Optimal decay estimates on the framework of Besov spaces for - - PowerPoint PPT Presentation
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References Optimal decay estimates on the framework of Besov spaces for hyperbolic systems with degenerate dissipation Jiang Xu Nanjing University
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References
Jiang Xu
Nanjing University of Aeronautics and Astronautics, China Joint work with Shuichi Kawashima Mathflows 2015 September 13-18th, 2015, Porquerolles
Jiang Xu hyperbolic dissipative systems
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References
1
Hyperbolic systems with symmetric dissipation Introduction New decay framework L2(Rn) ∩ ˙ B−s
2,∞(Rn)
Littlewood-Paley pointwise estimates Nonlinear applications
2
Hyperbolic systems with Non-symmetric dissipation Examples Frequency-localization time-decay inequality An open problem
3
References
Jiang Xu hyperbolic dissipative systems
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References Introduction New decay framework L2(Rn) ∩ ˙ B−s
2,∞(Rn)
Littlewood-Paley pointwise estimates Nonlinear applications
1
Hyperbolic systems with symmetric dissipation Introduction New decay framework L2(Rn) ∩ ˙ B−s
2,∞(Rn)
Littlewood-Paley pointwise estimates Nonlinear applications
2
Hyperbolic systems with Non-symmetric dissipation Examples Frequency-localization time-decay inequality An open problem
3
References
Jiang Xu hyperbolic dissipative systems
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References Introduction New decay framework L2(Rn) ∩ ˙ B−s
2,∞(Rn)
Littlewood-Paley pointwise estimates Nonlinear applications
Consider the following hyperbolic system A0∂tw + n
j=1 Aj∂xjw + Lw = 0,
w(0, x) = w0, (1) for (t, x) ∈ [0, +∞) × Rn. w(t, x) : RN-valued function; Aj(j = 0, 1, · · ·, n) and L are constant matrices of order N;
Jiang Xu hyperbolic dissipative systems
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References Introduction New decay framework L2(Rn) ∩ ˙ B−s
2,∞(Rn)
Littlewood-Paley pointwise estimates Nonlinear applications
Assume that the equation of (1) is “symmetric hyperbolic" in the following sense: (A1) Matrices Aj(j = 0, · · ·, n) are real symmetric and, in addition, A0 is positive definite. L is real symmetric and nonnegative definite. Also, assume (1) satisfies the “Shizuta-Kawashima" condition ( [Shizuta & Kawashima, Hokkaido Math. J., 1985)]) (A2) Let φ ∈ RN and (λ, ω) ∈ R × Sn−1. If Lφ = 0 and λA0φ + A(ω)φ = 0, then φ = 0.
Jiang Xu hyperbolic dissipative systems
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References Introduction New decay framework L2(Rn) ∩ ˙ B−s
2,∞(Rn)
Littlewood-Paley pointwise estimates Nonlinear applications
[Umeda, Kawashima & Shizuta (Japan J. Appl. Math., 1984)] The dissipative structure of (1) satisfies Reλ(iξ) ≤ −cη1(ξ) with η1(ξ) = |ξ|2 1 + |ξ|2 for c > 0, which leads to the optimal decay estimate wL2(Rn) w0L2(Rn)∩Lp(Rn)(1 + t)− n
2 ( 1 p − 1 2)
(2) for 1 ≤ p < 2.
Jiang Xu hyperbolic dissipative systems
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References Introduction New decay framework L2(Rn) ∩ ˙ B−s
2,∞(Rn)
Littlewood-Paley pointwise estimates Nonlinear applications
1
Hyperbolic systems with symmetric dissipation Introduction New decay framework L2(Rn) ∩ ˙ B−s
2,∞(Rn)
Littlewood-Paley pointwise estimates Nonlinear applications
2
Hyperbolic systems with Non-symmetric dissipation Examples Frequency-localization time-decay inequality An open problem
3
References
Jiang Xu hyperbolic dissipative systems
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References Introduction New decay framework L2(Rn) ∩ ˙ B−s
2,∞(Rn)
Littlewood-Paley pointwise estimates Nonlinear applications
In [X.-Kawashima (ARMA, 2015)], we give a new decay framework for (1): L2(Rn) ∩ ˙ B−s
2,∞(Rn)(0 < s ≤ n/2)
which can be regarded as the natural generalization, since Lp(Rn) ֒ → ˙ B−s
2,∞(Rn)
with s + n/2 = n/p. In particular, L1(Rn) ֒ → ˙ B0
1,∞(Rn) ֒
→ ˙ B−n/2
2,∞ (Rn).
