L_{2} and L_{∞} Estimates of the Solutions for the Compressible Navier-Stokes Equations in a 3D Exterior Domain

Takayuki KOBAYASHI

Department of Mathematics, Kyushu Institute of Technology, 1-1 Sensui-cho, Tobata-ku, Kitakyushu 804-8550, Japan. e-mail : kobayasi@mns.kyutech.ac.jp

We consider the equation of the motion of compressible viscous fluid in a 3D exterior domain. The equation is given by the following system for the density ρ(t, x) and the velocity v(t, x) = (v_{1}(t, x), v_{2}(t, x), v_{3}(t, x)),

ρ_{t} + ∇ · (ρv) = 0  in (0, ∞) × Ω,
ρ(v_{t} + (v · ∇)v) + ∇P (ρ) = μΔv + (μ + ν)∇(∇ · v)  in (0, ∞) × Ω,
v|_{∂Ω} = 0  on (0, ∞) × ∂Ω,
ρ(0, x) = ρ_{0}(x), v(0, x) = v_{0}(x)  in Ω,

(1)

where Ω is an exterior domain in R^{3} with the compact smooth boundary ∂Ω, P = P (ρ) the pressure, μ ≥ 0, \frac{2}{3}μ + ν > 0 the viscosity coefficients. The unique existence of smooth solutions globally in time near constant state (ρ̄_{0}, 0), where ρ̄_{0} is a positive constant was proved by the employing the same argument as in Matsumura and Nishida [11, 12] for the Cauchy problem in R^{3} ; Matsumura and Nishida[13, 14, 15] for the exterior domain in R^{3}. Concerning the decay property of solutions (ρ(t, x), v(t, x)), if the initial data (ρ_{0}(x) − ρ̄_{0}, v_{0}(x)) belongs to H^{4} and L_{1}, then as t → ∞

∥(ρ(t, ·) − ρ̄_{0}, v(t, ·))∥_{L_{∞}} = O(t^{−3/2}),
∥(ρ(t, ·) − ρ̄_{0}, v(t, ·))∥_{L_{2}} = O(t^{−3/4}),
∥(ρ(t, ·) − ρ̄_{0}, v(t, ·))∥_{L_{2}} = O(t^{1/2}).

This fact was investigated by Hoff and Zumbrun [3, 4], Liu and Wang [9], Matsumura and Nishida [11, 12], Ponce [16] and Weike [17] for the Cauchy problem case ; Kobayashi [7], Kobayashi and Shibata [8] for the exterior domain case. On the other hand, if the initial data belongs to H^{3} only, namely we do not assume that the initial data belongs to L_{1}, then, Deckelnick [1, 2] showed that as t → ∞

∥(ρ_{t}(t, ·), v_{t}(t, ·))∥_{L_{2}(Ω)} = O(t^{−1/2}),
∥∂_{x}(ρ(t, ·), v(t, ·))∥_{L_{2}(Ω)} = O(t^{−1/4}),
∥(v(t, ·))∥_{C^{0}(Ω̄)} = O(t^{−1/4}),
∥ρ(t, ·) − ρ̄_{0}∥_{C^{0}(Ω̄)} = O(t^{−1/8}),

in the exterior domain case ; Matsumura [10] showed that as t → ∞

∥(ρ_{t}(t, ·), v_{t}(t, ·))∥_{L_{2}(R^{3})} = O(t^{−1/2}),
∥∂_{x}(ρ(t, ·), v(t, ·))∥_{L_{2}(R^{3})} = O(t^{−1/2}),
∥∂_{x}^{2}(ρ(t, ·), v(t, ·))∥_{L_{2}(R^{3})} = O(t^{−1}),
∥(ρ(t, ·) − ρ̄_{0}, v(t, ·))∥_{L_{∞}(R^{3})} = O(t^{−3/4}),

in the Cauchy problem case. In this lecture, we shall investigate the exterior problem of the system (1) and give the better decay rate than the rate obtained by Deckelnick [1, 2] in the case that the initial data belongs to H^{3} or H^{4} only. In particular, the L_{2}-decay rate of the first derivative with respect to the spacial variable x for the solutions corresponds to the rate obtained by Matsumura [10].

Now, in order to explain our main results, we shall introduce the notations and assumptions. Let L_{p} denotes the usual L_{p} space on Ω with norm ∥ · ∥_{L_{p}}. Put

W_{p}^{m} = {u ∈ L_{p} | ∥u∥_{W_{p}^{m}} < ∞}, ∥u∥_{W_{p}^{m}} = ∑_{|α|≤m} ∥∂_{x}^{α}u∥_{L_{p}},

H^{m} = W_{2}^{m}, W_{p}^{0} = L_{p}, H^{0} = L_{2}.

