Compressible primitive equations

Transcription

1 Compressible primitive equations M. Ersoy, IMATH, Toulon 1 8e mes journe es scientifiques de l Universite de Toulon Toulon, April 16, joint work with T. Ngom and M. Sy (LANI, Senegal)

2 Outline of the talk Outline of the talk 1 Mathematical and physical background 2 A Global existence result 3 A quick numerical computation 4 Concluding remarks and perspectives M. Ersoy (IMATH) CPEs Toulon, April 16, / 20

3 Outline Outline 1 Mathematical and physical background 2 A Global existence result 3 A quick numerical computation 4 Concluding remarks and perspectives M. Ersoy (IMATH) CPEs Toulon, April 16, / 20

4 Context : compressible equation in a thin layer domain Navier-Stokes equations on Ω = {(x, y) R 2 R; H L} M. Ersoy (IMATH) CPEs Toulon, April 16, / 20

5 Context : compressible equation in a thin layer domain Navier-Stokes equations on Ω = {(x, y) R 2 R; H L} [Ped] Hydrostatic approximation (ε = H/L = W/V 1) Primitive equations J. Pedlowski Geophysical Fluid Dynamics. 2nd Edition, Springer-Verlag, New-York, M. Ersoy (IMATH) CPEs Toulon, April 16, / 20

6 Context : compressible equation in a thin layer domain Navier-Stokes equations on Ω = {(x, y) R 2 R; H L} [Ped] Hydrostatic approximation (ε = H/L = W/V 1) Primitive equations [EP14] Averaged Primitive Equations wr to y Pressurized model J. Pedlowski Geophysical Fluid Dynamics. 2nd Edition, Springer-Verlag, New-York, M. Ersoy A pressurized model for compressible pipe flow. Submitted, M. Ersoy (IMATH) CPEs Toulon, April 16, / 20

7 Atmosphere dynamic : assumptions Dynamic : M. Ersoy (IMATH) CPEs Toulon, April 16, / 20

8 Atmosphere dynamic : assumptions Dynamic : Compressible fluid Modeling (neglecting phenomena such as the evaporation and solar heating) : Compressible Navier-Stokes equations d dt ρ + ρdivu = 0 ρ d dt u + xp = div x (σ x ) + f t (ρv) + div(ρuv) + y p(ρ) = ρg + div y (σ y ) p(ρ) = c 2 ρ with U = (u, v), d dt := t + u x + v y, σ := 2Σ.D(U) + λdiv(u) M. Ersoy (IMATH) CPEs Toulon, April 16, / 20

9 Atmosphere dynamic : assumptions Dynamic : Compressible fluid vertical scale horizontal scale Modeling (neglecting phenomena such as the evaporation and solar heating) : Compressible Navier-Stokes equations Hydrostatic approximation Compressible Primitive Equations (CPEs) d dt ρ + ρdivu = 0 ρ d dt u + xp = div x (σ x ) + f t (ρv) + div(ρuv) + y p(ρ) = ρg + div y (σ y ) p(ρ) = c 2 ρ M. Ersoy (IMATH) CPEs Toulon, April 16, / 20

10 Atmosphere dynamic : assumptions Dynamic : Compressible fluid vertical scale horizontal scale Modeling (neglecting phenomena such as the evaporation and solar heating) : Compressible Navier-Stokes equations Hydrostatic approximation Compressible Primitive Equations (CPEs) d dt ρ + ρdivu = 0 ρ d dt u + xp = div x (σ x ) + f y p(ρ) = ρg p(ρ) = c 2 ρ M. Ersoy (IMATH) CPEs Toulon, April 16, / 20

11 Stratified density p = ξ(t, x)e g/c2 y Atmosphere dynamic : assumptions Dynamic : Compressible fluid vertical scale horizontal scale Modeling (neglecting phenomena such as the evaporation and solar heating) : Compressible Navier-Stokes equations Hydrostatic approximation Compressible Primitive Equations (CPEs) d dt ρ + ρdivu = 0 ρ d dt u + xp = div x (σ x ) + f M. Ersoy and T. Ngom y p(ρ) = ρg p(ρ) = c 2 ρ Existence of a global weak solution to a Compressible Primitive Equations. C. R. Acad. Sci. Paris, Ser. I, http ://dx.doi.org/ /j.crma , M. Ersoy, T. Ngom and M. Sy Compressible primitive equations : formal derivation and stability of weak solutions. Nonlinearity, 24(1), pp 79-96, M. Ersoy (IMATH) CPEs Toulon, April 16, / 20

