This research is a joint work with Professor Shinya Nishibata, Souhei Sugizaki of Tokyo institute of technology and Akitaka Matsumura of Osaka university. The present talk is concerned with the existence and asymptotic behaviors of radially symmetric stationary solutions for the compressible Navier-Stokes equation, describing the motion of viscous barotropic gas without external forces, where boundary and far field data are prescribed on the exterior domain in ℝ[n], n ≥ 3. We clear that for both inflow and outflow problems, there exist non trivial stationary solution, and for outflow problem we show that the stationary wave are asymptotic stable in a suitably small neighborhood of the initial data. Furthermore, detailed decay rate of the stationary solutions are derived

Hashimoto, Itsuko

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RIGHT:URL:CITATION:AUTHOR(S):ISSUE DATE:TITLE:Asymptotic behavior towardradially symmetric stationarysolutions of the compressibleNavier-Stokes equation (Nonlinearand Random Waves)Hashimoto, ItsukoHashimoto, Itsuko. Asymptotic behavior toward radially symmetric stationary solutions of the compressible Navier-Stokes equation (Nonlinear and Random Waves). 数理解析研究所講究録 2023, 2247: 31-392023-04http://hdl.handle.net/2433/28539031Asymptotic behavior toward radially symmetric stationary solutions of the compressible N avier-Stokes equation Itsuko Hashimoto College of Science and Engineering, Kanazawa University/ OCAMI 1. ABSTRACT This research is a joint work with Professor Shinya Nishibata, Souhei Sugizaki of Tokyo institute of technology and Akitaka Matsumura of Osaka university. The present talk is concerned with the existence and asymptotic behaviors of radially symmetric stationary solutions for the compressible N avier-Stokes equation, describ-ing the motion of viscous barotropic gas without external forces, where boundary and far field data are prescribed on the exterior domain in ]Rn, n 2': 3. We clear that for both inflow and outflow problems, there exist non trivial stationary solution, and for outflow problem we show that the stationary wave are asymptotic stable in a suitably small neighborhood of the initial data. Furthermore, detailed decay rate of the stationary solutions are derived. 2. INTRODUCTION In this talk, we consider the compressible N avier-Stokes equation which describes a barotropic motion of viscous gas in the exterior domain S1 to a ball in ]Rn (n 2': 2): { Pt + div(pU) = 0, (pU)t + div(pU ® U) + "v'p = v 6. U + (v + A)"v(divU), t > 0, x En, (l) where O = {x E ]Rn (n 2': 2); lxl > 1}, 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 A are the viscosity coefficients. In this talk, we focus our attention on the radially symmetric solutions, which have the form X p(t, .T) = p(t, r), U(t, .T) = - u(t, r), r = lxl, (2) r where u(t, r) is a scalar function. By plugging (2) to (1), we can rewrite (1) as in the form { (rn-1P)t + (rn-lpu)r = 0, pu2 (rn-lu) (3) (pu)l + (pu2 + p(p))r + (n - 1)7 = µ( rn-l rt, t > 0, r > l, whereµ= 2v +A> 0. Now, we consider the initial boundary value problems to (3) under the initial condition (p,u)(0,r) = (Po,uo)(r), r > l, (4) the far field condition lim (p, u)(t, r) = (P+, u+), t > 0, r➔oo (5) 32and also the following two types of boundary conditions depending on the sign of