Global Smooth Solutions in R 3 to Short Wave-Long Wave Interactions Systems for Viscous Compressible Fluids

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1 Global Smooth Solutions in R 3 to Short Wave-Long Wave Interactions Systems for Viscous Compressible Fluids Instituto de Matemática Pura e Aplicada - IMPA Rio de Janeiro, RJ, , Brazil joint with Ronghua Pan & Weizhe Zhang, School of Mathematics, Georgia Tech Junxiong Jia, Xi an Jiaotong University Taiwan, December 21, 2014

2 Abstract The short wave-long wave interactions for viscous compressible heat conductive fluids is modeled, following Dias & Frid (2011), by a Benney-type system coupling Navier-Stokes equations with a nonlinear Schrödinger equation along particle paths. We study the global existence of smooth solutions to the Cauchy problem in R 3 when the initial data are small smooth perturbations of an equilibrium state. This is a joint work with Roghua Pan and Weizhe Zhang. Recently, this analysis has been extended to the magnetohydrodynamics equations, in a joint work with Ronghua Pan and Junxiong Xia, currently in preparation.

3 The model The system describing short wave-long wave interactions for compressible fluids with which we are concerned reads as follows ρ t + div (ρu) = 0, (ρu) t + div (ρu u) = µ u + ν div u p(ρ, θ) + α (g ( 1 ρ )h( w Y 2 )), θ t + u θ + θp θ c ϑ ρ div u = 1 (κ θ + Ψ), c ϑ ρ i w t + y w = w 2 w + αg(v)h ( w 2 ) w, (1) whose terms will be explained subsequently, and for which it is the main purpose of this paper to solve the Cauchy problem in R 3, so (t, x) [0, ) R 3, for prescribed initial data (ρ(0, x), u(0, x), θ(0, x), w(0, y)) = (ρ 0 (x), u 0 (x), θ 0 (x), w 0 (y)), (2) which are small and smooth perturbations of a constant state, say, ρ = ρ 0, u = 0, θ = θ 0, w = 0.

4 Navier-Stokes So let us recall the Navier-Stokes system for a heat-conductive gas ρ t + div (ρu) = 0, (ρu) t + div (ρu u) = µ u + ν div u p(ρ, θ) + ρf, θ t + u θ + θp θ c ϑ ρ div u = 1 (κ θ + Ψ), c ϑ ρ (3) where, as usual, ρ, u, θ, p(ρ, θ) are, respectively, density, velocity, temperature, pressure, F is the external force and we denote p θ = p θ. Further, µ is the first viscosity coefficient, µ = ν µ is the second viscosity coefficient, with µ > 0, ν > 0. The latter follows from the usual conditions in classical fluid dynamics µ > 0, µ + 2 µ 0. (4) 3

5 Navier-Stokes, cont. Also, κ > 0 is the heat conduction coefficient. For simplicity, we assume µ, ν, κ to be constant. Moreover, c ϑ is the specific heat at constant volume, which, in general, is a positive function of (ρ, θ). Finally, Ψ = µ u 2 + ν div u 2, where, as usual, for a d-vector V = (v 1,, v d ), V 2 = v v d 2, and for a d d-matrix A = (a ij ), i, j = 1,..., d, we denote d A 2 := aij. 2 Here we will always assume the space dimension d = 3. i,j=1 As usual we assume that the pressure function p(ρ, θ) satisfies p ρ (ρ, θ) > 0, p θ (ρ, θ) > 0. (5)

6 Schrödinger Equation We also recall the nonlinear Schrödinger equation describing the propagation of the short waves, referred to an observer with the group velocity. The latter is taken to be equal to the fluid velocity u, in accordance to Benney s general prescription in [Be], and the equation then reads i w t + y w = w 2 w + G w, (6) where w is the complex-valued wave function, G is the potential due to the interaction with the fluid, and y denotes the Lagrangian coordinate which we define precisely as follows.