Jiang Xu hyperbolic dissipative systems
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References Introduction New decay framework L2(Rn) ∩ ˙ B−s
2,∞(Rn)
Littlewood-Paley pointwise estimates Nonlinear applications
Let the assumptions (A1)-(A2) hold. Suppose w0 ∈ L2(Rn) ∩ ˙ B−s
2,∞(Rn) for s > 0, then the solution of (1) has
the decay estimate wL2(Rn) w0L2(Rn)∩ ˙
B−s
2,∞(Rn)(1 + t)−s/2.
(3) In particular, suppose w0 ∈ L2(Rn) ∩ Lp(Rn)(1 ≤ p < 2), one further has wL2(Rn) w0L2(Rn)∩Lp(Rn)(1 + t)− n
2 ( 1 p − 1 2 ).
(4)
Jiang Xu hyperbolic dissipative systems
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References Introduction New decay framework L2(Rn) ∩ ˙ B−s
2,∞(Rn)
Littlewood-Paley pointwise estimates Nonlinear applications
The proof is divided into several steps. Step 1. Pointwise energy estimte gives d dt E[ ˆ w] + (c0 − κC)|(I − P) ˆ w|2 + c1κ|ξ|2 1 + |ξ|2 | ˆ w|2 ≤ 0 (5) with E[ ˆ w] = 1 2(A0 ˆ w, ˆ w) + κ 2Im
1 + |ξ|2 K(ω)A0 ˆ w, ˆ w
where we chosen κ > 0 so small that c0 − κC ≥ 0 and E[ ˆ w] ≈ | ˆ w|2, since A0 is positive definite.
Jiang Xu hyperbolic dissipative systems
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References Introduction New decay framework L2(Rn) ∩ ˙ B−s
2,∞(Rn)
Littlewood-Paley pointwise estimates Nonlinear applications
Step 2. The low- and high-frequency decompositions (the unit decomposition): w = wL + wH with wL = F−1[φ(ξ) ˆ w(ξ)] and wH = F−1[ϕ(ξ) ˆ w(ξ)], where 1 ≡ φ(ξ) + ϕ(ξ), and φ, ϕ ∈ C∞
c (Rn) (0 ≤ φ(ξ), ϕ(ξ) ≤ 1)
satisfy φ(ξ) ≡ 1, if |ξ| ≤ R; φ(ξ) ≡ 0, if |ξ| ≥ 2R with R > 0.
Jiang Xu hyperbolic dissipative systems
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References Introduction New decay framework L2(Rn) ∩ ˙ B−s
2,∞(Rn)
Littlewood-Paley pointwise estimates Nonlinear applications
Step 3. The high-frequency estimate d dt (ϕ2E[ ˆ w]) + c1R2 1 + R2 |ϕ ˆ w|2 ≤ 0, (6) which implies that wHL2 ≤ Ce−c2tw0L2, (7) for c2 > 0 depending on R.
Jiang Xu hyperbolic dissipative systems
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References Introduction New decay framework L2(Rn) ∩ ˙ B−s
2,∞(Rn)
Littlewood-Paley pointwise estimates Nonlinear applications
Step 4. The low-frequency estimate d dt (˜ E[ ˆ w]2) + c1 1 + R2 |ξ|2| ˆ wL|2 ≤ 0, (8) where ˜ E[ ˆ w] := {φ2E[ ˆ w]}1/2 and ˜ E[ ˆ w] ≈ | ˆ wL|. Furthermore, Plancherel’s theorem gives d dt ˜ E2
L + c3∇wL2 L2 ≤ 0,
(9) where ˜ EL ≈ wLL2 and the constant c3 > 0 depends on R.