the velocity on the boundary { (p,u)(t,-ro) = (Pb,ub), t > 0, (ub > 0), u(t, ro) - ub, t > 0, (ub:::; 0), (6) where Pb, P+ > 0, ub and u+ are given constants. The case ub > 0 is known as "inflow problem", the case ub = 0 as "impermeable wall problem", and the case ub < 0 as "outflow problem". When the problems are multi-dimensional (n 2: 2), Jiang [2] and Nakamura-Nishibata-Yanagi [8] studied the case U± = 0. They study more general compress-ible N avier-Stokes equation, describing the motion of viscous polytropic ideal 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. Recently, Hashimoto-Matsumura [4] treated the multi-dimensional problems in more general cases ub =/- 0 and they showed the existence of the radially symmetric stationary solution in a suitably small neighborhood of the far field state. 3. MAIN RESULT In this talk, we consider the asymptotic stability of the radially symmetric sta-tionary solution obtained in Matsumura-H [5]. The stationary solution (p, u) is a solution to (3)-(6) which is independent of the time variable. Thus, (p, u) satisfy the equations written as (rn-lpu)r = 0, ( n-1 ~) {JUUr + P(P)r = µ( r n ~ r)r, r 2': 1, r - (7) lim (p, u)(r) = (P+, U+ ), r--+oo (p, u) (ro) = (Pb, Ub) ( Ub > 0), u(ro) = Ub ( Ub :::; 0). From the first equation in (7), we easily see it holds rn- 1p(r)u(r) = m, r 2: 1, (8) for some constant m, and it also holds from the boundary conditions that m = PbUb (ub > 0), m = p(l)ub (ub:::; 0), where note that in the case ub :::; 0, m includes the unknown p(l) which should be determined later. The formula (8) implies that if n 2: 2, m u+ = lim u(r) = lim _1 = 0. r--+oo r--+oo rn P+ Hence, we need to assume u+ = 0 for the existence of multi-dimensional stationary solutions. Now we state the statement of existence Theorem. Theorem 3.1. Let n 2: 2 and u+ = 0. Then, for any P+ > 0, there exist positive constants Eo and C satisfying the following: 33(I) Let 'U_ > 0. If lu- I+ IP- - P+ I :S Ea, there exists a unique smooth solution (p, u) of the problem (7) satisfying lp(r) - P+I s cr-2(n-l)(lu-l 2 + IP- - P+I), c-lr-(n-l)lu-1 s lu(r)I s cr-(n-l)lu-1, r 2 ra. Furthermore, for any positive constant h, there exists a positive constant Ch such that it holds sup lp(r) - P+I S Chlu-1 2 - (9) r::,,:ro+h (II) Let u_ :S 0. If lu- I :S Ea, there exists a unique smooth solution (p, u) of the problem (7) satisfying lp(r) - P+I s cr-2(n-l)lu-l 2 , r 2 ro. Now we are ready to state the result for the asymptotic stability of stationary solution obtained for outflow problem in the following Theorem. Theorem 3.2. Let ub < 0, n 2 2 and u+ = 0. We assume that II be an arbitrary constant satisfying O < II < 1. Assume that the initial data (4) belongs to the function space Po E Cl+<T[l, oo), ua E C2+0"[1, oo). Then there exists a constant Ea such that if lu- I, llr n2\po - p, ua - ii) IIH1 :S Ea, then the initial boundary value problem (3)-(6) has a unique solution (p, u)(t, r) satisfying (p,u) E Cl+<T/2,1+0" x Cl+<T/2 ,2+0"([0,T] x [1,oo)), for an arbitrary T > 0 and (p - P+, u - u+) E C([0, oo); H 1 (ffi.)). Moreover, the so-lution (p, u) ( t, r) converges to the stationary wave (p, ii) (r) as time tends to infinity. Precisely, it holds that lim sup l(P, u)(t, r) - (,a, ii)(r)I = 0. t➔+00 rElR+ In the proof on the Theorems, we use spacial weighted energy method proposed by Nakamura-Nishibata-Yanagi [8] and decay rate of stationary solution. Ck+lT de-notes the Holder space of continuous functions which have the kth order derivatives of Hoolder continuity with exponent II. Remark 1. When u_ = 0, by (3)-(5), we easily see the solution is the trivial constant state (p, u) = (P+, 0). Therefore, when u_ > 0 and P- =/- P+, the estimate (11) shows a boundary layer for mass density does appear as u_ -+ +0. 