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(1)FOR A. A S IMPLIFIED M ODEL L OW M ACH N UMBER D IPHASIC F LOW. Yohan P ENEL1,2 , Stéphane D ELLACHERIE1 , Olivier L AFITTE2 1. DEN/DANS/DM2S/SFME/LETR, CEA Saclay, France. 2. LAGA, Institut Galilée, University of Paris 13, France. ICPDE 2010 - F OR THE 60 TH B IRTHDAY OF M. C HIPOT Poitiers – February, 20th, 2010.

(2) 1. DLMN 2. Theoretical results 3. Interface computations. I NTRODUCTION G OAL Modeling of the evolution of bubbles in a nuclear reactor at the scale of bubbles (DNS). 2 / 26 Y. Penel (CEA). The ABV Model. :.

(3) 1. DLMN 2. Theoretical results 3. Interface computations. I NTRODUCTION G OAL Modeling of the evolution of bubbles in a nuclear reactor at the scale of bubbles (DNS). Navier-Stokes equations for a diphasic compressible immiscible flow. Mach Number supposed to be small (still compressible flow but without acoustic wave effects). Diphasic Low Mach Number system (DLMN). Simplification but keeping a similar mathematical structure. Abstract Bubble Vibration model (ABV). Theoretical study. Numerical simulations. 2 / 26 Y. Penel (CEA). The ABV Model. :.

(4) 1. DLMN 2. Theoretical results 3. Interface computations. O UTLINE. 1. Derivation of the model. 2. Theoretical results. 3. Interface computations. 3 / 26 Y. Penel (CEA). The ABV Model. :.

(5) 1. DLMN 2. Theoretical results 3. Interface computations. PARTIAL O UTLINE. 1. Derivation of the model. 2. Theoretical results. 3. Interface computations. 4 / 26 Y. Penel (CEA). The ABV Model. :.

(6) 1. DLMN 2. Theoretical results 3. Interface computations. C OMPRESSIBLE D IPHASIC N AVIER -S TOKES S YSTEM. D IPHASIC C OMPRESSIBLE N AVIER -S TOKES. Starting point.  ∂t (ρ Y1 ) + ∇ · (ρ Y1 u) = 0 ,    ∂ ρ + ∇ · (ρ u) = 0 , t  ∂ (ρ u) + ∇ · (ρ u ⊗ u) = −∇P + ∇ · σ + ρ g , t    ∂t (ρ E ) + ∇ · (ρ uE ) = −∇ · (Pu) + ∇ · (κ∇T ) + ∇ · (σ u) + ρ g · u ,. together with transmission and boundary conditions. Nomenclature. • T : temperature. • ρ : density • g : gravity field. • P : pressure. • σ : Cauchy stress tensor. • Y1 : mass fraction of Fluid 1. • u : global velocity. • κ : thermal conductivity. • E : total energy. • θ = ( Y1 , T , P ) 5 / 26. Y. Penel (CEA). The ABV Model. :.

(7) 1. DLMN 2. Theoretical results 3. Interface computations. M ODELING. BUBBLES ( 0. Y1 (t = 0, x ) = Y (x ) =. Initial condition Ω. 1, if x ∈ Ω1 (0), 0, if x ∈ Ω2 (0).. Diphasic flow = non-miscible bi-fluid flow = Y1 not regular. 0. Y =0. Ω2 (0) Y0 = 1. Discontinuity of Y1 = interface of the bubble Ω1 (0). Ω ⊂ Rd , d ∈ {1, 2, 3}: open, bounded, smooth. The resolution of this equation with that initial condition amounts to determining for a domain Ω1 (t ).. 6 / 26 Y. Penel (CEA). The ABV Model. :.

(8) 1. DLMN 2. Theoretical results 3. Interface computations. H YPOTHESES. AND. TOOLS. P HYSICS. M ATHEMATICS. Bounded domain (reactor). Ω = [−1, 1]d , d ∈ {2, 3}. No void. ρ >0. Linear elasticity. Linearized Cauchy tensor. Common physical properties for both fluids. Single nondimensioned system. Low Mach Number. Asymptotic expansion w.r.t. M∗  1. Existence of an entropy. −T ds = dε + P dτ. Based on earlier Majda’s & Embid’s works on combustion (’84).. 7 / 26 Y. Penel (CEA). The ABV Model. :.

