Propagation of Singularities

Transcription

1 Title: Name: Affil./Addr.: Propagation of Singularities Ya-Guang Wang Department of Mathematics, Shanghai Jiao Tong University Shanghai, , P. R. China; Propagation of Singularities Motivations The system of classical thermoelasticity is a hyperbolic-parabolic coupled system describing the elastic and the thermal behavior of an elastic medium ([4, 16, 25]). It is well known that with respect to the decay of solutions as time tends to infinity, and also with respect to the existence of global smooth solutions for small data in one space variable, or of the potential part of displacement and temperature in multidimensional thermoelasticity, the system behaves like a purely parabolic one, see for example [15, 16, 17, 25] and references cited there. As in nonlinear wave equations, the smooth solutions to the nonlinear thermoelasticity shall blowup in a finite time in general, cf. [11, 14, 22]. In [21], it was observed that solutions to the linear problem do not have a smoothing effect, like in a parabolic problem. Moreover, in [40], the author proved that for the linear equations of thermoelasticity, even when the initial displacement and velocity are smooth, the nonsmooth initial temperature shall produce singularities for the elastic wave when the time t > 0. In more detail, consider the following Cauchy problem for the homogeneous linear thermoelastic system in one space variable: u tt α 2 u xx + γθ x = 0 θ t β 2 θ xx + γu tx = 0, (1) u(0, x) = u t (0, x) = 0, θ(0, x) = θ 0 (x)

2 with α, β and γ being positive constants for simplicity. If θ 0 lies in H s (IR) but not in H s+1 (IR) (the classical Solobev spaces) for a fixed s IR, then for any fixed T > 0, we 2 have: u C([0, T ], H s+2 (IR)) C 1 ([0, T ], H s+1 (IR)). (2) u / L 2 (0, T ; H s+3 (IR)) H 1 (0, T ; H s+2 (IR)) Therefore, it is very interesting and important to describe how the hyperbolic and parabolic operators in thermoelasticity influence the regularities of states, and to study how singularities of solutions are propagated. This problem was first studied in [26, 33, 34] for the propagation of weak singularities by using the Fourier analysis. Later on, a microlocally decoupling argument was successfully found in [38, 39] for the hyperbolic-parabolic coupled equations, from which one can obtain a complete picture on the propagation of weak singularities in thermoelasticity. Propagation of Weak Singularities First, let us present the general argument developed in [38, 39] to decouple the thermoelastic equations of hyperbolic-parabolic coupled type microlocally. Consider the following semilinear system of thermoelasticity in one space variable x IR: u tt α 2 (t, x)u xx + γ 1 (t, x)θ x = f(u, θ) (3) θ t β 2 (t, x)θ xx + γ 2 (t, x)u tx = g(u, θ) in which u represents the displacement, and θ = T a T 0 the temperature difference, with T a and T 0 being the absolute temperature and reference temperature respectively, α α 0 > 0 and β β 0 > 0 for two positive constants α 0, β 0. Denote by Ψ1,0(IR) k the set of pseudodifferential operators of order k with symbols in S k 1,0(IR) ([13, 7, 37]) and t being a parameter. Denote by Λ = (1 2 x) 1/2 the classical positive, self-adjoint operator with the symbol < ξ >= (1 + ξ 2 ) 1 2. Setting u + = ( t + iαλ)u and u = ( t iαλ)u, from (3) we know that U = (u +, u, θ) t satisfies the following system:

3 t U + A 2 (t, x, D x )U + A 1 (t, x, D x )U + A 0 (t, x, D x )U = F (U) (4) 3 where A 2 (t, x, D x ) = diag[0, 0, β 2 Λ 2 ] Ψ 2 1,0(IR), iαλ 0 γ 1 x A 1 (t, x, D x ) = 0 iαλ γ 1 x Ψ1,0(IR) 1 γ 2 x 2 A 0 (t, x, D x ) Ψ 0 1,0(IR) and F (U) = (f, f, g) t. γ 2 x 2 0 As in [36, 38, 39], by using the diagonal form of the highest order operator A 2 (t, x, D x ) given in (4), for any m IN, we can construct K m (t, x, D x ) Ψ 1 1,0 (IR), smooth in t, with the principal symbol being K 1 (t, x, ξ) = iξ β 2 < ξ > γ γ 1 γ 2 γ S 1 1,0(IR) (5) such that V = (I + K m (t, x, D x ))(u +, u, θ) t satisfies the following decoupled system modulo Ψ m 1,0 : t V + A 2 V + A 1 (t, x, D x )V + A 0 (t, x, D x )V + R m (t, x, D x )V = F(V ) (6) where A 1 (t, x, D x ) = diag[ iαλ, iαλ, 0] Ψ 1 1,0(IR), A 0 (t, x, D x ) = diag[a (0) 11, A (0) 22 ] Ψ 0 1,0(IR) with A (0) 11 being a 2 2 matrix and A (0) 22 being a scalar, R m Ψ m 1,0 and F(V ) = (I + K m )F (P (1) 1 V 1, P (2) 1 V 2, P 0 V 3 ), with P (k) 1 (t, x, D x ) Ψ 1 1,0 (IR) (k = 1, 2) and P 0 (t, x, D x ) Ψ 0 1,0(IR) depending smoothly on t. From (5) and (6), it is easy to see that V 1 = ( t α x )u γ 1 β 2 Λ 2 x θ + Ψ1,0 0 u + Ψ1,0 2 θ, and V 2 = ( t + α x )u γ 1 β 2 Λ 2 x θ + Ψ 0 1,0 u + Ψ 2 1,0 θ V 3 = θ + γ 2 β 2 Λ 2 2 xtu + Ψ 1 1,0 u + Ψ 1 1,0 θ mainly satisfy the equations of hyperbolic part and parabolic part respectively in (3) with coupled terms being lower order.

4 By applying the classical theory of hyperbolic equations and parabolic equations in the weakly coupled equations (6), we can obtain (refer to [38, 39, 40] for the detail): 4 Theorem 1. Consider the thermoelastic equations (3) with initial data u t=0 = u 0 (x), u t t=0 = u 1 (x), θ t=0 = θ 0 (x), x IR. (7) (1) When f = g = 0 in (3), if for a fixed s IR and a closed subset ω IR, (u 0, θ 0 ) H s (IR) C (IR ω), u 1 H s 1 (IR) C (IR ω), with H s (IR) being the classical Sobolev space based on L 2 (IR), then, we have (u, θ) C (((0, + ) IR) S) (8) with S being the union of characteristic curves of the operator 2 t α 2 (t, x) 2 x issuing from (0, x 0 ) for any x 0 ω. (2) For the semilinear equations (3), if (u 0, θ 0 ) H s (IR) C (IR 0) and u 1 H s 1 (IR) C (IR 0) for a fixed s > 3/2, then the local solution (u, θ) of (3)-(7) satisfies, (u, θ) C ((0, T ] ({x < x 1 (t)} {x > x 2 (t)})) u C((0, T ], H 2s 1/2 loc ({x 1 (t) < x < x 2 (t)})) C 1 ((0, T ], H 2s 3/2 loc ({x 1 (t) < x < x 2 (t)})), (9) θ C((0, T ], H 2s 1/2 loc ({x 1 (t) < x < x 2 (t)})) where x = x 1 (t) and x = x 2 (t) are two characteristic curves of t 2 α 2 (t, x) x 2 with x 1 (0) = x 2 (0) = 0 and x 1 (t) < x 2 (t) for t > 0. Remark 1. (1) The restriction of the regularity order 2s 1 2 given in (9) mainly results from the hyperbolic theory (cf. [1, 2, 8, 30, 31]). (2) Consider the following semilinear problem of thermoelasticity in three space variables x = (x 1, x 2, x 3 ) IR 3 u tt (2μ + λ) divu + μrot(rotu) + γ 1 θ = f(u, u t, u, θ) θ t β 2 Δθ + γ 2 divu t = g(u, u t, u, θ), (10) u t=0 = u 0 (x), u t t=0 = u 1 (x), θ t=0 = θ 0 (x)