4. PRELIMINARY AND PROPERTY OF THE STATIONARY SOLUTION In this section we translate the problem (7) and we use the symbol u(r) and p( r) as the stationary solution of ii( r) and ,a( r) for less confusion. We introduce the 34specific volume v by v = l/ p (accordingly, denote V± by 1/ P±)- Then, by (8)), the velocity u is given in terms of v as m u(r) = rn-l v(r), r 2: 1, (10) where m = ub/vb (ub > 0), and m = ub/v(l) (ub ~ 0). Substituting (10) into the second equation of (7), we have rrr:~l c:-lt + p(v)r = mµ c~~lt' (ll) where p(v) := p(l/v), and it holds p'(v) < 0 (v > 0) by the assumption on p(v). Now, we further introduce a new unknown function 77, as the deviation of v from the far field state v+, by 77(r) = v(r) - v+, Plugging (12) into (11), we have r 2: 1. (12) ( 77r ) _ mv+ ( 1 ) m2 ( 77 ) mµ rn-1 r = p(v+ + 77)r + -2- r2(n-1) r + rn-1 rn-1 r, (13) where m = ub/v_ (ub > 0), and m = ub/(v+ + 77(1)) (ub < 0). Under the far field condition 77( oo) = 0, the equation (13) is also equivalent to ( 77r _ m2v+ m277 2 100 77( s) ) mµ rn-1 - p(v+ + 77) - 2r2(n-1) - r2(n-1) + m (n - 1) r s2n-1 ds r = 0, (14) which implies that the function in the parenthesis in the left hand side of (14) is identically equals to a constant c0 for r > l. Then, it follows from the far field condition that which concludes c0 = -p( V+), otherwise it contradicts the far field condition again. Thus, we finally have the following reformulated problem in terms of 77: n-1 77r=-r -(p(v++77)-p(v+)) mµ + mv+ _l_ + m77 _ m(n - l)rn-l 100 77(s) ds, r > l, 2µ rn-1 µrn-1 µ r 8 2n-l (15) lim 77(r) = 0, r--+oo 77(1) = 17b := vb - V+ (ub > 0), no boundary condition (ub < 0), where m = ub/v_ (ub > 0), and m = ub/(v+ + 77(1)) (ub < 0). Once the desired solution 77 of (15) is obtained, the velocity u is immediately obtained by (10) as u(r) = ub(v+ + 77(r)) (ub > 0), u(r) = ub(v+ + 77(r)) (ub < 0). v_rn-1 (v+ +77(l))rn-1 The existence theorem for the reformulated problem (15) is the following. 35Theorem 4.1 (Hashimoto-Matsumura [5]). Let n 2: 2. Then, for any v+ > 0, there exist positive constants Eo and C satisfying the following: (I) Let ub > 0. If lubl + l77bl ~ Eo, there exists a unique smooth solution 17 of the problem (15) satisfying l11(r)I ~ cr-2<n-l)(lubl 2 + l77bl), r 2: 1. (II) Let ub < 0. If lubl ~ Eo, there exists a unique smooth solution 17 of the problem (15) satisfying r 2: 1. Theorem 4.1 ensures that if lubl is sufficiently small, then it holds that V+ 3 2 ~ V+ + 17(1) ~ :?+· In addition to the Theorem 4.1, we obtain the following lemma for the property of stationary solution. Lemma 4.2. Let u_ ~ 0 and n 2: 2. Then, for any V+ > 0, if lubl « 1 we obtain the following properties. 