12 Examples of applications Atmosphere modeling (left : density, right : velocity field) Pipe flow modeling (left : velocity profile in a horse shoe pipe, right : piezometric line) M. Ersoy (IMATH) CPEs Toulon, April 16, / 20

13 Outline Outline 1 Mathematical and physical background 2 A Global existence result 3 A quick numerical computation 4 Concluding remarks and perspectives M. Ersoy (IMATH) CPEs Toulon, April 16, / 20

14 the two-dimensional problem [EN10] Let us consider the equations on Ω = {(x, y); 0 < x < l, 0 < y < h} t ρ + x (ρu) + y (ρv) = 0 t (ρu) + x (ρu 2 ) + y (ρuv) + c 2 x ρ = x (ν 1 (t, x, y) x u) + y (ν 2 (t, x, y) y u) c 2 y ρ = ρg with the following boundary conditions : u x=0 = u x=l = 0, v y=0 = v y=h = 0, y u y=0 = y u y=h = 0 and the initial data u t=0 (x, y) = u 0 (x, y), ρ t=0 (x, y) = ξ 0 (x)e g/c2y, with 0 < m ξ 0 M < M. Ersoy and T. Ngom Existence of a global solution to a Compressible Primitive Equations. C. R. Acad. Sci. Paris, Ser. I, http ://dx.doi.org/ /j.crma , M. Ersoy (IMATH) CPEs Toulon, April 16, / 20

15 the two-dimensional problem [EN10] Let us consider the equations on Ω = {(x, y); 0 < x < l, 0 < y < h} t ρ + x (ρu) + y (ρv) = 0 t (ρu) + x (ρu 2 ) + y (ρuv) + c 2 x ρ = x (ν 1 (t, x, y) x u) + y (ν 2 (t, x, y) y u) c 2 y ρ = ρg with the following boundary conditions : u x=0 = u x=l = 0, v y=0 = v y=h = 0, y u y=0 = y u y=h = 0 and the initial data u t=0 (x, y) = u 0 (x, y), ρ t=0 (x, y) = ξ 0 (x)e g/c2y, with 0 < m ξ 0 M < keeping in mind the stratified structure of ρ = ξ(t, x)e g/c2y and M. Ersoy and T. Ngom Existence of a global solution to a Compressible Primitive Equations. C. R. Acad. Sci. Paris, Ser. I, http ://dx.doi.org/ /j.crma , M. Ersoy (IMATH) CPEs Toulon, April 16, / 20

16 the two-dimensional problem [EN10] Let us consider the equations on Ω = {(x, y); 0 < x < l, 0 < y < h} t ρ + x (ρu) + y (ρv) = 0 t (ρu) + x (ρu 2 ) + y (ρuv) + c 2 x ρ = x (ν 1 (t, x, y) x u) + y (ν 2 (t, x, y) y u) c 2 y ρ = ρg with the following boundary conditions : u x=0 = u x=l = 0, v y=0 = v y=h = 0, y u y=0 = y u y=h = 0 and the initial data u t=0 (x, y) = u 0 (x, y), ρ t=0 (x, y) = ξ 0 (x)e g/c2y, with 0 < m ξ 0 M < keeping in mind the stratified structure of ρ = ξ(t, x)e g/c2y and assume ν 1 (t, x, y) = ν 1 e g c 2 y, M. Ersoy and T. Ngom Existence of a global solution to a Compressible Primitive Equations. C. R. Acad. Sci. Paris, Ser. I, http ://dx.doi.org/ /j.crma , ν 2 (t, x, y) = ν 2 e g c 2 y, with (ν 1, ν 2 ) R 2 +. M. Ersoy (IMATH) CPEs Toulon, April 16, / 20

