn)g(s)dsl ~ ~suplsn-lg(s)I, r2". l. (3.33) r nK,V+ s::>r Proof. It holds that l oo ,w+ ( n n) loo n1<v+ n-1( ) I r e-~s-r g(s)dsl~ r e-lu=Tr s-rs-(n-l)sCn-l)lg(s)lds ~ r-2(n-1)~ sup lsn-lg(s)I, r 2". 1. nK,V+ s::>r 118Thus, the proof of Lemma 3.2 is completed. By using Lemma 3.2, we can show that for any fixed V+ > 0, there exist positive constants Eo and C such that if lu- I :::; Eo, then there exists a positive constant M satisfying (3.34) as in the same manner of proof for the inequality (3.7). So we omit the details of the proofs of (3.34) and that {77(m)}m>O forms a Cauchy sequence in Y. References [1] P. Germain, T. Iwabuchi, Self-similar solutions for compressible Navier-Stokes equations, to appear, arXiv:1903.09958. [2] S. Jiang, Global spherically symmetric solutions to the equations of a viscous polytropic ideal gas in an exterior domain, Comm. Math. Phys. 178 (1996), 339-374. [3] S. Kawashima, S. Nishibata, and P. Zhu, Asymptotic stability of the stationary solution to the compressible Navier-Stokes equations in the half space, Comm. Math. Phys. 240 (2003), 483-500. [4] I. Hashimoto, A. Matsumura, Asymptotic behavior toward nonlinear waves for radially symmetric solutions of the multi-dimensional Burgers equation, J. Differential Equations, 266 (2019), 2805-2829. [5] I. Hashimoto, A. Matsumura, Existence of radially symmetric station-ary solutions for the compressible N avier-Stokes equation, to appear in Methods and Applications of Analysis. [6] A. Matsumura, Inflow and outflow problems in the half space for a one-dimensional isentropic model system of compressible viscous gas, Methods and Applications of Analysis 8 (2001), 645-666. [7] A. Matsumura and K. Nishihara, Large-time behaviors of solutions to an inflow problem in the half space for a one-dimensional system of compressible viscous gas, Commun. Math. Phys. 222 (2001), 449-474. [8] T. Nakamura, S. Nishibata, and S. Yanagi, Large-time behavior of spher-ically symmetric solutions to an isentropic model of compressible viscous fluid in a field of potential forces, Math. Models Methods Appl. Sci. 14 (2004), 1849-1879. 119College of science and engineering Kanazawa university /OCAMI Kanazawa 920-1192 JAPAN E-mail address: itsuko@se.kanazawa-u.ac.jp 

Existence of radially symmetric stationary solutions for the compressible Navier-Stokes equation (Mathematical Analysis of Viscous Incompressible Fluid)

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Existence of radially symmetric stationary solutions for the compressible Navier-Stokes equation (Mathematical Analysis of Viscous Incompressible Fluid)

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