5 where all coefficients (μ, λ, γ 1, γ 2, β) are smooth functions of (t, x) with μ, λ + μ and β being positive. By taking the decomposition u = u p + u s, with u p being the potential part, rot u p = 0, and u s the solenoidal part, div u s = 0, it is easy to know that u s satisfies a wave equation with speed μ, and (u p, θ) satisfy a hyperbolic-parabolic coupled system similar to the thermoelastic equations (3) in one space variable case, with the wave speed 2μ + λ. Let {t = g 1 (x)} and {t = g 2 (x)} be the forward characteristic cones issuing from the origin for the operators 2 t (2μ+λ)Δ and 2 t μδ respectively, and denote by C + = {t > g 1 (x)} {t = g 2 (x)}. If (u 0, θ 0 ) H s (IR 3 ) C (IR 3 {0}) and u 1 H s 1 (IR 3 ) C (IR 3 {0}) for s > 5, then the local solution (u, θ) to the 2 problem (10) satisfies (u, θ) C ((0, T ] {x : t < g 1 (x)}) u C((0, T ], H 2s 5 2 ε loc (C + )) C 1 ((0, T ], H 2s 7 2 ε loc (C + )) θ C([0, T ], H 2s 5 2 ε loc (C + )) for any ε > 0. These results can be obtained by using the same decoupling idea for the system of (u p, θ) as that given from (3) to (6). One can refer to [38, 39] for the detail. (3) By using the para-differential operators ([3, 37]) and the above decoupling argument, one can obtain a similar result on the propagation of weak singularities of solutions to quasilinear thermoelastic equations ([42]). The problem of weak singularities in viscous compressible flow had been studied in [9]. 5 The results in this section show that weak singularities in thermoelasticity are propagated mainly by the elastic fields, while the parabolic impact of heat conduction still exists. The above decoupling idea was also used to study other related problems in [10, 35, 41] etc..

6 Propagation of Discontinuities in Thermoelasticity 6 In previous section, by introducing a microlocally decoupling argument, we have studied the propagation of singularities in hyperbolic-parabolic coupled systems of thermoelasticity. However, all these results did not deal with the problem of strong singularities, such as discontinuities, because we have used the theory of pseudo-differential operators and para-differential operators in these works. Obviously, it is very interesting and important to study the strong singularity problems. It is not difficulty to see that the heat signal is propagated in the infinite speed in hyperbolic-parabolic coupled systems of thermoelasticity. To overcome this paradox in some cases, one approach is to use the Cattaneo law instead of the Fourier law for heat conduction, the equations of thermoelasticity become purely hyperbolic, in which the thermal disturbance is transmitted as a wave-like pulse with a finite speed ([5, 6]). This kind equations are so-called the thermoelastic system with second sound. The well-posedness and asymptotics of solutions to the thermoelastic equations with second sound had been studied in [29, 23, 24]. For the related mathematical results, see the review paper [25]. With respect to the Cauchy problem (3)-(7), we consider the following problem for the semilinear thermoelastic system with second sound in {t > 0, x IR}: u tt α 2 u xx + γ 1 θ x = f(u, θ) θ t + δq x + γ 2 u tx = g(u, θ) τq t + q + κθ x = 0 (11) u t=0 = u 0 (x), u t t=0 = u 1 (x), θ t=0 = θ 0 (x), q t=0 = q 0 (x) where u, θ and q denote the displacement, temperature and heat flux respectively, all coefficients in (11) are supposed to be constants for simplicity, with α, κδ, τ, γ 1 γ 2 being positive. Formally, the equations given in (11) can be regarded as a relaxation approx-

7 7 imation of the classical thermoelastic equations (3). As the thermoelastic equations (11) with second sound are purely hyperbolic, one can study the behavior of strong singularities in (11) as in [20, 32], our main idea is to derive the behavior of strong singularities (e.g. discontinuities) in the hyperbolic-parabolic coupled equations (3) of thermoelasticity by investigating the asymptotic behavior of strong singularities in the thermoelastic equations (11) as the relaxation parameter τ goes to zero. In the following discussion, we shall always assume that the initial data given in (11) satisfy u 0(x), u 1 (x), θ 0 (x), q 0 (x) piecewise smooth with a possible jump at {x = 0}. (H) Additionally, f and g are assumed to be smooth and globally Lipschitz in their arguments for simplicity (These assumptions are not optimal in general.). Denoting by u ± = ( t ± α x )u, and U = (U 1, U 2, U 3, U 4, U 5 ) (u, u +, u, θ, q) obviously, from (11) we obtain the following first order system for U: t U + B 1 x U + B 0 U = F (U) U0 (x), x < 0 U(0, x) = U 0 (x) U 0 + (x), x > 0 with U 0 (x) = (u 0, u 1 + αu 0, u 1 αu 0, θ 0, q 0 ), obvious matrices B 1 and B 0, and (12) F (U) := ( 0, f(u 1, U 4 ), f(u 1, U 4 ), g(u 1, U 4 ), 0 ). It is easy to see that the eigenvalues of B 1 are λ 1 = 0, λ 2,3 = α(1 δβ 2κγ τ) + O(τ 2 ), κγ λ 4,5 = τ + O( τ). (13) Denote by r k and l k the right and left eigenvectors of B 1 with respect to λ k for 1 k 5, i.e. (λ k Id B 1 )r k = l k (λ k Id B 1 ) = 0, with the normalization l j r k = δ jk.