Set

W_{p}^{k,m} = {(ρ, v) = (ρ, v_{1}, v_{2}, v_{3}) | ρ ∈ W_{p}^{k}, v_{j} ∈ W_{p}^{m}, j = 1, 2, 3},
∥(ρ, v)∥_{W_{p}^{k,m}} = ∥ρ∥_{W_{p}^{k}} + ∥v∥_{W_{p}^{m}},

and

H^{k,m} = W_{2}^{k,m}, ∥u∥_{H^{k,m}} = ∥u∥_{W_{2}^{k,m}}.

Let ρ̄_{0} be a positive constant. We assume that

A1. P is a smooth function in a neighborhood of ρ̄_{0} and \frac{∂P}{∂ρ} > 0.

A2. The initial data (ρ_{0}, v_{0}) satisfies the compatibility condition and regularity, namely (ρ_{0} − ρ̄_{0}, v_{0}) ∈ H^{3}, v_{0}|_{∂Ω} = 0 and (ρ_{1}, v_{1}) = (ρ_{t}, v_{t})|_{t=0} satisfies

ρ_{1} = −∇ · (ρ_{0}v_{0}),

v_{1} = −(v_{0} · ∇)v_{0} + \frac{μ}{ρ_{0}} Δv_{0} + \frac{ν}{ρ_{0}} ∇(∇ · v_{0}) − \frac{∇P (ρ_{0})}{ρ_{0}},

and

ρ_{1} ∈ H^{2}, v_{1} ∈ H^{1}, v_{1}|_{∂Ω} = 0.

Put

X(0, ∞) = {U = (ρ, v) | ρ − ρ̄_{0} ∈ ⋂_{j=0}^{1} C^{j}([0, ∞); H^{3−j}),

∂_{x}ρ ∈ L_{2}((0, ∞); H^{2}), ρ_{t}, v_{t} ∈ L_{2}((0, ∞); H^{2}),

v ∈ ⋂_{j=0}^{1} C^{j}([0, ∞); H^{3−2j}), ∂_{x}v ∈ L_{2}((0, ∞); H^{3})},

and

N(0, ∞)^{2} = sup_{0≤t<∞} (∥U(t) − Ū_{0}∥_{H^{3}}^{2} + ∥U_{t}(t)∥_{H^{2,1}}^{2}) + ∫_{0}^{∞} (∥∂_{x}U(s)∥_{H^{2,3}}^{2} + ∥U_{s}(s)∥_{H^{2}}^{2}) ds,

where Ū_{0} = (ρ̄_{0}, 0). Then, we have

Proposition 1 (Matsumura and Nishida [13, 14, 15]) Assume that the assumptions A.1 and A.2 hold. Then, there exists an ϵ_{0} such that if ∥(ρ_{0} − ρ̄_{0}, v_{0})∥_{H^{3}} ≤ ϵ_{0}, then (1) admits a unique solution (ρ, v) ∈ X(0, ∞).

Moreover, there exists a constant C such that

N(0, ∞) ≤ C∥(ρ_{0} − ρ̄_{0}, v_{0})∥_{H^{3}}.

Remark. If the initial data (ρ_{0} −ρ̄_{0}, v_{0}) ∈ H^{4} and satisfies the second order compatibility condition and regularity, namely (ρ_{2}, v_{2}) = (ρ_{tt}, v_{tt})|_{t=0} is determined successively by the initial data (ρ_{0}, v_{0}) through the system (1), then we have

Ñ(0, ∞)^{2} = sup_{0≤t<∞} (∥U(t) − Ū_{0}∥_{H^{4}}^{2} + ∥U_{t}(t)∥_{H^{3,2}}^{2})
+ ∫_{0}^{∞} (∥∂_{x}U(s)∥_{H^{3,4}}^{2} + ∥U_{s}(s)∥_{H^{3}}^{2}) ds,
≤ C∥(ρ_{0} − ρ̄_{0}, v_{0})∥_{H^{4}}^{2}.

Now, we shall state our main results.

Theorem 1 Assume that the assumptions A.1 and A.2 hold. Then, there exists an ϵ_{1} such that if ∥(ρ_{0} − ρ̄_{0}, v_{0})∥_{3,2} ≤ ϵ_{1}, the solution (ρ, v) of the system (1) has the following asymptotic behavior as t → ∞ :

∥(ρ_{t}(t, ·), v_{t}(t, ·))∥_{L_{2}} = O(t^{−1/2}),
∥∂_{x}v(t, ·)∥_{H^{1}} = O(t^{−1/2}),
∥∂_{x}ρ(t, ·)∥_{L_{2}} = O(t^{−1/2}),
∥∂_{x}^{2}ρ(t, ·)∥_{L_{2}} = O(t^{−3/4}log t),
∥(ρ(t, ·) − ρ̄_{0}, v(t, ·))∥_{L_{∞}} = O(t^{−3/4}log t).

Corollary 1 The assumptions in Theorem 2.1 hold. Moreover, if the initial data (ρ_{0} − ρ̄_{0}, v_{0}) ∈ H^{4} satisfies the second order compatibility condition and regularity in Remark, then we have

∥(ρ(t, ·) − ρ̄_{0}, v(t, ·))∥_{L_{∞}} = O(t^{−3/4}) as t → ∞.

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