17 A useful change of variables [EN10] Multiplying the previous equations by e g/c2y, we get : ) t (ξ) + x (ξu) + e g/c2y y (ξe g/c2y v = 0 ( t (ξu) + x ξu 2 ) ) + e g/c2y y (ξue g/c2y v + c 2 x (ξ) = x (ν 1 x u)+ e g/c2y y (ν 2 e g/c2y c 2 y (ξe g/c2 y ) = ξe g/c2y g setting 1 w = e g/c2y v M. Ersoy and T. Ngom Existence of a global solution to a Compressible Primitive Equations. C. R. Acad. Sci. Paris, Ser. I, http ://dx.doi.org/ /j.crma , M. Ersoy (IMATH) CPEs Toulon, April 16, / 20

18 A useful change of variables [EN10] Multiplying the previous equations by e g/c2y, we get : t (ξ) + x (ξu) + e g/c2y y (ξw) = 0 ( t (ξu) + x ξu 2 ) + e g/c2y y (ξuw) + c 2 x (ξ) = x (ν 1 x u)+ ) e g/c2y y (ν 2 e g/c2y y u c 2 y (ξe ) g/c2 y = ξe g/c2y g setting 1 w = e g/c2y v 2 e g/c2y y = z with z = 1 c2 2 g e g/c y M. Ersoy and T. Ngom Existence of a global solution to a Compressible Primitive Equations. C. R. Acad. Sci. Paris, Ser. I, http ://dx.doi.org/ /j.crma , M. Ersoy (IMATH) CPEs Toulon, April 16, / 20

19 A useful change of variables [EN10] Multiplying the previous equations by e g/c2y, we get : t (ξ) + x (ξu) + z (ξw) = 0 t (ξu) + x ( ξu 2 ) + z (ξuw) + c 2 x (ξ) = x (ν 1 x u)+ z (ν 2 z u) z ξ = 0 setting 1 w = e g/c2y v 2 e g/c2y y = z with z = 1 c2 2 g e g/c y M. Ersoy and T. Ngom Existence of a global solution to a Compressible Primitive Equations. C. R. Acad. Sci. Paris, Ser. I, http ://dx.doi.org/ /j.crma , M. Ersoy (IMATH) CPEs Toulon, April 16, / 20

20 A useful change of variables [EN10] Finally, we get a simplified compressible primitive model (SCP) on Ω = {(x, z); 0 < x < l, 0 = 1 c 2 /g < y < H = 1 c 2 /ge g/c2h } t (ξ) + x (ξu) + z (ξw) = 0 t (ξu) + x ( ξu 2 ) + z (ξuw) + c 2 x (ξ) = x (ν 1 x u)+ z (ν 2 z u) z ξ = 0 with the following boundary conditions : u x=0 = u x=l = 0, w z= 0 = w z=h = 0, z u z= 0 = z u z=h = 0 and the initial data u t=0 (x, z) = u 0 (x, z), ξ t=0 (x) = ξ 0 (x), with 0 < m ξ 0 M <. N. E. Kochin On simplification of the equations of hydromechanics in the case of the general circulation of the atmosphere. Trudy, Glavn. Geofiz. Observator., 4 :21 45, M. Ersoy (IMATH) CPEs Toulon, April 16, / 20

21 As a consequence, firstly, Assume c 2 = g = h = 1, H note u = 1 u(t, x, z) dz, H 0 note ξ for ξh and ν 1 for ν 1 H then the averaged (SCP) (ASCP) system becomes { t (ξ) + x (ξu) = 0 t (ξu) + x (ξ(u 2 )) + x (c 2 ξ) = x (ν 1 x u) using the boundary conditions : u x=0 = u x=l = 0, w z= 0 = w z=h = 0, z u z= 0 = z u z=h = 0 M. Ersoy and T. Ngom Existence of a global solution to a Compressible Primitive Equations. C. R. Acad. Sci. Paris, Ser. I, http ://dx.doi.org/ /j.crma , M. Ersoy (IMATH) CPEs Toulon, April 16, / 20