(9) 1. DLMN 2. Theoretical results 3. Interface computations. DLMN S YSTEM Diphasic Low Mach Number System: At order 0 in the asymptotic expansion, the system reads:.   ∂t Y1 + u · ∇Y1 = 0 ,       ∇ · u = Gθ , ρ(∂t u + (u · ∇)u) = −∇Π + 2∇ · [µ D (u)] + ρ g ,    ρ cp (∂t T + u · ∇T ) = α TP 0 (t ) + ∇ · (κ∇T ) ,     P 0 (t ) = H (t ), θ where P is the thermodynamic pressure, Π the dynamic pressure and: Dt ρ 1 P 0 (t ) β ∇ · (κ∇T ) Gθ (t , x ) := − =− + , ρ Γ P (t ) P (t ) Z   β (θ )∇ · κ(θ )∇T dx Z Hθ (t ) := Ω . 1 dx Ω. Γ(θ ). 8 / 26 Y. Penel (CEA). The ABV Model. :.

(10) 1. DLMN 2. Theoretical results 3. Interface computations. DLMN S YSTEM Diphasic Low Mach Number System: At order 0 in the asymptotic expansion, the system reads:.   ∂t Y1 + u · ∇Y1 = 0 ,    6=0   =⇒ compressibility, elliptic contribution  ∇ · u = Gθ , ρ(∂t u + (u · ∇)u) = −∇Π + 2∇ · [µ D (u)] + ρ g ,    ρ cp (∂t T + u · ∇T ) = α TP 0 (t ) + ∇ · (κ∇T ) ,     P 0 (t ) = H (t ), θ where P is the thermodynamic pressure, Π the dynamic pressure and: Dt ρ 1 P 0 (t ) β ∇ · (κ∇T ) Gθ (t , x ) := − =− + , ρ Γ P (t ) P (t ) Z   β (θ )∇ · κ(θ )∇T dx Z Hθ (t ) := Ω . 1 dx Ω. Γ(θ ). 8 / 26 Y. Penel (CEA). The ABV Model. :.

(11) 1. DLMN 2. Theoretical results 3. Interface computations. D ERIVED S YSTEMS. FOR PRELIMARY STUDIES. Hodge Decomposition: u = ∇φ + w with ∇ · w = 0 and boundary conditions. w→0. Potential DLMN System.  ∂t Y1 + ∇φ · ∇Y1 = 0 ,     ∆φ = G , θ  ρ c (∂ T  p t + ∇φ · ∇T ) = α TP 0 (t ) + ∇ · (κ∇T ) ,    0 P (t ) = Hθ (t ).. Gθ → GY. Abstract Bubble Vibration Model.   ∂t Y1 + ∇φ · ∇Y1 = 0 ,   Z 1  Y1 (t , y) dy . ∆φ (t , x) = ψ(t ) Y1 (t , x) − |Ω| Ω 9 / 26 Y. Penel (CEA). The ABV Model. :.

(12) 1. DLMN 2. Theoretical results 3. Interface computations. PARTIAL O UTLINE. 1. Derivation of the model. 2. Theoretical results General study 1D. 3. Interface computations. 10 / 26 Y. Penel (CEA). The ABV Model. :.

(13) 1. DLMN 2. Theoretical results 3. Interface computations. G LOBAL. 2.1. General study. ISSUES. System of Partial Differential Equations 1. 2. Properties of solutions Existence of solutions depending on the regularity of Y 0 Strong solutions when Y 0 ∈ H s , s large enough Weak solutions when Y 0 ∈ L∞. 3. Uniqueness. 4. Numerical simulations. Main tool: Energy estimates. Set: 1 2 3. Y = {Y ∈ L∞ (Ω) : Y (x ) ∈ [0, 1] for almost every x ∈ Ω}   Ws,T (Ω) = C 0 [0, T ], L2 (Ω) ∩ L∞ [0, T ], H s (Ω)  ZT (Ω) = L∞ [0, T ], W 1,∞ (Ω) 11 / 26 Y. Penel (CEA). The ABV Model. :.

(14) 1. DLMN 2. Theoretical results 3. Interface computations. A LGEBRAIC. 2.1. General study. PROPERTIES.   ∂t Y + ∇φ · ∇Y = 0,        Y (0, ·) = Y 0 ,    Z  − 1 0 0   ∆φ (t , x) = ψ(t ) Y (t , x) − |Ω| Y (t , x ) dx ,   Ω     ∇φ · n|∂ Ω = 0.. 12 / 26 Y. Penel (CEA). The ABV Model. :.

