Decay profiles of a linear artificial viscosity system

Transcription

1 Decay profiles of a linear artificial viscosity system Gene Wayne, Ryan Goh and Roland Welter Boston University rwelter@bu.edu July 2, 2018 This research was supported by funding from the NSF. Roland Welter (BU) Decay profiles July 2, / 22

2 Overview 1 Introduction 2 Theorem 3 Sketch of the proof Roland Welter (BU) Decay profiles July 2, / 22

3 The Compressible Navier-Stokes Equations In their 1995 paper, Hoff and Zumbrun study the compressible Navier-Stokes equations over R d : m j t + div ( m j m ρ ρ t + div m = 0 ) + P(ρ)xj = ɛ ( m j ρ ) (m ) + η div ρ x j These equations govern the density ρ and momentum m of a fluid and are particularly relevant for fluids having high Mach number. Roland Welter (BU) Decay profiles July 2, / 22

4 The Compressible Navier-Stokes Equations pt 2 ( ) ρ They examined perturbations away from the constant state = showing that if u 0 = (ρ 0, m 0 ) T is such that m ( ) 1, 0 E = u 0 L 1 + u 0 H [d/2]+l for l 3 is sufficiently small, then the solution u(x, t) = (ρ, m) T satisfies u(t) L p C(l)EL(t)(1 + t) rp for 2 p, where r p = d 2 (1 1 p ) is the rate of decay of the heat kernel and { log(1 + t) if d = 2 L(t) 1 otherwise Roland Welter (BU) Decay profiles July 2, / 22

5 The Linear Artificial Viscosity System In the course of obtaining their result, Hoff and Zumbrun analyze the following system: ρ t + div m = 1 (ɛ + η) ρ 2 m t + c 2 ρ = ɛ m + 1 (η ɛ) div m 2 Hoff and Zumbrun refer to this system as the effective linear artificial viscosity system, and show that the solution u(x, t) = (ρ, m) T of the CNSE are time asymptotic to those of the artificial viscosity system: u(t) G(t) u 0 L p C(l)E(1 + t) rp 1/2 Roland Welter (BU) Decay profiles July 2, / 22

6 Higher order asymptotics Kagei and Okita expanded on these results by computing a higher order profile of the asymptotic behavior. For d 3, they showed u(t) G(t) u 0 d i G 1 (t, ) i=1 0 R d F 0 i dyds L p CK(t)(1 + t) rp 3 4 We aim to provide a method by which the asymptotic behavior can be computed out to any desired order. To do so we ll study the linear artificial viscosity system. Roland Welter (BU) Decay profiles July 2, / 22

7 Fourier Analysis and Helmholtz Decomposition We assume our solutions are such that we can take the Fourier transform. The transformed solutions satisfy ˆρ t + iξ T ˆm = ν ξ 2 ˆρ ˆm t + ic 2 ξ ˆρ = ɛ ξ 2 ˆm 1 2 (η ɛ)ξ( ξ T ˆm ) where ν = 1 2 (ɛ + η). We then make use of the Helmholtz projection, which separates the divergence and divergence free parts of m: ˆm = iξ ξ 2 â + ˆb Here â = iξ T ˆm and ˆb satisfies ξ T ˆb = 0. Roland Welter (BU) Decay profiles July 2, / 22

8 Hyperbolic-parabolic system By making this projection, one must then consider the resulting system: t ρ = ν ρ a t a = c 2 ρ + ν a t b = ɛ b The incompressible part is decoupled from the rest of the system and has already been analyzed in [Gallay and Wayne, 2002], hence we study the resulting hyperbolic-parabolic system for ρ and a. We ll work over R 2. Roland Welter (BU) Decay profiles July 2, / 22