22 As a consequence, firstly, Assume c 2 = g = h = 1, H note u = 1 u(t, x, z) dz, H 0 note ξ for ξh and ν 1 for ν 1 H mean-oscillation u = u + ũ with H 0 ũdz = 0 then the averaged (SCP) (ASCP) system becomes { t (ξ) + x (ξu) = 0 t (ξu) + x (ξ(u 2 )) + x (c 2 ξ) = x (ν 1 x u) using the boundary conditions : u x=0 = u x=l = 0, w z= 0 = w z=h = 0, z u z= 0 = z u z=h = 0 M. Ersoy and T. Ngom Existence of a global solution to a Compressible Primitive Equations. C. R. Acad. Sci. Paris, Ser. I, http ://dx.doi.org/ /j.crma , M. Ersoy (IMATH) CPEs Toulon, April 16, / 20

23 As a consequence, firstly, Assume c 2 = g = h = 1, H note u = 1 u(t, x, z) dz, H 0 note ξ for ξh and ν 1 for ν 1 H mean-oscillation u = u + ũ with H 0 ũdz = 0 then the averaged (SCP) (ASCP) system becomes { t (ξ) + x (ξu) = 0 t (ξu) + x (ξu 2 ) + x (c 2 ξ + ξϕ(ũ)) = x (ν 1 x u) with ϕ(f) = H 0 f 2 dz. M. Ersoy and T. Ngom Existence of a global solution to a Compressible Primitive Equations. C. R. Acad. Sci. Paris, Ser. I, http ://dx.doi.org/ /j.crma , M. Ersoy (IMATH) CPEs Toulon, April 16, / 20

24 As a consequence, secondly, multiplying the SCP model by H (using the previous notations) the SCP system becomes t (ξ) + x (ξu) + z (ξw) = 0 t (ξu) + x (ξu 2 ) + z (ξuw) + x (c 2 ξ) = D z (ξ) = 0 with D = x (ν 1 x u) + z (ν 2 z u) M. Ersoy and T. Ngom Existence of a global solution to a Compressible Primitive Equations. C. R. Acad. Sci. Paris, Ser. I, http ://dx.doi.org/ /j.crma , M. Ersoy (IMATH) CPEs Toulon, April 16, / 20

25 As a consequence, secondly, multiplying the SCP model by H (using the previous notations) inserting u = u + ũ in the SCP model the SCP system becomes t (ξ) + x (ξu) + z (ξw) = 0 t (ξu) + x (ξu 2 ) + z (ξuw) + x (c 2 ξ) = D z (ξ) = 0 with D = x (ν 1 x u) + z (ν 2 z u) M. Ersoy and T. Ngom Existence of a global solution to a Compressible Primitive Equations. C. R. Acad. Sci. Paris, Ser. I, http ://dx.doi.org/ /j.crma , M. Ersoy (IMATH) CPEs Toulon, April 16, / 20

26 As a consequence, secondly, multiplying the SCP model by H (using the previous notations) inserting u = u + ũ in the SCP model the SCP system becomes t (ξ) + x (ξu) + x (ξũ) + z (ξw) = 0 t (ξu) + t (ξũ) + x (ξu 2 ) + x (ξ(2uũ + ũ 2 )) + z (ξuw) + x (c 2 ξ) = D z (ξ) = 0 with D = x (ν 1 x u) + x (ν 1 x ũ) + z (ν 2 z ũ) M. Ersoy and T. Ngom Existence of a global solution to a Compressible Primitive Equations. C. R. Acad. Sci. Paris, Ser. I, http ://dx.doi.org/ /j.crma , M. Ersoy (IMATH) CPEs Toulon, April 16, / 20

27 As a consequence, secondly, multiplying the SCP model by H (using the previous notations) inserting u = u + ũ in the SCP model keeping in mind the ASCP model the SCP system becomes t (ξ) + x (ξu) + x (ξũ) + z (ξw) = 0 t (ξu) + t (ξũ) + x (ξu 2 ) + x (ξ(2uũ + ũ 2 )) + z (ξuw) + x (c 2 ξ) = D z (ξ) = 0 with D = x (ν 1 x u) + x (ν 1 x ũ) + z (ν 2 z ũ) where t (ξu) + x (ξu 2 ) + x (c 2 ξ) x (ν 1 x u) = x (ξϕ(ũ)) M. Ersoy and T. Ngom Existence of a global solution to a Compressible Primitive Equations. C. R. Acad. Sci. Paris, Ser. I, http ://dx.doi.org/ /j.crma , M. Ersoy (IMATH) CPEs Toulon, April 16, / 20