Jiang Xu hyperbolic dissipative systems
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References Introduction New decay framework L2(Rn) ∩ ˙ B−s
2,∞(Rn)
Littlewood-Paley pointwise estimates Nonlinear applications
Step 5. Applying the interpolation inequality fL2 ∇fθ
L2f1−θ ˙ B−s
2,∞
s 1 + s
(10) to the low-frequency part wL, we obtain the differential equality from (9): d dt ˜ E2
1 + Cw0−2/s ˙ B−s
2,∞ wL2(1+1/s)
L2
≤ 0, (11) which implies that wLL2 w0 ˙
B−s
2,∞(1 + t)−s/2.
(12)
Jiang Xu hyperbolic dissipative systems
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References Introduction New decay framework L2(Rn) ∩ ˙ B−s
2,∞(Rn)
Littlewood-Paley pointwise estimates Nonlinear applications
Step 6. Combining the low-frequency estimate and high-frequency estimate: wL2 ≤ wLL2 + wHL2
B−s
2,∞(1 + t)−s/2.
(13) Finally, note that the embedding Lp(Rn) ֒ → ˙ B−s
2,∞(Rn)(s = n(1/p − 1/2)), we arrive at
wL2 w0L2∩Lp(1 + t)− n
2 ( 1 p − 1 2),
(14) which coincides with the corresponding result (2) in the classical framework.
Jiang Xu hyperbolic dissipative systems
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References Introduction New decay framework L2(Rn) ∩ ˙ B−s
2,∞(Rn)
Littlewood-Paley pointwise estimates Nonlinear applications
Consider the hyperbolic-parabolic composite system A0Ut + n
j=1 AjUxj = n j,k Bj,kUxjxk,
U(0, x) = U0. (15) Similarly, it can be shown that the solution of (15) admits the decay estimate UL2(Rn) U0L2(Rn)∩ ˙
B−s
2,∞(Rn)(1 + t)−s/2,
(16) if U0 ∈ L2(Rn) ∩ ˙ B−s
2,∞(Rn) for s > 0.
Jiang Xu hyperbolic dissipative systems
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References Introduction New decay framework L2(Rn) ∩ ˙ B−s
2,∞(Rn)
Littlewood-Paley pointwise estimates Nonlinear applications
In the proofs of Theorems 1.1, we see that the new framework allows to pay less attention on the traditional spectral analysis. Then, Littlewood-Paley decomposition ⇒ Unit decomposition which gives us the main MOTIVATION to study decay problems in spatially Besov spaces for nonlinear dissipative systems.
Jiang Xu hyperbolic dissipative systems
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References Introduction New decay framework L2(Rn) ∩ ˙ B−s
2,∞(Rn)
Littlewood-Paley pointwise estimates Nonlinear applications
1
Hyperbolic systems with symmetric dissipation Introduction New decay framework L2(Rn) ∩ ˙ B−s
2,∞(Rn)
Littlewood-Paley pointwise estimates Nonlinear applications
2
Hyperbolic systems with Non-symmetric dissipation Examples Frequency-localization time-decay inequality An open problem
3
References
Jiang Xu hyperbolic dissipative systems
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References Introduction New decay framework L2(Rn) ∩ ˙ B−s
2,∞(Rn)
Littlewood-Paley pointwise estimates Nonlinear applications
If w0 ∈ ˙ Bσ
2,1 ∩ ˙
B−s
2,∞ for σ ≥ 0 and s > 0, then the solutions
w(t, x) of (1) has the decay estimate ΛℓwBσ−ℓ
2,1
w0 ˙
Bσ
2,1∩ ˙
B−s
2,∞(1 + t)− ℓ+s 2
(17) for 0 ≤ ℓ ≤ σ. In particular, if w0 ∈ ˙ Bσ
2,1 ∩ Lp(1 ≤ p < 2), one
further has ΛℓwBσ−ℓ
2,1
w0 ˙
Bσ
2,1∩Lp(1 + t)− n 2 ( 1 p − 1 2)− ℓ 2
(18) for 0 ≤ ℓ ≤ σ.