(15) 1. DLMN 2. Theoretical results 3. Interface computations. A LGEBRAIC. 2.1. General study. PROPERTIES. Restrictions on the potential: φ0 is prescribed by Y 0 . In Z φ (t , x) dx = 0 addition, we impose Ω   ∂t Y + ∇φ · ∇Y = 0,        Y (0, ·) = Y 0 ,    Z  − 1 0 0   ∆φ (t , x) = ψ(t ) Y (t , x) − |Ω| Y (t , x ) dx ,   Ω     ∇φ · n|∂ Ω = 0.. 12 / 26 Y. Penel (CEA). The ABV Model. :.

(16) 1. DLMN 2. Theoretical results 3. Interface computations. A LGEBRAIC. 2.1. General study. PROPERTIES. Restrictions on the potential: φ0 is prescribed by Y 0 . In Z φ (t , x) dx = 0 addition, we impose Ω   ∂t Y + ∇φ · ∇Y = 0,     There is at most one weak solution in the class ZT (Ω)    Y (0, ·) = Y 0 ,    Z  − 1 0 0   ∆φ (t , x) = ψ(t ) Y (t , x) − |Ω| Y (t , x ) dx ,   Ω     ∇φ · n|∂ Ω = 0.. 12 / 26 Y. Penel (CEA). The ABV Model. :.

(17) 1. DLMN 2. Theoretical results 3. Interface computations. A LGEBRAIC. 2.1. General study. PROPERTIES. Restrictions on the potential: φ0 is prescribed by Y 0 . In Z φ (t , x) dx = 0 addition, we impose Ω   ∂t Y + ∇φ · ∇Y = 0,     There is at most one weak solution in the class ZT (Ω)    Y (0, ·) = Y 0 ,    Z If Y 0 ∈ Y, then every bounded solution in the class  − 1 0 0   in tY) (maximum ∆φ (t ,ZxT)(Ω) = isψ( Y (t , x)principle) − |Ω| Y (t , x ) dx ,   Ω     ∇φ · n|∂ Ω = 0.. 12 / 26 Y. Penel (CEA). The ABV Model. :.

(18) 1. DLMN 2. Theoretical results 3. Interface computations. A LGEBRAIC. 2.1. General study. PROPERTIES. Restrictions on the potential: φ0 is prescribed by Y 0 . In Z φ (t , x) dx = 0 addition, we impose Ω   ∂t Y + ∇φ · ∇Y = 0,     There is at most one weak solution in the class ZT (Ω)    Y (0, ·) = Y 0 ,    Z If Y 0 ∈ Y, then every bounded solution in the class  − 1 0 0   in tY) (maximum ∆φ (t ,ZxT)(Ω) = isψ( Y (t , x)principle) − |Ω| Y (t , x ) dx ,   Ω     If Ω ·isnsymmetric ∇φ . Y 0 is even, then every solution be|∂ Ω = 0and. longing to ZT (Ω) is even. 12 / 26 Y. Penel (CEA). The ABV Model. :.

(19) 1. DLMN 2. Theoretical results 3. Interface computations. A LGEBRAIC. 2.1. General study. PROPERTIES. T HEOREM (L AFITTE & D ELLACHERIE , ’05; P ENEL , ’10) Assume Y ∈ H (Ω) with s > bd /2c + 1 and ψ ∈ C [0, +∞) .  Then there exists T > 0 depending on kY k and ψ such that the  ∂t Y +A∇φmodel · ∇has Y a= 0,  unique classical solution Y ∈ W ,T (Ω) for some   T at least greater than T .    0  Y (0, ·) = Y ,    Z  − 1 0 0   ∆φ (t , x) = ψ(t ) Y (t , x) − |Ω| Y (t , x ) dx ,   Ω     ∇φ · n|∂ Ω = 0. 0. s. . 0. 0. 0. s. BV. 1. s. 0. 12 / 26 Y. Penel (CEA). The ABV Model. :.