7 The Lagrangian Transformation For (t, x) [0, ) R d, let Φ(t; x) be the solution of the initial value problem dφ (t; x) = u(t, Φ(t; x)), dt (7) Φ(0; x) = x. It is an easy exercise to deduce that the Jacobian J Φ (t; x) = det( Φ x (t; x)) of the transformation x Φ(t; x) satisfies dj Φ (t; x) = div u(t, Φ(t; x))j Φ (t; x), dt J Φ (0; x) = 1. We define the Lagrange transformation Y (t, z) = (t, y(t, z)) by the relation y(t, Φ(t; x)) = y 0 (x), (9) for some given (diffeomorphic) transformation y 0 : R d R d, and we choose xd y 0 (x) := (x 1,, x d 1, ρ(0, x 1,, x d 1, s) ds). (10) 0 (8)

8 The Lagrangian Transformation, cont. From the relations (8), (9), (10), J y (t) := det( y z (t, Φ(t; x))) satisfies dj y (t) = div u(t, Φ(t; x))j y (t), dt (11) J y (0) = ρ(0, x). d ρ(t, Φ(t; x)) dt J y (t) = (ρ t(t, Φ(t; x)) + u(t, Φ(t; x)) ρ(t, Φ(t; x)))j y (t) J y (t)ρ(t, Φ(t; x)) J 2 y (t) = div u(t, Φ(t; x))ρ(t, Φ(t; x))j y (t) + div u(t, Φ(t; x))j y (t)ρ(t, Φ(t; x)) J 2 y (t) = 0, ( y det z i.e., det ) (t, Φ(t, x)) ( y (t, z) z = J y (t) = ρ(t, Φ(t, x)), ) = ρ(t, z), for all (t, z) [0, ) R d. (12) (13)

9 Justification of the model As in [DF], when the only force acting on the fluid is the one due to the interaction with the short wave, we postulate that F and G have the form F = α ρ (g ( 1 ρ )h( w Y 2 )), G = αg(v(y, t))h ( w 2 ), (14) where α, α are positive constants, Y (t, x) = (t, y(t, x)) is the Lagrangian transformation as above, v(y, t) is the specific volume defined by the relation 1 v(t, y(t, x)) = (15) ρ(t, x) and g, h : [0, ) [0, ) are nonnegative smooth functions with h(0) = h (0) = 0. As explained in [DF], the formulas for F and G, in (14), extend the simplest forms for the coupling terms which would be ρf = x ( w 2 ) and G(t, y) = v(t, y), which resembles the coupling for the Burgers equation proposed in [Be]. The introduction of the functions g(v) and h( w 2 ) serves to guarantee the invariance of the physical domain ρ > 0 and to cut off, if necessary, the coupling for large values of v, ρ, w 2, w 2. The specific forms of the interaction force F and interaction potential G is then imposed for the validity of the energy identity, again in accordance with the general prescription in [Be].

10 The Energy Equation Differential form of the energy equation: ( (ρ(e + u u ))t + div (ρu(e + div = α α e(ρ, θ) is the internal energy, which satisfies c ϑ = e )) + div (pu κ x θ) 2 ( ( µu u + ν(div u)u + α ug ( 1 ))) ρ )h( w Y 2 ) dx ( ( div y ( w t y w + w t y w) αg(v(t, y))h( w(t, y) 2 ) e y w ) ) 2 w(t, y) 4 dy, (16) t θ, by definition of c ϑ, and the Maxwell s relation ρ = 1 ρ 2 (p(ρ, θ) θp θ), (17) which follows from the Second Law of Thermodynamics. The latter may be expressed synthetically by the well known Gibbs relation de = θds pdv, (18) where S is the entropy and v = 1/ρ is the specific volume.

