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September  2012, 11(5): 1661-1697. doi: 10.3934/cpaa.2012.11.1661

A class of large amplitude oscillating solutions for three dimensional Euler equations

1. 

UMR6625, Université Rennes 1, Campus de Beaulieu, 263 avenue du Général Leclerc CS 74205, 35042 Rennes

2. 

IRMAR, UMR 6625-CNRS, Universite de Rennes I,Campus de Beaulieu, 35042 Rennes, France

Received  December 2010 Revised  December 2011 Published  March 2012

In this article, we construct large amplitude oscillating waves, noted $ (u ^ {\varepsilon}) _ {\varepsilon} $ where $ \varepsilon \in] 0,1] $ is a parameter going to zero, which are devised to be local solutions on some open domain of the time-space $ R_+ \times R^3$ of both the three dimensional Burger equations (without source term), the compressible Euler equations (with some constant pressure) and the incompressible Euler equations (without pressure). The functions $ u^\varepsilon (t,x) $ are characterized by the fact that the corresponding Jacobian matrices $ D_x u^\varepsilon (t,x) $ are nilpotent of rank one or two. Our purpose is to describe the interesting geometrical features of the expressions $ u^\varepsilon (t,x) $ which can be obtained by this way.
Citation: Christophe Cheverry, Mekki Houbad. A class of large amplitude oscillating solutions for three dimensional Euler equations. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1661-1697. doi: 10.3934/cpaa.2012.11.1661
References:
[1]

A. Biryuk, On multidimensional Burgers type equations with small viscosity, in "Contributions to Current Challenges in Mathematical Fluid Mechanics," Adv. Math. Fluid Mech., (2004), 1-30. doi: 10.1007/978-3-0348-7877-7_1.

[2]

W. M. Boothby, "An Introduction to Differentiable Manifolds and Riemannian Geometry," Pure and Applied Mathematics, No. 63, Academic Press, 1975.

[3]

C. Cheverry, A deterministic model for the propagation of turbulent oscillations, J. Differ. Equations, 9 (2009), 2637-2679. doi: 10.1016/j.jde.2009.08.009.

[4]

C. Cheverry and O. Guès, Counter-examples to concentration-cancellation, Arch. Ration. Mech. Anal., 189 (2008), 363-424. doi: 10.1007/s00205-008-0132-6.

[5]

C. Cheverry, O. Guès and G. Métivier, Large-amplitude high-frequency waves for quasilinear hyperbolic systems, Adv. Differential Equations, 9 (2004), 1065-1089.

[6]

C. Cheverry and M. Houbad, Compatibility conditions to allow some large amplitude WKB analysis for Burger's type systems, Phys. D, 237 (2008), 1429-1443. doi: 10.1016/j.physd.2008.03.022.

[7]

C. Cheverry, Cascade of phases in turbulent flows, Bull. Soc. Math. France, 134 (2006), 33-82.

[8]

C. Cheverry, Recent results in large amplitude monophase nonlinear geometric optics, in "Instability in Models Connected with Fluid Flows," Int. Math. Ser. (N. Y.), 6 (2008), 267-288. doi: 10.1007/978-0-387-75217-4_5.

[9]

J.-F. Coulombel and A. Morando, Stability of contact discontinuities for the nonisentropic Euler equations, Ann. Univ. Ferrara Sez. VII (N.S.), 50 (2004), 79-90.

[10]

R. J. DiPerna and A. J. Majda, Oscillations and concentrations in weak solutions of the incompressible fluid equations, Comm. Math. Phys., 108 (1987), 667-689. doi: 10.1007/BF01214424.

[11]

I. Gallagher and L. Saint-Raymond, On pressureless gases driven by a strong inhomogeneous magnetic field, SIAM J. Math. Anal., 36 (2005), 1159-1176. doi: 10.1137/S0036141003435540.

[12]

E. Grenier, On the nonlinear instability of Euler and Prandtl equations, Comm. Pure Appl. Math., 53 (2000), 1067-1091. doi: 10.1002/1097-0312(200009)53:9<1067::AID-CPA1>3.0.CO;2-Q.