28 As a consequence, secondly, multiplying the SCP model by H (using the previous notations) inserting u = u + ũ in the SCP model keeping in mind the ASCP model the SCP system becomes x (ξũ) + z (ξw) = 0 t (ξũ) + x (ξ(2uũ + ũ 2 )) x (ξϕ(ũ)) + z (ξuw) = D z (ξ) = 0 with D = x (ν 1 x ũ) + z (ν 2 z ũ) M. Ersoy and T. Ngom Existence of a global solution to a Compressible Primitive Equations. C. R. Acad. Sci. Paris, Ser. I, http ://dx.doi.org/ /j.crma , M. Ersoy (IMATH) CPEs Toulon, April 16, / 20

29 As a consequence, secondly, multiplying the SCP model by H (using the previous notations) inserting u = u + ũ in the SCP model keeping in mind the ASCP model the SCP system becomes x (ξũ) + z (ξw) = 0 t (ξũ) + x (ξ(2uũ + ũ 2 )) x (ξϕ(ũ)) + z (ξuw) = D z (ξ) = 0 with D = x (ν 1 x ũ) + z (ν 2 z ũ) i.e. with w = 1 ξ t (ξũ) + x (ξ(2uũ + ũ 2 ) ξϕ(ũ)) + z (ξuw) = D H 0 x (ξũ) dz. M. Ersoy and T. Ngom Existence of a global solution to a Compressible Primitive Equations. C. R. Acad. Sci. Paris, Ser. I, http ://dx.doi.org/ /j.crma , M. Ersoy (IMATH) CPEs Toulon, April 16, / 20

30 Finally, Formally, solving the SCP model is then equivalent to find a fix point to : Given ũ 1, find (ξ, u) := T 1 (ũ 1 ) such that { t (ξ) + x (ξu) = 0 t (ξu) + x (ξu 2 ) + x (c 2 ξ + ξϕ(ũ)) = x (ν 1 x u) M. Ersoy and T. Ngom Existence of a global solution to a Compressible Primitive Equations. C. R. Acad. Sci. Paris, Ser. I, http ://dx.doi.org/ /j.crma , M. Ersoy (IMATH) CPEs Toulon, April 16, / 20

31 Finally, Formally, solving the SCP model is then equivalent to find a fix point to : Given ũ 1, find (ξ, u) := T 1 (ũ 1 ) such that { t (ξ) + x (ξu) = 0 t (ξu) + x (ξu 2 ) + x (c 2 ξ + ξϕ(ũ)) = x (ν 1 x u) Given (ξ, u), find ũ 2 = T 2 (ξ, u) such that t (ξũ) + x (ξ(2uũ + ũ 2 ) ξϕ(ũ)) + z (ξuw) = x (ν 1 x ũ) + z (ν 2 z ũ) M. Ersoy and T. Ngom Existence of a global solution to a Compressible Primitive Equations. C. R. Acad. Sci. Paris, Ser. I, http ://dx.doi.org/ /j.crma , M. Ersoy (IMATH) CPEs Toulon, April 16, / 20

32 Schauder fix point Following [GK05] (suitable a priori estimates for u (hence for ũ) and ξ bounded), one can define the space K = { f; f L2 (0,T,H 1 (Ω) k < } B. V. Gatapov and A. V. Kazhikhov Existence of a global solution to one model problem of atmosphere dynamics. Siberian Mathematical Journal, 46(5), pp , S. N. Antontsev and A. V. Kazhikhov and V. N. Monakhov Boundary value problems of nonhomogeneous fluid mechanics. Studies in Mathematics and its Applications,North-Holland Publishing Co., Amsterdam, M. Ersoy (IMATH) CPEs Toulon, April 16, / 20