11 The Energy Equation, cont. where S 0 = S( ρ 0, θ 0 ). (ρ(s S 0 )) t + div (ρu(s S 0 )) + κ log θ = Ψ θ + κ θ 2 θ 2, (19) e (ρ, S) := e(ρ, S) e( ρ 0, S 0 ) ρ 0 ρ eρ( ρ 0, S 0 )(ρ ρ 0 ) e S ( ρ 0, S 0 )(S S 0 ). Thus, denoting c S = e S ( ρ 0, S 0 ), using (16) and (19), ( (ρ(e + u u ))t + div (ρu(e + )) + div (pu κ x θ) 2 ( ( div ρ 20 u + µu u + ν(div u)u + α ug ( 1 )) ρ )h( w Y 2 ) = α α Ψ c S κ log θ + c S θ + c S κ θ ) 2 θ 2 dx ( ( div y ( w t y w + w t y w) αg(v(t, y))h( w(t, y) 2 ) y w ) ) 2 w(t, y) 4 dy, (20) t By Gibbs relation (18), we see that e S = θ > 0. As usual, we impose that e, as a function of (v, S), with v = 1/ρ, satisfies 2 e v 2 2 e v S 2 e 2 > 0, (21) e v S S 2 in the sense of quadratic forms.

12 The Energy Equation in Integral Form Integrating (20) over R 3, we obtain d dt + d dt ) (ρ(e (ρ, S) + u 2 2 ) dx + c S ( α 1 R 3 α 2 y w 2 R 3 R 3 ( ) Ψ θ + κ θ 2 θ 2 dx + 1 ) 2 w(t, y) 4 + αg(v(t, y))h( w(t, y) 2 ) dy = 0. (22) d w(t, y) 2 dy = 0, (23) dt R 3

13 Main Theorem Theorem Assume that for some positive constants ρ 0, θ 0, (ρ 0 ρ 0, u 0, θ 0 θ 0, w 0 ) H 3 (R 3 ) L 1 (R 3 ). (24) There exists 0 < ɛ < 1 such that if δ 0 := (ρ 0 ρ 0, u 0, θ 0 θ 0, w 0 ) H 3+ (ρ 0 ρ 0, u 0, θ 0 θ 0, w 0 ) L 1 < ɛ, (25) then there exists a unique global smooth solution (ρ, u, θ, w) to the Cauchy problem (1),(2), (ρ ρ 0 ) C([0, ), H 3 (R 3 )) C 1 ([0, ), H 2 (R 3 )), (u, θ θ 0, w) C([0, ), H 3 (R 3 )) C 1 ([0, ), H 1 (R 3 )), such that, for some C 0 = C 0 (ε) > 0, it holds that (ρ ρ 0, u, θ θ 0, w) C([0, ),H 3 ) C 0 δ 0. (26)

14 Main Theorem, cont. Moreover, we have the following decay estimates w(t) Cδ 0 (1 + t) 3 2, (ρ ρ 0, u, θ θ 0 ) L 2 Cδ 0 (1 + t) 3 4, (27) x (ρ ρ 0, u, θ θ 0 ) H 2 Cδ 0 (1 + t) 5 4.

15 Relevant References Be Ca D.J. Benney. A general theory for interactions between short and long waves. Studies in Applied Mathematics 56 (1977), T. Cazenave. Semilinear Schrödinger Equations. Courant Lecture Notes 10, American Mathematical Society, Da C. Dafermos. Hyperbolic Conservation Laws in Continuum Physics. Third Edition. Springer, DFF DF DLUY DUYZ FPZ KS MN J. P. Dias, M. Figueira, and H. Frid. Vanishing viscosity with short wave long wave interactions for systems of conservation laws, Archive for Rational Mechanics and Analysis, 196 (2010), no. 3, J. P. Dias and H. Frid. Short wave-long wave interactions for compressible Navier-Stokes equations, SIAM Journal on Mathematical Analysis, 43 (2011), no. 2, R. Duan, H. Liu, S. Ukai, and T. Yang. Optimal L p -L q convergence rates for the compressible Navier-Stokes equations with potential force, Journal of Differential Equations, 238 (2007), no. 1, R. Duan, S. Ukai, T. Yang, H. Zhao. Optimal convergence rates for the compressible Navier-Stokes equations with potential forces. Mathematical Models and Methods in Applied Sciences 17 (2007), No. 5, Frid, H.; Pan, R.; Zhang, W. Global smooth solutions in R3 to short wave-long wave interactions systems for viscous compressible fluids. SIAM J. Math. Anal. 46 (2014), no. 3, T. Kobayashi, Y. Shibata. Decay estimates of of solutions for the equations of motion of compressible viscous and heat conductive gases in an exterior domain in R 3. Comm. Math. Phys. 200 (1999), A. Matsumura and T. Nishida. The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids. Proc. Japan Acad. 55, Ser. A (1979),

