GLOBAL EXISTENCE OF SMOOTH SOLUTIONS FOR A

Abstract. The bitemperature Euler model describes a crucial step of Inertial Confinement 4 Fusion (ICF) when the plasma is quasineutral while ionic and electronic temperatures remain distinct. 5 The model is written as a first-order hyperbolic system in non-conservative form with partially 6 dissipative source terms. We consider the polytropic case for both ions and electrons with different 7 γ-law pressures. The system does not fulfill the Shizuta-Kawashima condition and the physical 8 entropy, which is a strictly convex function, does not provide a symmetrizer of the system. In this 9 paper we exhibit a symmetrizer to apply the result on the local existence of smooth solutions in 10 several space dimensions. In the one-dimensional case we establish energy and dissipation estimates 11 leading to global existence for small perturbations of equilibrium states. 12

Inertial Confinement Fusion (ICF). It was derived from a kinetic model by using a 20 hydrodynamic limit and the Boltzmann entropy. For this kinetic model, a Discrete 21 Velocity Model (DVM) method with an asymptotic preserving discretization toward 22 Euler equations was obtained. The kinetic approach also allows to design numerical 23 schemes for the bitemperature Euler equations. See [1,5]. 24 We denote by ρ e and ρ i the electronic and ionic densities, ρ = ρ e + ρ i the total

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Quasineutrality is assumed, so that the ionization ratio Z = n e /n i is a constant. This 29 implies that the electronic and ionic mass fractions are constant and given by 30 (1.2) c e = Zm e m i + Zm e , c i = m i m i + Zm e . 31 We suppose that the ionic and electronic velocities are equal: u e = u i = u, and the 32 pressure of each species satisfies a gamma-law with its own γ exponent :

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(1.3) p e = (γ e − 1)ρ e ε e = n e k B T e , p i = (γ i − 1)ρ i ε i = n i k B T i , γ e > 1, γ i > 1, 34 where k B is the Boltzmann constant (k B > 0), ε α and T α represent respectively the 35 internal specific energy and the temperature of species α for α = e, i.
where "·" stands for the inner product in R d . This is a non-conservative hyperbolic 55 system which can be written in the synthetic form 56 (1.6) ∂ t W + Moreover, any smooth solution of the system satisfies the entropy equality 66 (1.9) ∂ t η(ρ, ρu, E e , E i ) + div φ(ρ, ρu, E e , which is a partially dissipative condition of the system. It is known that the second-68 order derivative of a strictly convex entropy provides a symmetrizer of a hyperbolic 69 system in conservative form (see [9,3] The last equation in (1.10) shows that the total energy is a conservative variable. If 86 γ e = γ i , we introduce a total internal specific energy ε by ε = c e ε e + c i ε i . Then In what follows, we consider the Cauchy problem for (1.5) near constant equilib-98 rium states in case γ e = γ i . Let us introduce An equilibrium stateV is a constant solution of (1.5). We consider in particular an 101 equilibrium state with zero velocity. Let with relations (1.1)-(1.4) and (1.8) and

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Now we write the system with variables (ρ, u, ε e , ε i ). We first remark that Then, for ρ > 0, the first two equations in (2.1) give By the definition of E α and the first two equations in (2.1) together with (2.3), we We also have These equalities imply that It follows that Finally, by the expression of p α and the last two equations in (2.1), we obtain which are the equations for ε e and ε i . Thus, system (2.1) is equivalent to 212 (2.5) where the superscript T denotes the transpose of a vector. Since p = ρε 1 and with u = (u 1 , · · · , u d ) T , I d being the unit matrix and (e 1 , · · · , e d ) being the standard 226 basis of R d .

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By a symmetrizer B 0 (V) for system (2.7) we mean that B 0 (V) is a symmetric 228 positive definite matrix such that B 0 (V)B j (W) is symmetric for all j ∈ {1, 2, · · · , d} 229 (see [15]). Now we introduce a diagonal matrix Obviously, B 0 (V) is symmetric positive definite in Ω. Moreover, which is a symmetric matrix. Therefore, B 0 (V) is a symmetrizer and system (2.7) is 234 symmetrizable hyperbolic in the sense of Friedrichs. According to Lax [14] or Kato

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[12] (see also Majda [15]), for smooth initial data, the Cauchy problem for (2.1) admits 236 a unique smooth solution, locally in time. This result is stated as follows and it holds 237 in any space dimension.
respectively. From (2.6) and (2.13), we further obtain which imply that

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(2.14) 259 By (2.6) and the expression of µ 2 above, we see that µ 1 and µ 2 can further be expressed (2.16) where M is a constant invertible matrix given by By the expression of B j given in (2.8), we can calculate the eigenvalues λ i and 265 the eigenvectors r i of (2.13). They are given by ρ(t, ξ)dξ, with X 1 (t) = u(t, X 1 (t)).