Jiang Xu hyperbolic dissipative systems
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References Introduction New decay framework L2(Rn) ∩ ˙ B−s
2,∞(Rn)
Littlewood-Paley pointwise estimates Nonlinear applications
Following the similar procedure of proof of Theorem 1.1, we can obtain Theorem 1.2 by deducing that
2q(σ−ℓ)∆qΛℓwL2
q≥0
2q(σ−ℓ)∆qΛℓw0L2 e−c2tw0 ˙
Bσ
2,1,
(19) and Λℓ∆−1wL2 w0 ˙
B−s
2,∞(1 + t)− ℓ+s 2 .
(20)
Jiang Xu hyperbolic dissipative systems
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References Introduction New decay framework L2(Rn) ∩ ˙ B−s
2,∞(Rn)
Littlewood-Paley pointwise estimates Nonlinear applications
If w0 ∈ ˙ Bσ
2,1 ∩ ˙
B−s
2,∞ for σ ∈ R, s ∈ R satisfying σ + s > 0, then
the solution w(t, x) of (1) has the decay estimate w ˙
Bσ
2,1 w0 ˙
Bσ
2,1∩ ˙
B−s
2,∞(1 + t)− σ+s 2 .
(21) In particular, if w0 ∈ ˙ Bσ
2,1 ∩ Lp(1 ≤ p < 2), one further has
w ˙
Bσ
2,1 w0 ˙
Bσ
2,1∩Lp(1 + t)− n 2 ( 1 p − 1 2)− σ 2 .
(22)
Jiang Xu hyperbolic dissipative systems
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References Introduction New decay framework L2(Rn) ∩ ˙ B−s
2,∞(Rn)
Littlewood-Paley pointwise estimates Nonlinear applications
Due to the fact that the operator ˙ ∆q consists with ∆q for q ≥ 0, it suffices to show the low-frequency estimate. In this case, we use a different idea inspired by Sohinger & Strain [Adv. Math., 2014]. Precisely, 2qσwqL2 w0 ˙
B−s
2,∞(1 + t)− σ+s 2
t)σ+se− 1
2 c3(2q√
t)2
, (23) which implies that
2qσwqL2 w0 ˙
B−s
2,∞(1 + t)− σ+s 2 .
(24)
Jiang Xu hyperbolic dissipative systems
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References Introduction New decay framework L2(Rn) ∩ ˙ B−s
2,∞(Rn)
Littlewood-Paley pointwise estimates Nonlinear applications
1
Hyperbolic systems with symmetric dissipation Introduction New decay framework L2(Rn) ∩ ˙ B−s
2,∞(Rn)
Littlewood-Paley pointwise estimates Nonlinear applications
2
Hyperbolic systems with Non-symmetric dissipation Examples Frequency-localization time-decay inequality An open problem
3
References
Jiang Xu hyperbolic dissipative systems
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References Introduction New decay framework L2(Rn) ∩ ˙ B−s
2,∞(Rn)
Littlewood-Paley pointwise estimates Nonlinear applications
Let us consider generally hyperbolic systems of balance laws Ut +
n
F j(U)xj = G(U), (25) for (t, x) ∈ [0, +∞) × Rn, where U(t, x) : RN-valued function taking values in an open set OU ⊂ RN (the state space); F j, G : RN-valued smooth functions on OU; The system (25) is supplemented with the initial data U0 = U(0, x), x ∈ Rn. (26)
Jiang Xu hyperbolic dissipative systems
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References Introduction New decay framework L2(Rn) ∩ ˙ B−s
2,∞(Rn)
Littlewood-Paley pointwise estimates Nonlinear applications
G(U) ≡ 0, conservation laws system (classical solutions blow-up); G(U) = 0, hyperbolic response, or relaxation schemes. Its form is given by G(U) =
with 0 ∈ RN1 and g(U) ∈ RN2, where N1 + N2 = N(N1 = 0).