9 Hyperbolic-parabolic system pt 2 To this end, we introduce the following operators: W = ( 0 ) 1 c 2 0 and H = and hence our system can be written as ( ) ρ t = ( W + H ) ( ) ρ a a ( ) ν 0 0 ν We ll also be working in the function space L 2 (n) which is the closure of the smooth functions having compact support with respect the norm ( ) 1/2 f L 2 (n) = (1 + x 2 ) n f 2 dx R 2 (1) Roland Welter (BU) Decay profiles July 2, / 22

10 Main Result Theorem Choose n > 1, and suppose that ρ 0, a 0 C L 2 (n). If k Z is such that k + 1 < n k + 2, then the solution u(x, t) = (ρ, a) T of (1) with initial data u 0 = (ρ 0, a 0 ) T satisfies and u(, t) L C u 0 L 2 (n)(1 + νt) 5/4 1 u(, t) S(t)(H α, u (1 + νt) α 0 )φ α 2 +1 α k where S(t) is the semigroup generated by W. L C u 0 L 2(1 + t) 5 4 k 2 Roland Welter (BU) Decay profiles July 2, / 22

11 Sketch of the proof 1 Commutativity of the W and H operators 2 Hermite expansion for solutions of the heat equation 3 Decay of solutions to the wave equation Roland Welter (BU) Decay profiles July 2, / 22

12 Commutativity of the W and H operators In the Fourier domain, the system is ) (ˆρ t = ( ) (ˆρ Ŵ + Ĥ) â â hence the solution is found to be ( ) ρ(x, t) = 1 a(x, t) (2π) 2 e ix ξ exp [( Ŵ + Ĥ ) t ] ) (ˆρ0 (ξ) dξ R 2 â 0 (ξ) Since the matrices Ŵ and Ĥ commute for all ξ R 2, we can write this as ( ) ρ(x, t) = 1 a(x, t) (2π) 2 e ix ξ exp [ Ŵ t ] exp [ Ĥt ] ) (ˆρ0 (ξ) dξ R 2 â 0 (ξ) Roland Welter (BU) Decay profiles July 2, / 22

13 Commutativity of the W and H operators In the Fourier domain, the system is ) (ˆρ t = ( ) (ˆρ Ŵ + Ĥ) â â hence the solution is found to be ( ) ρ(x, t) = 1 a(x, t) (2π) 2 e ix ξ exp [( Ŵ + Ĥ ) t ] ) (ˆρ0 (ξ) dξ R 2 â 0 (ξ) Since the matrices Ŵ and Ĥ commute for all ξ R 2, we can write this as ( ) ρ(x, t) = 1 a(x, t) (2π) 2 e ix ξ exp [ Ŵ t ] exp [ Ĥt ] ) (ˆρ0 (ξ) dξ R 2 â 0 (ξ) Roland Welter (BU) Decay profiles July 2, / 22

14 Hermite Expansion As shown in [Gallay and Wayne, 2002], solutions of the heat equation in L 2 (n) have a special form which we can make use of here. Suppose that k Z is such that k + 1 < n k + 2. We define for α k, and let φ 0 (x, t) = 1 x 2 4π e 4(1+νt) φ α (x, t) = (1 + νt) α /2 α x φ 0 (x) H α (x) = 2 α /4 α! e x 2 x (e ) α x 2 /4 be the α-th Hermite polynomial. Roland Welter (BU) Decay profiles July 2, / 22

15 Hermite expansion pt 2 Lemma Let n 0, and suppose that f 0 L 2 (n). Let k Z be such that k + 1 < n k + 2. If S(t) is the semigroup associated with the heat equation, then the solution f (x, t) = S(t)f 0 has the form where for all ɛ > 0 f (x, t) = α k (H α, f 0 )(1 + νt) α 2 1 φ α + R(x, t) ( ) 1 R(, t) L 2 (n) C f 0 L 2 (n) 1 + νt 2 (1+n ɛ) Roland Welter (BU) Decay profiles July 2, / 22