8 8 Letting L and R be the matrices composed by left eigenvectors {l j } 5 j=1 and right eigenvectors {r j } 5 j=1 respectively, then V = LU satisfies the following problem with the principle part of the equations being diagonal: t V + Λ x V + B 0 V = F (V ) (14) V (0, x) = V 0 (x) where Λ = diag (λ 1, λ 2, λ 3, λ 4, λ 5 ), B0 = LB 0 R + L ( t R + B 1 x R), and F (V ) = L F (RV ). By Σ k = {(t, x) x λ k t = 0} we denote the characteristic lines for (14) issuing from the origin for 1 k 5, and by [V i ] Σk the jump of V i on Σ k, i.e. at (t, x ) with x = λ k t : [V i ] Σk (t, x ) := lim (t,x) (t,x ) x>λ k t If the solution V of (14) has a jump on Σ k V i (t, x) lim (t,x) (t,x ) x<λ k t V i (t, x). for some fixed 1 k 5, then for 1 i 5, from the differential equations (14), we know that ( t + λ i x )V i should be locally bounded everywhere. However, for any i = k, X i = t + λ i x is transversal to Σ k, we obtain ( t + λ i x ) V i Σk = const [V i ] Σk δ Σk where δ Σk is the Dirac measure supported on Σ k. The above contradiction implies that V i has no jump on Σ k if i = k. For a fixed 1 j 5, by noting that the vector field X j = t + λ j (t, x) x is tangential to Σ j, from (14) we know that the jump of V j on Σ j satisfies the following transport equation: ( t + λ j x )[V j ] Σj + b jj [V j ] Σj = [ F j (V )] Σj, (15) with b jj denoting the j th diagonal element of B 0. By the asymptotic properties of b jj in τ 0, and the inverse transform from V to U = (u, u t + αu x, u t αu x, θ, q), from (15) we can conclude

9 Theorem 2. Let (u, θ, q) be the solution to (11) with f, g being smooth and globally 9 Lipschitz in u and θ, and the initial data u 0, u 1, θ 0, q 0 be piecewise smooth with a possible jump at {x = 0}. Then, u(t, x) is continuous for all t > 0, and [θ] Σk 0 as τ 0 exponentially on Σ 4,5, of order O(τ) on Σ 2,3, [ (t,x) u] Σ4,5 0 exponentially, as τ 0 [u t + αu x ] Σ3 0 of order O(τ) [u t αu x ] Σ2 0 of order O(τ) and lim [u t + αu x ] Σ2 = [u 1 + αu τ 0 0] {0} e γ1γ2 2κδ t lim [u t αu x ] Σ3 = [u 1 αu τ 0 0] {0} e γ1γ2 2κδ t [q] Σ4,5 0 exponentially, as τ 0, is kept is kept. lim [q] Σ τ 0 2 = γ 2 [u 2δ 1 + αu 0] {0} e γ1γ 2 2κδ t lim [q] Σ τ 0 3 = γ 2 [u 2δ 1 αu 0] {0} e γ1γ 2 2κδ t is kept is kept. Remark 2. (1) From the above results, we see that the propagation of discontinuities in thermoelasticity is mainly dominated by the hyperbolic operators while the parabolic effect still exists as well. Especially, the temperature θ is smoothed out immediately as t > 0, and the heat flux q has the discontinuities on Σ 2,3 mainly arising from the discontinuities of the elastic waves, this gives information of θ x in the hyperbolicparabolic coupled equations of thermoelasticity (3). Furthermore, we see that the jumps on Σ 2,3 of t,x u and q decay more rapidly for small heat conduction coefficient κδ when t +, which is similar to a phenomenon observed by David Hoff [12] for the discontinuous solutions of the compressible Navier-Stokes equations. (2) The results in Theorem 2 was established in [27], in which the case that the nonlinear functions f, g depending on u t, u x had also been studied. This study was extended to the variable coefficient case in [19], in which we observed that the decay

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