Jiang Xu hyperbolic dissipative systems
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References Introduction New decay framework L2(Rn) ∩ ˙ B−s
2,∞(Rn)
Littlewood-Paley pointwise estimates Nonlinear applications
An important example in gas dynamics is the following damped compressible Euler equations: ∂tρ + ∇ · (ρu) = 0, ∂t(ρu) + ∇ · (ρu ⊗ u) + ∇P(ρ) = −ρu (27) for (t, x) ∈ [0, +∞) × Rn, where ρ(t, x) : the density of gas flow; u(t, x) : the velocity field of gas flow.
Jiang Xu hyperbolic dissipative systems
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References Introduction New decay framework L2(Rn) ∩ ˙ B−s
2,∞(Rn)
Littlewood-Paley pointwise estimates Nonlinear applications
In this position, decay results are stated as follows (σc = 1 + n/2, [X.-Kawashima (ARMA, 2015)]).
Suppose that w0 − ¯ w ∈ Bσc
2,1 ∩ ˙
B−s
2,∞(0 < s ≤ n/2) and the norm
E0 := w0 − ¯ wBσc
2,1∩ ˙
B−s
2,∞ is sufficiently small. Then it holds that
Λℓw(t)X1 E0(1 + t)− s+ℓ
2
(28) for 0 ≤ ℓ ≤ σc − 1, where X1 := Bσc−1−ℓ
2,1
if 0 ≤ ℓ < σc − 1 and X1 := ˙ B0
2,1 if ℓ = σc − 1;
Jiang Xu hyperbolic dissipative systems
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References Introduction New decay framework L2(Rn) ∩ ˙ B−s
2,∞(Rn)
Littlewood-Paley pointwise estimates Nonlinear applications
Λℓ(I − P)w(t)X2 E0(1 + t)− s+ℓ+1
2
(29) for 0 ≤ ℓ ≤ σc − 2, where X2 := Bσc−2−ℓ
2,1
if 0 ≤ ℓ < σc − 2 and X2 := ˙ B0
2,1 if ℓ = σc − 2.
Jiang Xu hyperbolic dissipative systems
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References Examples Frequency-localization time-decay inequality An open problem
1
Hyperbolic systems with symmetric dissipation Introduction New decay framework L2(Rn) ∩ ˙ B−s
2,∞(Rn)
Littlewood-Paley pointwise estimates Nonlinear applications
2
Hyperbolic systems with Non-symmetric dissipation Examples Frequency-localization time-decay inequality An open problem
3
References
Jiang Xu hyperbolic dissipative systems
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References Examples Frequency-localization time-decay inequality An open problem
Consider the Timoshenko system, which is a set of two coupled wave equations of the form
ψtt − σ(ψx)x − (ϕx − ψ) + γψt = 0 (30) for (t, x) ∈ [0, +∞) × R. The system (30) is supplemented with the initial data (ϕ, ϕt, ψ, ψt)(x, 0) = (ϕ0, ϕ1, ψ0, ψ1)(x). (31)
Jiang Xu hyperbolic dissipative systems
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References Examples Frequency-localization time-decay inequality An open problem
By the change of variable introduced by [Ide, Haramoto & Kawashima (M3AS, 2008)]: v = ϕx − ψ, u = ϕt, z = aψx, y = ψt, (32) with a > 0 being the sound speed defined by a2 = σ′(0). System (30)-(31) can be rewritten as a Cauchy problem for the first-order hyperbolic system of U = (v, u, z, y)⊤
U(x, 0) = U0(x), (33)
Jiang Xu hyperbolic dissipative systems
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References Examples Frequency-localization time-decay inequality An open problem
where A(U) = − 1 1 a
σ′(z/a) a
, L = 1 −1 γ . Note that A(U) is a real symmetrizable matrix due to σ′(z/a) > 0, the dissipative matrix L is nonnegative definite, however, L is not symmetric.
Jiang Xu hyperbolic dissipative systems
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References Examples Frequency-localization time-decay inequality An open problem
Consider the compressible isentropic Euler-Maxwell system in plasmas physics, which is given by ∂tn + ∇ · (nu) = 0, ∂t(nu) + ∇ · (nu ⊗ u) + ∇p(n) = −n(E + u × B) − nu, ∂tE − ∇ × B = nu, ∇ · E = n∞ − n, ∂tB + ∇ × E = 0, ∇ · B = 0, (34) for (t, x) ∈ [0, +∞) × R3.