17(r) > 0 , 17r(r) < 0, 17rr(r) > 0 (1 ~ r < oo), u(r) < 0 , Ur> 0 , Urr < 0 (l ~ r < oo) C31r-n+llubl ~ lu(r)I ~ C3r-n+ 1 1ubl, lurl ~ C4r-nlubl, lurrl ~ C5r-n-llubl, lp(r) - P+I ~ C5r-2n+2 1ubl 2 , 1:r(p(r)- P+)I ~ C1r-2n+llubl 2 , 1::2 (p(r)- P+)I ~ Csr-2nlubl 2 , where,C1 , • • • ,C8 are positive constants independent of r and lubl. 5. THE ENERGY ESTIMATES In this section, we derive the a priori estimate. u(r) and p(r) stand for the stationary solution for (7) here. In order to prove the stability result in Theorem 3.2, it is convenient to regard the solution (p, u) as a perturbation from the stationary solution (p, u). Thus, we define new unknown functions as ¢(r, r) := p(t, r) - p(r), 1/J(t, r) := u(t, r) - u(r). Subtracting (7) from (3) yields that { <Pt + U<Pr + P1/Jr = F, (16) p( 1/Jt + u1/Jr) + P' (p) <Pr - µ1/Jrr = G, where { F := -f5r1/J - Ure/>- n;1 (¢u + p'ljJ) G := -(¢1/J +ii¢+ ro'ljJ)ur - (P'(p) - P'(p))Pr + µ(n - l)(*)r (l 7) 36The initial and boundary conditions to the system (16) are derived from ( 4) and (6), ¢(0, r) = ¢o(r) := p(r) - p(r), 1/J(t, 0) = 0. 1/J(0, r) = 1/Jo(r) := u0 (r) - u(r) (18) The local existence of the solution ( ¢, 1/J) to the initial boundary value problem (16) and (18) is stated as follows.(Tani, [9]) Lemma 5.1. Assume that the same conditions in Theorem 3.2 hold. Then there exists a positive constant To, depending only on l¢ol1+u and l'I/Jol2+u, such that the initial boundary value problem (3) - (6) has a unique solution(¢, 1/J) in the space: (¢,1/J) E c1+u/2,1+u x c1+u/2,2+u([0,To] X [l,oo)), r n;-l ¢, r n2 11jJ, r n21 <Pr, r n 21 1Pr E G([0, To]; L 2 (ll~+)), r n 21 1Pr E £ 2 (0, To; L2 (ffi+)). Next, we proceed to the a priori estimate in the Sobolev space, which is stated in Proposition 5.2. To show this estimate, it is convenient to use notation N(t) := sup 11(¢,1/J)(T)lli-O'.:'.r::::t Proposition 5.2. Let(¢, 1/J) be a solution to (16), (17) and (18) in a time interval [0, T], which has the same regularity as in Lemma 5.1. Then there exist positive constants Eo and G, such that if N(T) ::; Eo, then the following estimate holds for an arbitrary t E [O, T]: 11(¢, 1/J)(t)lli + 1t 11¢r(T)ll 2 + 111/Jr(T)lli + 1(¢, <Pr)(T, l)l 2dT (19) Proof. The proof is divided into three steps, which are stated in Lemmas 5.3, 5.4 and 5.5. Combining the uniform estimates proved in these three Lemmas gives the desired estimate (19). □ The smallness assumption on N(T) in Proposition 5.2 ensures that if Eo is suff-ciently small, then there exist certain positive constants cp and Gp such that 0::; cp::; p(t, r) ::; Gp for t E [0, T]. Next, we introduce an energy form by C •= !n/,2 lp P(TJ) - P(p) (,,. 'f/ + 2 . 2 i5 TJ 37By using (16), (17) and (18), we see that p£ satisfies the equation by the straight-forward computation. t/;2 (p£)t + {pu£ + ( P(p) - P(j5) )t/J - µt/Jt/Jr - µ( n - 1) 2r }r 2 n - 1 ( pu 2 t/;2) + µt/Jr + -2- ----:;:-t/J + µ7i + (n -1) {pu [P P(ry) - P(j5) dry+ P(p) - P(j5) (¢u + pt/;)} r JP 772 rp + (n - 1/il (P(ry) - P(j5) - P'(p)P-= j5) r p p = - (pt/J2 + P(p) - P(p) - P'(p)(p - p)) Ur - ~</Jt/J ( (r~:~~)r) r We are in a position to state the basic energy estimate. Lemma 5.3. There exist positive constants c:1 and C such that if N(T) + lubl :::; c:1, then it holds that loo rn-1¢2 + rn-1t/J2dr + lubl 1t ¢(1, T)2dT + 1t loo rn-3t/J2 + rn-lt/J; + lublr-1t/J2drdT :=; C 100 rn-1¢6 + rn-ltj;Jdr + Clubl 31t 11¢r(T)ll 2dT, for an arbitrary t E [O, T]. Next, we state the L2-estimates for the first derivatives, <Pr and t/Jr- To derive this, we use a difference quotient of a function <p with respect to r defined as - ( ) ·- <p(r + h) - <p(r) for 1Ph - l.f)h r .