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It is clear that this change of variables is a diffeomorphism from R + × R to itself. For 286 simplicity, we use the same notation for unknown variables in Eulerian coordinates 287 (t, x) and in Lagrangian coordinates (t, y).

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Given a first-order partial differential equation By (2.18), in Lagrangian coordinates this equation is written equivalently as Applying this to (2.12), we obtain Similarly to (2.14), by (2.4), we obtain where µ 1 and µ 2 are given in (2.16).

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We consider a smooth solution U nearŪ, namely, U T is small. In the proof below, we 325 denote by C > 0 and c 0 > 0 generic constants independent of t and T .

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The global existence of smooth solutions to (2.24) and (2.25) will be proved in 327 the three steps shown in Introduction. where ν 1 = ν 1 (τ, p, v) given by 335 (3.1) . 336 It is clear that, for all τ > 0 and p > 0, ν 1 > 0 in a neighborhood of v = 0. We 337 introduce a new variable Hence, it is easy to check that (3.4) and (3.5) are satisfied.

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It follows from the definition of b α in (1.8) that .

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Using the fact that and δ(p, v) is defined in (3.9). By Lemma 3.3, in a neighborhood ofŪ, there are 417 positive constantsμ 1 ,μ 2 andμ 0 such that Then it is easy to check that, in a neighborhood ofŪ, A 0 (U ) is a symmetrizer of system 420 (3.8), namely, A 0 (U ) is symmetric positive definite and A 0 (U )A(U ) is symmetric. In 421 particular,

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Taking the inner product of (3.11) with A 0 (U )U k in L 2 (R), we obtain the Friedrichs 428 energy equality

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By the definition of A 0 andÃ, we have Next, a direct calculation yields Observe that each of three terms on the right-hand side of (3.15) is quadratic in 457 variables (u, p, v) with coefficients depending on derivatives of U −Ū up to order m.

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Moreover, using Lemma 3.3, we have δν ≥δν in a neighborhood ofŪ, whereν > 0 is 459 a constant. Thus, the Moser-type inequalities imply that where U 0 is the initial data of U . Finally, the change of variables U −→ U is a C ∞ -466 diffeomorphism in a neighborhood ofŪ . Then, U −Ū l is equivalent to U −Ū l for 467 all l ∈ N. Summing up this inequality for all k = 1, 2, · · · , m, and using Lemma 3.2, 468 we obtain (3.10).
. 476 We multiply the third equation in (3.18) by (γ i − γ e ) and take the inner product with

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By the Young inequality and the Moser-type inequalities, the last term above is 480 bounded by Moreover, by the first equation in (3.18) and an integration by parts, we have where β > 0 is a small constant to be chosen. This implies that Similarly, taking the inner product of the first equation in (3.18) with ∂ k+1 y p in 488 L 2 (R) and using an integration by parts, we have By the second equation in (3.18), we obtain as above Combining (3.19) and (3. 20), and choosing β > 0 to be sufficiently small, it yields Integrating (3.21) over [0, t] with t ∈ [0, T ], we obtain Summing this inequality for all k = 0, 1, · · · , m − 1 and using Lemma 3.4, we obtain 504 (3.17).
Since U T is sufficiently small, we further obtain  that Y 0 is a C m -diffeomorphism from R to R. We denote by X 0 the inverse C m -519 diffeomorphism of Y 0 and define 520 U 0 (y) = 1 ρ 0 , u 0 , 1 2 c e u 2 0 + c e ε e0 , 1 2 c i u 2 0 + c i ε i0 (X 0 (y)).

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On the other hand, the result in Theorem 2.3 also implies that U ∈ C 1 (R + × R) 528 and U is globally Lipschitzian on R with respect to y (in particular for τ and u). Then 529 the Cauchy problem to the following ordinary differential equation admits a unique global solution Y 1 ∈ C 2 (R + ). Let us further define a function X by 532 Finally, we define 538 V(t, x) = (ρ, u, ε e , ε i ) T t, Y (t, x) .

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It is proved in [18] (see also [25]) that entropy solutions of the Cauchy problem for 540 that the bitemperature Euler model, which is a non-conservative system, provides a 551 good example on this topic.

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More precisely, we consider the system in the form (1.5) or equivalently (1.6). De-553 note W = (ρ, ρu T , E e , E i ). Since η defined in (1.7) is a strictly convex entropy, η (W) 554 is a symmetric positive definite matrix. The result below implies that η (W)C j (W) 555 is not symmetric in one space dimension.

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Proposition. Consider the one dimensional system (2.12) and denote by C 1 (W) =