Jiang Xu hyperbolic dissipative systems
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References Examples Frequency-localization time-decay inequality An open problem
Set w = (n, u, E, B)⊤ (⊤ transpose). Then (34) can be written in the vector form A0(w)wt +
3
Aj(w)wxj + L(w)w = 0, (35) where the coefficient matrices are given explicitly as A0(w) = a(n) nI I I , L(w) = n(I − ΩB) nI −nI ,
Jiang Xu hyperbolic dissipative systems
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References Examples Frequency-localization time-decay inequality An open problem
3
Aj(w)ξj = a(n)(u · ξ) p′(n)ξ p′(n)ξ⊤ n(u · ξ)I −Ωξ Ωξ . Here, a(n) := p′(n)/n is the enthalpy function and ΩξE⊤ = (ξ × E)⊤. Clearly, (35) is a symmetric hyperbolic system, since Aj(w) = (Aj)(w)⊤(j = 0, 1, 2, 3), A0 > 0 and the dissipative matrix L(w) ≥ 0, however, L(w) is not symmetric.
Jiang Xu hyperbolic dissipative systems
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References Examples Frequency-localization time-decay inequality An open problem
1
Hyperbolic systems with symmetric dissipation Introduction New decay framework L2(Rn) ∩ ˙ B−s
2,∞(Rn)
Littlewood-Paley pointwise estimates Nonlinear applications
2
Hyperbolic systems with Non-symmetric dissipation Examples Frequency-localization time-decay inequality An open problem
3
References
Jiang Xu hyperbolic dissipative systems
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References Examples Frequency-localization time-decay inequality An open problem
It was shown that the non-symmetric dissipation affected the Timoshenko and Euler-Maxwell systems such that the weak dissipative mechanism of REGULARITY-LOSS was present. More precisely, their dissipative structure satisfies Reλ(iξ) ≤ −cη2(ξ) with η2(ξ) = |ξ|2 (1 + |ξ|2)2 for c > 0, which leads the decay property for the linearized solution zL: zLL2 (1 + t)−3/4z0L1 + (1 + t)−ℓ/2∂ℓ
xz0L2,
(36) where ℓ is a non-negative integer. [Ide & Kawashima (08’), Duan (11’), Ueda & Kawashima (11’)].
Jiang Xu hyperbolic dissipative systems
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References Examples Frequency-localization time-decay inequality An open problem
In this case, higher regularity than that for the global-in-time existence is usually assumed to obtain optimal decay rates. To
localization time-decay property to obtain the minimal decay regularity, which means the extra regularity is not necessary.
Jiang Xu hyperbolic dissipative systems
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References Examples Frequency-localization time-decay inequality An open problem
[X.-Mori-Kawashima (2015)]
Let η(ξ) be a positive, continuous and real-valued function in Rn satisfying η(ξ) ∼ |ξ|σ1, |ξ| → 0; |ξ|−σ2, |ξ| → ∞; for σ1, σ2 > 0. If f ∈ ˙ Bs+ℓ
r,α (Rn) ∩ ˙
B−̺
2,∞(Rn) for s ∈ R, ̺ ∈ R and 1 ≤ α ≤ ∞
such that s + ̺ > 0, then it holds that
Jiang Xu hyperbolic dissipative systems
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References Examples Frequency-localization time-decay inequality An open problem
˙ ∆qfe−η(ξ)tL2
q
− s+̺
σ1 f ˙
B−̺
2,∞
+ (1 + t)
− ℓ
σ2 +γσ2(r,2)f ˙
Bs+ℓ
r,α
, (37) for ℓ > n( 1
r − 1 2) a with 1 ≤ r ≤ 2, where γσ(r, p) := n σ( 1 r − 1 p).
aLet us remark that ℓ ≥ 0 in the case of r = 2. Jiang Xu hyperbolic dissipative systems
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References Examples Frequency-localization time-decay inequality An open problem
In Theorem 2.1, two points need to be noticed:
The low-frequency regularity is less restrictive than the usual Lp space, due to the embedding Lp(Rn) ֒ → ˙ B−̺
2,∞(Rn)(̺ = n(1/p − 1/2), 1 ≤ p < 2);
For the high-frequency part, it decays in time not only with algebraic rates of any order as long as the function is spatially regular enough, but also additional information related to the localized integrability is available.