[13]

M. Houbad, "Oscillations, feuilletages, lois de Burger," Ph.D thesis, University of Rennes 1, France, 2010.

[14]

J.-L. Joly, G. Métivier and J. Rauch, Generic rigorous asymptotic expansions for weakly nonlinear multidimensional oscillatory waves, Duke Math. J., 70 (1993), 373-404. doi: 10.1215/S0012-7094-93-07007-X.

[15]

A. J. Majda and A. L. Bertozzi, "Vorticity and Incompressible Flow," Cambridge Texts in Applied Mathematics, 27 (2002).

[16]

S. Schochet, Fast singular limits of hyperbolic PDEs, J. Differential Equations, 114 (1994), 476-512. doi: 10.1006/jdeq.1994.1157.

show all references

References:
[1]

A. Biryuk, On multidimensional Burgers type equations with small viscosity, in "Contributions to Current Challenges in Mathematical Fluid Mechanics," Adv. Math. Fluid Mech., (2004), 1-30. doi: 10.1007/978-3-0348-7877-7_1.

[2]

W. M. Boothby, "An Introduction to Differentiable Manifolds and Riemannian Geometry," Pure and Applied Mathematics, No. 63, Academic Press, 1975.

[3]

C. Cheverry, A deterministic model for the propagation of turbulent oscillations, J. Differ. Equations, 9 (2009), 2637-2679. doi: 10.1016/j.jde.2009.08.009.

[4]

C. Cheverry and O. Guès, Counter-examples to concentration-cancellation, Arch. Ration. Mech. Anal., 189 (2008), 363-424. doi: 10.1007/s00205-008-0132-6.

[5]

C. Cheverry, O. Guès and G. Métivier, Large-amplitude high-frequency waves for quasilinear hyperbolic systems, Adv. Differential Equations, 9 (2004), 1065-1089.

[6]

C. Cheverry and M. Houbad, Compatibility conditions to allow some large amplitude WKB analysis for Burger's type systems, Phys. D, 237 (2008), 1429-1443. doi: 10.1016/j.physd.2008.03.022.

[7]

C. Cheverry, Cascade of phases in turbulent flows, Bull. Soc. Math. France, 134 (2006), 33-82.

[8]

C. Cheverry, Recent results in large amplitude monophase nonlinear geometric optics, in "Instability in Models Connected with Fluid Flows," Int. Math. Ser. (N. Y.), 6 (2008), 267-288. doi: 10.1007/978-0-387-75217-4_5.

[9]

J.-F. Coulombel and A. Morando, Stability of contact discontinuities for the nonisentropic Euler equations, Ann. Univ. Ferrara Sez. VII (N.S.), 50 (2004), 79-90.

[10]

R. J. DiPerna and A. J. Majda, Oscillations and concentrations in weak solutions of the incompressible fluid equations, Comm. Math. Phys., 108 (1987), 667-689. doi: 10.1007/BF01214424.

[11]

I. Gallagher and L. Saint-Raymond, On pressureless gases driven by a strong inhomogeneous magnetic field, SIAM J. Math. Anal., 36 (2005), 1159-1176. doi: 10.1137/S0036141003435540.

[12]

E. Grenier, On the nonlinear instability of Euler and Prandtl equations, Comm. Pure Appl. Math., 53 (2000), 1067-1091. doi: 10.1002/1097-0312(200009)53:9<1067::AID-CPA1>3.0.CO;2-Q.

[13]

M. Houbad, "Oscillations, feuilletages, lois de Burger," Ph.D thesis, University of Rennes 1, France, 2010.

[14]

J.-L. Joly, G. Métivier and J. Rauch, Generic rigorous asymptotic expansions for weakly nonlinear multidimensional oscillatory waves, Duke Math. J., 70 (1993), 373-404. doi: 10.1215/S0012-7094-93-07007-X.

[15]

A. J. Majda and A. L. Bertozzi, "Vorticity and Incompressible Flow," Cambridge Texts in Applied Mathematics, 27 (2002).

[16]

S. Schochet, Fast singular limits of hyperbolic PDEs, J. Differential Equations, 114 (1994), 476-512. doi: 10.1006/jdeq.1994.1157.

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