(20) 1. DLMN 2. Theoretical results 3. Interface computations. A LGEBRAIC. 2.1. General study. PROPERTIES. T HEOREM (L AFITTE & D ELLACHERIE , ’05; P ENEL , ’10) Assume Y ∈ H (Ω) with s > bd /2c + 1 and ψ ∈ C [0, +∞) .  Then there exists T > 0 depending on kY k and ψ such that the  ∂t Y +A∇φmodel · ∇has Y a= 0,  unique classical solution Y ∈ W ,T (Ω) for some   T at least greater than T .    0  YL(EMMA 0, ·) = Y ,  Suppose that there exists a weak solution YZ(t , x) = 1Ω ( ) (x)  where Ω (t ) ⊂ Ω and ψ ∈ C (0, +∞), then the volume of the bubble   by:  ∆φ ( t ,isxgiven ) = ψ( t ) Y (t , x) − |Ω|−1 Y (t , x0 ) dx0 ,   Z Ω   |Ω (0)| exp ψ(τ)dτ   t )| = |Ω| ∇φ · n|∂|ΩΩ (= 0. |Ω (0)| + |Ω (0)| exp Z ψ(τ)dτ . 0. s. . 0. 0. 0. s. BV. 1. s. 0. 1. 1 t. 0. 1. t. 1. 0. 1. t. 2. 1. 0. 12 / 26 Y. Penel (CEA). The ABV Model. :.

(21) 1. DLMN 2. Theoretical results 3. Interface computations. P RELIMINARY. 1. 2.1. General study. RESULTS. Energy estimates for the transport equation: ∂t Y + u · ∇Y = f sup kY (t , ·)kr ≤ e. χr (T ).  Y. Z. 0 r. t ∈[0,T ]. Z with χr (t ) = Cadv ,0 (r , d , Ω) 0. +. T. e. −χr (t ). 0.  kf (t , ·)kr dt ,. t. k∇u(τ, ·)kmax(s0 ,r −1) dτ .. 2. Elliptic regularity results for the Poisson equation. 3. Embeddings Ws,T (Ω) ⊂ C 0 [0, T ], H. 4. Classical functional inequalities (Moser, interpolation, Gronwall, . . . ). s0. .  ⊂ C 0 [0, T ] × Ω for any s0 < s. 13 / 26 Y. Penel (CEA). The ABV Model. :.

(22) 1. DLMN 2. Theoretical results 3. Interface computations. S KETCH. OF PROOF. 2.1. General study. (1). Iterative scheme: 1. 2. ∆φ. (k ).  = ψ(t ) Y. (k ). (t , x) −. 1. |Ω|. Z Y. (k ). 0. 0. . (t , x ) dx. Ω. , ∇φ (k ) · n|∂ Ω = 0. ∂t Y (k +1) + ∇φ (k ) · ∇Y (k +1) = 0, Y (k +1) (0, ·) = Y 0. Objectives: 1. to ensure the convergence of (Y (k ) ). 2. to derive estimates in order to avoid a progressive loss of regularity as k → +∞. 3. to check that the limit is a solution of the A BV model. 14 / 26 Y. Penel (CEA). The ABV Model. :.

(23) 1. DLMN 2. Theoretical results 3. Interface computations. S KETCH. OF PROOF. 2.1. General study. (2). Proof of convergence: 1 Boundedness in Ws,T (Ω) which induces weak-? convergence to Ỹ ∈ Ws,T (Ω) via the Arzela-Ascoli theorem and compactness arguments 2 Strong convergence to Y ∈ W0,T (Ω) 3. Ws0 ,T (Ω). Finally, Y = Ỹ ∈ Ws,T (Ω) and Y (k ) − −−−−−→ Y , for any s0 < s. 15 / 26 Y. Penel (CEA). The ABV Model. :.

(24) 1. DLMN 2. Theoretical results 3. Interface computations. S KETCH. OF PROOF. 2.1. General study. (2). Proof of convergence: 1 Boundedness in Ws,T (Ω) which induces weak-? convergence to Ỹ ∈ Ws,T (Ω) via the Arzela-Ascoli theorem and compactness arguments 2 Strong convergence to Y ∈ W0,T (Ω) 3. Ws0 ,T (Ω). Finally, Y = Ỹ ∈ Ws,T (Ω) and Y (k ) − −−−−−→ Y , for any s0 < s. Using energy estimates, we get:. " sup kY t ∈[0,T ]. (k +1). 0. (t , ·)ks ≤ kY ks exp Cabv sup kY t ∈[0,T ]. (k ). (t , ·)ks. Z. T. # |ψ(t )| dt .. 0. Since the sequence un+1 = u0 eun converges iff u0 ≤ e−1 , we show that:. ∀ k ∈ N, sup kY (k ) (t , ·)ks ≤ ekY 0 ks t ∈[0,T ]. Z. T. under the hypothesis 0. |ψ(t )| dt ≤. 1 e · Cabv · kY 0 ks. .. 15 / 26 Y. Penel (CEA). The ABV Model. :.