16 Deformation Gradient Estimate Denote E(t) = F(t) F 0, where F = y x, F 0 = y y 1 0 is the initial deformation gradient, and let E i,j be the (i, j) entry of E. The following estimate, with k = 2, plays a central role in this paper. Lemma For some C = C(ε) > 0, and for k = 0, 1, 2,, the matrix E satisfies E(t) 2 H e C( t t k 0 u(τ) H dτ) k u(τ) H k dτ. (28) 0

17 Estimate for the Schrödinger Equation We recall that S 5 (t) = e it in L 2 (R 3 ) associated with the homogeneous Schrödinger equation. Lemma If p [2, ] and t 0, then S 5 (t) maps L p (R N ) continuously to L p (R N ) and S 5 (t)ϕ Lp (R N ) (4πt) N( p ) ϕ L p, for all ϕ Lp (R N ). (29) In what follows, we will assume that A(T ) := sup (1 + t) 5/4 u(t) H 2 σ 0, (30) 0 t T where 0 < σ 0 < 1 is suitably chosen in view of Lemma 2. The justification for the assumption (30) is obtained as a result of our decay estimates for the Navier-Stokes equations, and is one of the key points in the proof of Theorem 1.

18 Estimate for the Schrödinger Equation, cont.1 Lemma Let (ρ(t), u(t), θ(t), w(t)) X 3 (0, T ; ε) X 3 w (0, T ; ε) be a local solution of (1),(2). There exists C 0 = C 0 (ε) > 0 such that, for all t [0, T ], where the solution is defined and (ρ, u, θ, w) C([0, T ]; H 3 (R 3 )), w(t) C 0δ 0, (31) (1 + t) 3/2 w(t) H 3 C 0 w 0 H 3. (32)

19 Estimate for the Schrödinger Equation, cont. 2 By Duhamel s principle, we have t w(t) = S 5 (t)w 0 + S 5 (t s)( w 2 w(s) + g(v)h ( w 2 )w(s)) ds. (33) 0 Therefore, using Lemma 3, in (33) with p =, combined with the Sobolev embedding H 2 (R 3 ) L (R 3 ), and S 5 (t)ϕ H k = ϕ H k, (34) for any k N {0} (cf., e.g., [Ca]) we obtain, for t > 0, w 0 w(s) C min{ w 0 H 2, L 1 (1 + s) 3/2 } s + C min{ w(τ) 2 H 0 2, δ 0 C + CεMw (s) (1 + s) 3/2 0 w(τ) 2 δ 0 Mw (s) C + Cε (1 + s) 3/2 (1 + s) 3/2, } w(τ) dτ, (1 + s τ) 3/2 s 1 1 (1 + τ) 3/2 dτ (1 + s τ) 3/2 (35) we use that h is smooth, h (0) = 0, M w (s) := sup (1 + τ) 3/2 w(τ), 0 τ s M w (t) Cδ 0, which gives (31). Now, taking H 3 in (33), using (31), (34), t C w H 3 w 0 H (1 + s) 3/2 w(s) H 3 ds,

20 Estimates for the Navier-Stokes Equations with ρ ρ ρ 0, u ρ0 P 1 u, θ P2 ρ 0 P 1 P 3 (θ θ 0 ), P 1 = p ρ( ρ 0, θ 0 ) ρ 0, P 2 = p θ( ρ 0, θ 0 ) ρ 0, P 3 = θ 0 p θ ( ρ 0, θ 0 ) c ϑ ρ 0, the linear part of the Navier-Stokes system becomes ρ t + γdiv u = 0, u t µ 1 u µ 2 div u + γ ρ + λ θ = 0, θ t κ θ + λdiv u = 0, (36) where µ 1 = µ ρ, 0 µ 2 = ν ρ, 0 γ = P 1 ρ 0, λ = P 1 P 2, κ = κ c ϑ P 2 P 1 P 3 ρ 0.