33 Schauder fix point Following [GK05] (suitable a priori estimates for u (hence for ũ) and ξ bounded), one can define the space K = { f; f L2 (0,T,H 1 (Ω) k < } such that providing ũ 1 K, one can construct (ξ, u) such that T 1 (ũ 1 ) = (ξ, u) (see [AKM83]), B. V. Gatapov and A. V. Kazhikhov Existence of a global solution to one model problem of atmosphere dynamics. Siberian Mathematical Journal, 46(5), pp , S. N. Antontsev and A. V. Kazhikhov and V. N. Monakhov Boundary value problems of nonhomogeneous fluid mechanics. Studies in Mathematics and its Applications,North-Holland Publishing Co., Amsterdam, M. Ersoy (IMATH) CPEs Toulon, April 16, / 20

34 Schauder fix point Following [GK05] (suitable a priori estimates for u (hence for ũ) and ξ bounded), one can define the space K = { f; f L2 (0,T,H 1 (Ω) k < } providing ũ 1 K, one can construct (ξ, u) such that T 1 (ũ 1 ) = (ξ, u) (see [AKM83]), providing (ξ, u), one can construct ũ 2 K such that T 2 (ξ, u) = ũ 2 K. B. V. Gatapov and A. V. Kazhikhov Existence of a global solution to one model problem of atmosphere dynamics. Siberian Mathematical Journal, 46(5), pp , S. N. Antontsev and A. V. Kazhikhov and V. N. Monakhov Boundary value problems of nonhomogeneous fluid mechanics. Studies in Mathematics and its Applications,North-Holland Publishing Co., Amsterdam, M. Ersoy (IMATH) CPEs Toulon, April 16, / 20

35 Schauder fix point Following [GK05] (suitable a priori estimates for u (hence for ũ) and ξ bounded), one can define the space K = { f; f L2 (0,T,H 1 (Ω) k < } providing ũ 1 K, one can construct (ξ, u) such that T 1 (ũ 1 ) = (ξ, u) (see [AKM83]), providing (ξ, u), one can construct ũ 2 K such that T 2 (ξ, u) = ũ 2 K. As a consequence, the operator T = T 2 T 1 has at least one fixed point (Schauder fixed point theorem). B. V. Gatapov and A. V. Kazhikhov Existence of a global solution to one model problem of atmosphere dynamics. Siberian Mathematical Journal, 46(5), pp , S. N. Antontsev and A. V. Kazhikhov and V. N. Monakhov Boundary value problems of nonhomogeneous fluid mechanics. Studies in Mathematics and its Applications,North-Holland Publishing Co., Amsterdam, M. Ersoy (IMATH) CPEs Toulon, April 16, / 20

36 More precisely Following [GK05], Definition let us define (ξ, u, w) a weak solution of the SCP equations (i.e. satisfying SCPE in the distribution sense) such that ξ L (0, T ; H 1 (0, H)), t ξ L 2 (0, T ; L 2 (0, H)) u L 2 (0, T ; H 2 (Ω)) H 1 (0, T ; L 2 (Ω)), w L 2 (0, T ; L 2 (Ω)). B. V. Gatapov and A. V. Kazhikhov Existence of a global solution to one model problem of atmosphere dynamics. Siberian Mathematical Journal, 46(5), pp , M. Ersoy (IMATH) CPEs Toulon, April 16, / 20

37 More precisely Following [GK05], Definition let us define (ξ, u, w) a weak solution of the SCP equations (i.e. satisfying SCPE in the distribution sense) such that ξ L (0, T ; H 1 (0, H)), t ξ L 2 (0, T ; L 2 (0, H)) u L 2 (0, T ; H 2 (Ω)) H 1 (0, T ; L 2 (Ω)), w L 2 (0, T ; L 2 (Ω)). Then, Theorem For every (ξ 0, u 0 ), there exists a weak solution to SCP where ξ is a bounded strictly positive function. B. V. Gatapov and A. V. Kazhikhov Existence of a global solution to one model problem of atmosphere dynamics. Siberian Mathematical Journal, 46(5), pp , M. Ersoy (IMATH) CPEs Toulon, April 16, / 20