Jiang Xu hyperbolic dissipative systems
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References Examples Frequency-localization time-decay inequality An open problem
Based on the decay inequality (37) in Theorem 2.1, we prove that ( [X.-Mori-Kawashima (JDE, 2015)]) The classical solution U(t, x) of Timoshenko system admits UL2 I0(1 + t)− 1
4 ,
(38) where I0 := U0B3/2
2,1 (R)∩ ˙
B−1/2
2,∞ (R) is sufficiently small; Jiang Xu hyperbolic dissipative systems
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References Examples Frequency-localization time-decay inequality An open problem
For the Euler-Maxwell system, we prove that ( [X.-Kawashima (2015)]) The classical solution w(t, x) admits w − w∞L2 I1(1 + t)− 3
4 ,
(39) where I1 := w0 − w∞B5/2
2,1 (R3)∩ ˙
B−3/2
2,∞ (R3) is sufficiently small. Jiang Xu hyperbolic dissipative systems
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References Examples Frequency-localization time-decay inequality An open problem
1
Hyperbolic systems with symmetric dissipation Introduction New decay framework L2(Rn) ∩ ˙ B−s
2,∞(Rn)
Littlewood-Paley pointwise estimates Nonlinear applications
2
Hyperbolic systems with Non-symmetric dissipation Examples Frequency-localization time-decay inequality An open problem
3
References
Jiang Xu hyperbolic dissipative systems
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References Examples Frequency-localization time-decay inequality An open problem
Inspired by the recent efforts on Timoshenko and Euler-Maxwell systems, we consider the following general form A0∂tw + n
j=1 Aj∂xjw + Lw = 0,
w(0, x) = w0, (40) for (t, x) ∈ [0, +∞) × Rn, where Aj = (Aj)⊤(j = 0, · · ·, n), A0 > 0 and L ≥ 0 but it is non-symmetric. Up to now, there is only two decay results for (40): L2-framework, [Ueda, Duan & Kawashima (ARMA, 2012)]; Lp-framework (2 ≤ p ≤ ∞), [X.-Mori-Kawashima (JMPA, 2015)]. However, the corresponding nonlinear application is left OPEN!
Jiang Xu hyperbolic dissipative systems
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References Examples Frequency-localization time-decay inequality An open problem
Inspired by the recent efforts on Timoshenko and Euler-Maxwell systems, we consider the following general form A0∂tw + n
j=1 Aj∂xjw + Lw = 0,
w(0, x) = w0, (40) for (t, x) ∈ [0, +∞) × Rn, where Aj = (Aj)⊤(j = 0, · · ·, n), A0 > 0 and L ≥ 0 but it is non-symmetric. Up to now, there is only two decay results for (40): L2-framework, [Ueda, Duan & Kawashima (ARMA, 2012)]; Lp-framework (2 ≤ p ≤ ∞), [X.-Mori-Kawashima (JMPA, 2015)]. However, the corresponding nonlinear application is left OPEN!
Jiang Xu hyperbolic dissipative systems
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References
hyperbolic system of balance laws, Arch. Rational Mech. Anal. 211 (2014) 513–553.
Besov spaces for generally dissipative systems, Arch. Rational Mech. Anal. 218 (2015) 275–315.
regularity for compressible Euler-Maxwell equations, J. Math. Pures Appl., (2015), in press.
the Timoshenko system: The case of equal wave speeds, J. Differential Equations 258 (2015) 1494–1518.
for the Timoshenko system: The case of non-equal wave speeds, J. Differential Equations (2015), in press.
Jiang Xu hyperbolic dissipative systems
Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References
Jiang Xu hyperbolic dissipative systems