21 Estimates for the Navier-Stokes Equations, cont.1 In vector notation, (36) reads U t + A U = 0, (37) where U = (ρ, u, θ) and 0 γdiv 0 A := γ µ 1 µ 2 div λ. 0 λdiv κ The operator A is closed and accretive in, say, D( A) = {(ρ, u, θ) L 2 (R 3 ) : ρ H 1 (R 3 ), (u, θ) H 2 (R 3 )}, with Ran(λ 0 A) = L 2 (R 3 ), for any λ 0 > 0, and so it generates a contraction semigroup E(t) := e t A, t 0.

22 Estimates for the Navier-Stokes Equations, cont.2 Lemma (KS) For t 0, k = 0, 1, 2,, U 0 H k (R 3 ) L 1 (R 3 ), it holds k E(t)U 0 C(k)(1 + t) 3 4 k 2 ( U0 L 1 + U 0 k ). (38) Using the above change of variables, Navier-Stokes part of the system (1) may be written as ρ t + γdiv u = F 1, u t µ 1 u µ 2 div u + γ ρ + λ θ = F 2 + G, (39) θ t κ θ + λdiv u = F 3,

23 Estimates for the Navier-Stokes Equations, cont.3 where F 1, F 2, F 3 are the same as in [DUYZ], having the following equivalence property F 1 ( i ρ)u i + ρ i u i F j 2 ui i u j + ρ i i u j + ρ j i u i + ρ j ρ + θ j ρ + ρ j θ + θ j θ, F 3 u i i θ + ρ i i θ + ρ i u i + θ i u i + ρψ(u) + Ψ(u), and so G = α ( g 1 ( )h( w 2 ) ). ρ + ρ 0 ρ + ρ 0 G j ρ( j ρ) w 4 + ( j ρ) w 4 + ρ w 2 j ( w 2 ). (40)

24 Estimates for the Navier-Stokes Equations, cont.4 Lemma Let V = (U, w) C([0, T ]; H 3 (R 3 )), with U = (ρ, u, θ), be the local solution of (1),(2), and assume (30). Under the assumptions of Theorem 1, we have { U(t) Cδ 0 (1 + t) Cε t 0 (1 + t s) 3 4 U(s) 2 ds, U(t) Cδ 0 (1 + t) Cε t 0 (1 + t s) 5 4 U(s) 2 ds. (41)

25 Estimates for the Navier-Stokes Equations, cont.4 Lemma Let V = (U, w) C([0, T ]; H 3 (R 3 )), with U = (ρ, u, θ), be the local solution of (1),(2), and assume (30). Under the assumptions of Theorem 1, we have { U(t) Cδ 0 (1 + t) Cε t 0 (1 + t s) 3 4 U(s) 2 ds, U(t) Cδ 0 (1 + t) Cε t 0 (1 + t s) 5 4 U(s) 2 ds. (41) Lemma Under the same hypotheses of Lemma 6, if ε > 0 is small, then dh(t) +D 2 ( 2 ρ (u, θ) 2 dt 2) Cε U(t) 2 +Cε w(t) 2, (42) where H(t) is equivalent to U(t) 2 2, that is, there exists a positive constant C 2 such that 1 C 2 U(t) 2 2 H(t) C 2 U(t) 2 2, t 0. (43)