38 More precisely Following [GK05], Definition let us define (ξ, u, w) a weak solution of the SCP equations (i.e. satisfying CPE in the distribution sense) such that ρ L (0, T ; H 1 (Ω)), t ρ L 2 (0, T ; L 2 (Ω)) u L 2 (0, T ; H 2 (Ω)) H 1 (0, T ; L 2 (Ω)), v L 2 (0, T ; L 2 (Ω)). Then, Theorem For every (ρ 0 = ξ 0 e y, u 0 ), there exists a weak solution to SCP and therefore to the CP equations where ρ is a bounded strictly positive function. B. V. Gatapov and A. V. Kazhikhov Existence of a global solution to one model problem of atmosphere dynamics. Siberian Mathematical Journal, 46(5), pp , M. Ersoy and T. Ngom Existence of a global weak solution to a Compressible Primitive Equations. C. R. Acad. Sci. Paris, Ser. I, http ://dx.doi.org/ /j.crma , M. Ersoy (IMATH) CPEs Toulon, April 16, / 20

39 Outline Outline 1 Mathematical and physical background 2 A Global existence result 3 A quick numerical computation 4 Concluding remarks and perspectives M. Ersoy (IMATH) CPEs Toulon, April 16, / 20

40 Numerical approximation t n+1 = t n + δt be the discrete time with δt be the time step, U = (u, w) t and d dt = t + U. Let us write the SCP equations as follows : d ξ = ξdiv(u) (1) dt ξ d dt u = x(c 2 ξ) + D (2) z (ξ) = 0 (3) with D = x (ν 1 x u) + z (ν 1 z u) M. Ersoy (IMATH) CPEs Toulon, April 16, / 20

41 Numerical approximation t n+1 = t n + δt be the discrete time with δt be the time step, U = (u, w) t and d dt = t + U. Let us write the SCP equations as follows : with d ξ = ξdiv(u) (1) dt ξ d dt u = x(c 2 ξ) + D (2) z (ξ) = 0 (3) D = x (ν 1 x u) + z (ν 1 z u) and remark that using (3) with (1), one has zz w = 1 ξ x(ξ z u) M. Ersoy (IMATH) CPEs Toulon, April 16, / 20

42 Numerical approximation t n+1 = t n + δt be the discrete time with δt be the time step, U = (u, w) t and d dt = t + U. Given ξ n and u n at time t n, one can approximate equations as follows : zz w n = 1 ξ n x(ξ n z u n ) ξ n+1 ξ n X n δt = ξ n div(u n ) u n+1 u n X n δt = 1 ( x ξ n (c 2 ξ n ) + D n) with D = x (ν 1 x u) + z (ν 1 z u) and d ds X = U which can easily implemented in a FreeFem++ a code. a. M. Ersoy (IMATH) CPEs Toulon, April 16, / 20

43 A numerical test numerical parameters : H = 1/2, g = c = 1, δt = 0.05, Tf = 2, 5, ν1 = ν2 = 1, u0 (x, z) = xz density ξ (top left), density ρ (top right), divergence div(ρu ) (bottom left), velocity field U (bottom right), M. Ersoy (IMATH) CPEs Toulon, April 16, / 20

44 Outline Outline 1 Mathematical and physical background 2 A Global existence result 3 A quick numerical computation 4 Concluding remarks and perspectives M. Ersoy (IMATH) CPEs Toulon, April 16, / 20

45 1 uniqueness of the 2D problem? M. Ersoy (IMATH) CPEs Toulon, April 16, / 20

46 1 uniqueness of the 2D problem? 2 existence and uniqueness of the 3D problem? M. Ersoy (IMATH) CPEs Toulon, April 16, / 20

47 1 uniqueness of the 2D problem? 2 existence and uniqueness of the 3D problem? Partially [ENS11] M. Ersoy, T. Ngom, M.Sy Compressible primitive equations : formal derivation and stability of weak solutions. Nonlinearity, 46(5), 24(1), pp 79-96, M. Ersoy (IMATH) CPEs Toulon, April 16, / 20

48 1 uniqueness of the 2D problem? 2 existence and uniqueness of the 3D problem? Partially [ENS11] 3 what happens in the case p(ρ) = ρ γ, γ 1? M. Ersoy, T. Ngom, M.Sy Compressible primitive equations : formal derivation and stability of weak solutions. Nonlinearity, 46(5), 24(1), pp 79-96, M. Ersoy (IMATH) CPEs Toulon, April 16, / 20

49 Thank you ou Thank y for your for your attention ttention a M. Ersoy (IMATH) CPEs Toulon, April 16, / 20