(25) 1. DLMN 2. Theoretical results 3. Interface computations. S KETCH. OF PROOF. 2.1. General study. (2). Proof of convergence: 1 Boundedness in Ws,T (Ω) which induces weak-? convergence to Ỹ ∈ Ws,T (Ω) via the Arzela-Ascoli theorem and compactness arguments 2 Strong convergence to Y ∈ W0,T (Ω) 3. Ws0 ,T (Ω). Finally, Y = Ỹ ∈ Ws,T (Ω) and Y (k ) − −−−−−→ Y , for any s0 < s. 15 / 26 Y. Penel (CEA). The ABV Model. :.

(26) 1. DLMN 2. Theoretical results 3. Interface computations. S KETCH. OF PROOF. 2.1. General study. (2). Proof of convergence: 1 Boundedness in Ws,T (Ω) which induces weak-? convergence to Ỹ ∈ Ws,T (Ω) via the Arzela-Ascoli theorem and compactness arguments 2 Strong convergence to Y ∈ W0,T (Ω) 3. Ws0 ,T (Ω). Finally, Y = Ỹ ∈ Ws,T (Ω) and Y (k ) − −−−−−→ Y , for any s0 < s. On the other hand, we prove that: e−χ (t ). . . Y (k +1) − Y (k ) (t , ·) 0. ≤ Cabv. Z. t. e−χ (τ ). . . Y (k ) − Y (k −1) (τ, ·) 0. 0. Iterating the process, we show that the series ∑ Y (k +1) − Y (k ). 0,T. dτ. satisfies the Cauchy. criterion in W0,T (Ω) which is complete. Hence the strong convergence of Y (k ) in W0,T (Ω).. 15 / 26 Y. Penel (CEA). The ABV Model. :.

(27) 1. DLMN 2. Theoretical results 3. Interface computations. S KETCH. OF PROOF. 2.1. General study. (2). Proof of convergence: 1 Boundedness in Ws,T (Ω) which induces weak-? convergence to Ỹ ∈ Ws,T (Ω) via the Arzela-Ascoli theorem and compactness arguments 2 Strong convergence to Y ∈ W0,T (Ω) 3. Ws0 ,T (Ω). Finally, Y = Ỹ ∈ Ws,T (Ω) and Y (k ) − −−−−−→ Y , for any s0 < s. 15 / 26 Y. Penel (CEA). The ABV Model. :.

(28) 1. DLMN 2. Theoretical results 3. Interface computations. S KETCH. OF PROOF. 2.1. General study. (2). Proof of convergence: 1 Boundedness in Ws,T (Ω) which induces weak-? convergence to Ỹ ∈ Ws,T (Ω) via the Arzela-Ascoli theorem and compactness arguments 2 Strong convergence to Y ∈ W0,T (Ω) 3. Ws0 ,T (Ω). Finally, Y = Ỹ ∈ Ws,T (Ω) and Y (k ) − −−−−−→ Y , for any s0 < s. We rewrite the iterative system under an intergal form and we conclude applying the dominated convergence theorem to show that the limit (Y1 , ∇φ ) is actually a solution.. 15 / 26 Y. Penel (CEA). The ABV Model. :.

(29) 1. DLMN 2. Theoretical results 3. Interface computations. T IME. INTERVAL. 2.1. General study. (1). We proved the existence of a solution under the assumption: 0. e · Cabv · kY ks. 1. 2. 3. 4. Z. T. |ψ(t )| dt ≤ 1.. 0. We should bear in mind that this condition is sufficient and specific to the method used in the course of the proof: T is not necessarily optimal Given an initial datum Y 0 , we have a global existence for all ψ ∈ L1 (0, +∞) such 1 that kψkL1 ≤ e · Cabv · kY 0 ks Given a pulse ψ and a time T , there is a local existence for all initial data such 1 that kY 0 ks ≤ e · Cabv · kψkL1 (0,T ) If ψ ≡ 0, the solution is trivially Y ≡ Y 0 and T = +∞ 16 / 26 Y. Penel (CEA). The ABV Model. :.