26 Conclusion of the Proof of Theorem 1 To conclude the proof of Theorem 1, it remains to prove the decay estimates (27), which imply the global existence and establish, in particular, (30). For this part we follow the arguments in [DLUY]. So, we define M(t) = sup (1 + s) 5 2 H(s). (44) 0 s t Observe that U(s) 2 C H(s) C(1 + s) 5 4 M(t), 0 s t. It follows from (41) that t U(t) Cδ 0 (1 + t) Cε M(t) (1 + t s) 5 4 (1 + s) 5 4 ds C(1 + t) 5 4 (δ0 + ε M(t)), 0 (45)

27 Conclusion of the Proof of Theorem 1, cont.1 Now, from (42), we have dh(t) dt + D 2 2 U(t) 2 1 Cε U(t) 2 + Cε w 2, so, adding D 2 U(t) 2 to both sides, we get dh(t) dt with D 2 = D2 C 2. Therefore, + D 2H(t) C U(t) 2 + Cε w 2, (46) H(t) H(0)e D 2 t + C t 0 e D 2 (t s) U(s) 2 ds t + Cε e D 2 (t s) w(s) 2 ds. 0 (47) Now, using (45) and the decay estimate for w (under the assumption (30), to be justified) into (47), we get

28 Conclusion of the Proof of Theorem 1, cont.2 t H(t) H(0)e D 2 t + C(δ0 2 + ε 2 M(t)) + Cεδ 2 0 t 0 e D 2 (t s) (1 + s) 3 ds 0 e D 2 (t s) (1 + s) 5 2 ds H(0)e D 2 t + C(1 + t) 5 2 (δ ε 2 M(t)) + Cεδ 2 0(1 + t) 3, (48) and, hence, M(t) C(H(0) + δ 2 0) + Cε 2 M(t), which, for ε > 0 sufficiently small, implies and so we get the decay estimate M(t) Cδ 2 0, (49) U(t) 2 Cδ 0 (1 + t) 5 4, (50) which, in particular, justifies the assumption (30), if δ 0 > 0 is sufficiently small.

29 Conclusion of the Proof of Theorem 1, cont.3 Finally, applying (50) in the first inequality in (41), we obtain the last decay estimate U(t) Cδ 0 (1 + t) 3 4. (51) Clearly, (50) and (51) imply the global existence as long as we take δ 0 > 0 sufficiently small, and the proof of Theorem 1 is complete.

30 Magnetohydrodynamics ρ t + div(ρu) = 0, (ρu) t + div(ρu u) = µ u + (λ + µ) divu P(ρ) + H H 1 2 ( H 2 ) + ρf, H t (u H) = (ν H), divh = 0, (52)

31 Ionosphere The D region is the lowest in altitude, though it absorbs the most energetic radiation, hard x-rays. The D region doesn t have a definite starting and stopping point, but includes the ionization that occurs below about 90km. The E region peaks at about 105km. It absorbs soft x-rays. The F region starts around 105km and has a maximum around 600km. It is the highest of all of the regions. Extreme ultra-violet radiation (EUV) is absorbed there. On a more practical note, the D and E regions reflect AM radio waves back to Earth. Radio waves with shorter lengths are reflected by the F region. Visible light, television and FM wavelengths are all too short to be reflected by the ionosphere. So your t.v. stations are made possible by satellite transmissions.

32 Auroras The aurora is formed when protons and electrons from the Sun travel along the Earth s magnetic field lines. These particles from the Sun are very energetic. We are talking major-league energy, much more than the power of lightning: 20 million amps at 50,000 volts is channeled into the auroral oval. It s no wonder that the gases of the atmosphere light up like the gases of a streetlamp! The aurora is also known as the northern and southern lights.

33 Aurora Ovals From the ground, they can usually be seen where the northern and southern auroral ovals are on the Earth. The northern polar auroral oval usually spans Fairbanks, Alaska, Oslo, Norway, and the Northwest Territories. Sometimes, when the Sun is active, the northern auroral oval expands and the aurora can be seen much farther south. The lights of the aurora come in different colors. Oxygen atoms give off green light and sometimes red. Nitrogen molecules glow red, blue, and purple.

34 The Van Allen Radiation Belt

35 THE END

36 THE END THANKS!!!