- h , h > 0. In addition, the following identity holds (1.pw)h = i.()hWh + l.f)Wh, where we have defined <ph := cp(r + h). The statement of the L2-estimates for the first derivatives, <Pr is the following. 38Lemma 5.4. There exist positive constants c:2 and C such that if N(T) + l'ubl ::::; c:2, then the following estimate holds for any t E [0, T]: 100 rn-1¢;dr + lubl 1t <Pr(T, l)2dT 1tfoo ¢2 + lubl---2:. + rn-1¢; + rn-lur¢;drdT o 1 r '.SC loo rn-1¢6 + rn-lt/J&dr + loo rn-1¢6,rdr + Clubl31t 11¢r(T)ll2dT + N(t) 1t lubl 2</J(T, l)2dT + CN(t) 1t llt/JrllidT + C(lubl + N(t)) 1t loo rn-1¢; + rn-3t/J2 + rn-lt/J;drdT. The statement of the £ 2-estimates for the first derivatives, t/Jr is the following. Lemma 5.5. There exist positive constants c:3 and C such that if N(T) + lubl ::::; c:3 , then it holds that for t E [0, T]: loo rn-lt/J;dr + 1t loo rn-lt/J;rdrdT '.SC loo rn-1¢6 + rn-lt/J6dr + loo rn-1(¢6,r + W6,r)dr + Clubl31t 11¢r(T)ll2dT + C(lubl + N(t)) 1t loo rn-1¢; + rn-3t/J2 + rn-lt/J;drdT + CN(t) 1t rn-lt/J;rdT. Once the Sobolev estimates are obtained, we can proceed to derive the Holder estimates. A priori estimate in the Sobolev space (19) in Lemma 5.2 ensure the first step of the Hi::iolder estimate: IPlf/4,1/2 , lulf/4,1/2 < C(Eo), (20) where E0 is defined by Eo := 100 rn-1(¢6 + W6 + ¢6,r + t/J&,r)dr. Once we obtain the basic estimate (20), the higher order estimate in Hi::iolder space is derived by the same strategy as in Kawashima-Nishibata-Zhu [3], Nakamura-Nishibata-Yanagi [8]. The higher order estimate in Hi::iolder space is described as the following proposition. Proposition 5.6. Let O < a < l. Under the same assumptions in Lemma 5.1, it holds IPli+p+a-, lulf+p+a- < C(T), where C(T) is a constant depending only on T, IPoli+a-, luol2+a- and 11(¢,t/J)lli-39We can derive the above estimate by using shauder estimate in Lagrangean coordinate. (See [3] and [8] for details.) Finally, combining Lemma 5.1, Proposition 5.2 and Proposition 5.6, we obtain the Theorem 3.2. 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 sym-metric solutions of the multi-dimensional Burgers equation, J. Differential Equations, 266 (2019), 2805-2829. [5] I. Hashimoto, A. Matsumura, Existence of Radially Symmetric Stationary Solutions for the Compressible Navier-Stokes Equation, Methods and Applications of Analysis, 28 (2021), to appear. [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 spherically 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. [9] A. Tani, On the first initial-boundary problem of compressible viscous fluid motion, Pub!. RIMS Kyoto Univ. 13 (1977), 193-253. College of Science and Engineering Kanazawa university/ OCAMI Kanazawa 920-1192 JAPAN E-mail address: itsuko@se.kanazawa-u.ac.jp 

Asymptotic behavior toward radially symmetric stationary solutions of the compressible Navier-Stokes equation (Nonlinear and Random Waves)

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Asymptotic behavior toward radially symmetric stationary solutions of the compressible Navier-Stokes equation (Nonlinear and Random Waves)

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