(30) 1. DLMN 2. Theoretical results 3. Interface computations. T IME. INTERVAL. (2) –. 2.1. General study. STILL IN PROGRESS. Continuation principle: if Y1 (T , ·) ∈ H s , we can apply the local existence theorem to the system:   ∂t Z + ∇ϕ · ∇Z = 0,.       . Z (0, ·) = Y1 (T , ·),.   Z  1 0 0   Z (t , x ) dx , ∆ϕ(t , x) = ψ̃(t ) Z (t , x) −   |Ω| Ω     ∇ϕ · n = 0, |∂ Ω. with ψ̃(t ) = ψ(T + t ). Thus, we can extend the solution Y1 to a new time interval [0, T + TZ ]. Let us denote Tk the time of existence at Step k in the continuation process. If the sequence is globally defined, there are two possibilities: either the series ∑ Tk converges (and there is a local existence theorem), or it diverges (and we obtain a global solution). 17 / 26 Y. Penel (CEA). The ABV Model. :.

(31) 1. DLMN 2. Theoretical results 3. Interface computations. 2.2. 1D. R EMARKS In one space dimension, the Poisson equation is trivially solved and the A BV Model reads:  Z x Z x +L L Y (t , y ) dy ∂x Y (t , x ) = 0. ∂t Y (t , x ) + ψ(t ) Y (t , y ) dy − 2L −L −L. =⇒ continuity of the velocity field due to the embedding H 1 ⊂ C 0 in 1D. Specificities of the 1D-case: 1. A single nonlinear integro-differential equation. 2. Explicit solution in the irregular case. 3. Finite propagation speed. Open problems: does the regularized solution converge to the explicit irregular solution? 18 / 26 Y. Penel (CEA). The ABV Model. :.

(32) 1. DLMN 2. Theoretical results 3. Interface computations. E XPLICIT. 2.2. 1D. CALCULATIONS Y 0 (x ) Yε0 (x ) 1. x. −β0. −L. β0. −β0 − ε. L. β0 + ε. Explicit irregular solution in 1D Y1 (t , x ) = 1[−β (t ),β (t )] (x ) with. β (t ) =  1−. β0 β0 L.  exp Ψ(t ) +. β0. .. L 19 / 26. Y. Penel (CEA). The ABV Model. :.

(33) 1. DLMN 2. Theoretical results 3. Interface computations. PARTIAL O UTLINE. 1. Derivation of the model. 2. Theoretical results. 3. Interface computations. 20 / 26 Y. Penel (CEA). The ABV Model. :.

(34) 1. DLMN 2. Theoretical results 3. Interface computations. D ISCRETIZATION Mismatch between the discrete and continuous levels. 21 / 26 Y. Penel (CEA). The ABV Model. :.

(35) 1. DLMN 2. Theoretical results 3. Interface computations. D ISCRETIZATION Mismatch between the discrete and continuous levels. 21 / 26 Y. Penel (CEA). The ABV Model. :.

(36) 1. DLMN 2. Theoretical results 3. Interface computations. T EST ABV. ABV model:.  ∂t Y1 + ∇φ · ∇Y1 = 0,        Y (0, x) = Y 0 (x), 1. 1   Y1 (t , y) dy, ∆φ = Y1 −   4 [−1,1]2    ∇φ · n|∂ Ω = 0.. ZZ. ψ ≡ 1: constant growth. The initial datum Y 0 consists of two disjoint circles of different radii.. 22 / 26 Y. Penel (CEA). The ABV Model. :.

(37) 1. DLMN 2. Theoretical results 3. Interface computations. T EST ABV. 100 × 100 grid with a refinement rate equal to 6. 22 / 26 Y. Penel (CEA). The ABV Model. :.

(38) 1. DLMN 2. Theoretical results 3. Interface computations. C ONCLUSION. Done Existence and Uniqueness Theorem with an approximation of the time interval Study of the 1D-case Derivation of a numerical scheme to preserve interfaces. To do Approximating the time of existence for the DLMN system Enrichment with physical content Theoretical studies with irregular initial data. 23 / 26 Y. Penel (CEA). The ABV Model. :.

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However, in the last decade, both from the theoretical and the numerical sides, significant progresses have been made in the understanding, analysis and approximation of the

The mathematical theory of low Mach number flows

In particular, even in the fast case solutions of the Euler equations with uniformly bounded initial data exist for a time independent of ε?. Nevertheless, the equations