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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, Universite 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.

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##### 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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