16 Commutativity of the W and H operators In the Fourier domain, the system is ) (ˆρ t = ( Ŵ + Ĥ ) (ˆρ ) â â hence the solution is found to be ( ) ρ(x, t) = 1 a(x, t) (2π) 2 e ix ξ exp [( Ŵ + Ĥ) t ] ) (ˆρ0 (ξ) dξ R 2 â 0 (ξ) Since the matrices Ŵ and Ĥ commute for all ξ R2, we can write this as ( ) ρ(x, t) = 1 a(x, t) (2π) 2 e ix ξ exp [ Ŵ t ] ( ) ˆf (ξ, t) dξ R 2 ĝ(ξ, t) Roland Welter (BU) Decay profiles July 2, / 22

17 Decay of solutions to the wave equation Following [Constantin, 2012], we make use of the method of oscillatory integrals for bounding solutions of the wave equation. Write where ρ(x, t) = 1 [ ] (2π) 2 e ix ξ sin(ct ξ ) ξ 1 ĝ(ξ) + cos(ct ξ )ˆf (ξ) dξ R 2 = ρ + + ρ ρ ± = 1 2(2π) 2 e i(x ξ±ct ξ )(ˆf i ξ 1 ĝ ) dξ R 2 Without loss of generality, assume x = (0, x ). Roland Welter (BU) Decay profiles July 2, / 22

18 Decay of solutions to the wave equation pt 2 Using a smooth cutoff function χ, we can write ρ + (x, t) = 1 2(2π) 2 [ R 2 e i( x ξ 2+ct ξ ) [ˆf i ξ 1 ĝ ] (1 χ)dξ + δ 0 0 eicrt(λ cos θ+1)[ˆf ir 1 ĝ ] χdθrdr + 0 π π δ eicrt(λ cos θ+1)[ˆf ir 1 ĝ ] χdθrdr ] = T 1 + T 2 + T 3 where λ = x ct, r = ξ and cos θ = ξ 2 ξ. Roland Welter (BU) Decay profiles July 2, / 22

19 Decay of solutions to the wave equation pt 3 One can bound each of the terms separately. For T 1 T 1 = 1 2(2π) 2 e i( x ξ 2+ct ξ ) [ˆf i ξ 1 ĝ ] (1 χ)dξ R 2 write e ict ξ = ( 1 ξ ) j j ict ξ 1 ξ j e ict ξ 1 By integrating by parts we can obtain the desired decay. Similarly, for T 2 write T 2 = 1 2(2π) 2 δ 0 0 e icrt(λ cos θ+1)[ˆf ik 1 ĝ ] χdθrdr ( e icrt(λ cos θ+1) 1 ) j j = ict(λ cos θ + 1) cos θ+1) eicrt(λ r j Roland Welter (BU) Decay profiles July 2, / 22

20 Decay of solutions to the wave equation pt 4 For T 3, there are two cases: when x ct and when x is far from the light cone. When x is far from the light cone, the same integration by parts technique can be used. When x ct, one needs to explicitly use the Hermite expansion and bound the integral using the properties of each term. Roland Welter (BU) Decay profiles July 2, / 22

21 Future Directions 1 Weaken the regularity assumptions. 2 Extend to the nonlinear artificial viscosity case. 3 Obtain results in higher dimensions. 4 Relate expansion to the compressible Navier-Stokes. Roland Welter (BU) Decay profiles July 2, / 22

22 References David Hoff and Kevin Zumbrun (1995) Multi-dimensional Diffusion Waves for the Navier-Stokes Equations of Compressible Flow Indiana University Mathematics Journal Vol. 44, No 2 pp Yoshiyuki Kagei and Masatoshi Okita (2017) Asymptotic profiles for the compressible Navier-Stokes equations on the whole space Journal of Mathematical Analysis and its Applications Vol. 445, No 1 pp Thierry Gallay and Eugene Wayne (2002) Invariant Manifolds and the Long-Time Asymptotics of the Navier-Stokes and Vorticity Equations on R 2 Archive for Rational Mechanics and Analysis 162(3): Peter Constantin (2012) Introduction to the Wave Equation Roland Welter (BU) Decay profiles July 2, / 22