# American Institute of Mathematical Sciences

• Previous Article
Coexistence and extinction in the Volterra-Lotka competition model with nonlocal dispersal
• CPAA Home
• This Issue
• Next Article
Regularizing rate estimates for mild solutions of the incompressible Magneto-hydrodynamic system
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 & Applied Analysis, 2012, 11 (5) : 1661-1697. doi: 10.3934/cpaa.2012.11.1661
##### References:

show all references

##### References:
 [1] Sasho Popov, Jean-Marie Strelcyn. The Euler-Poisson equations: An elementary approach to integrability conditions. Journal of Geometric Mechanics, 2018, 10 (3) : 293-329. doi: 10.3934/jgm.2018011 [2] Zhongkai Guo. Invariant foliations for stochastic partial differential equations with dynamic boundary conditions. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5203-5219. doi: 10.3934/dcds.2015.35.5203 [3] Gang Bao. Mathematical modeling of nonlinear diffracvtive optics. Conference Publications, 1998, 1998 (Special) : 89-99. doi: 10.3934/proc.1998.1998.89 [4] Baruch Cahlon. Sufficient conditions for oscillations of higher order neutral delay differential equations. Conference Publications, 1998, 1998 (Special) : 124-137. doi: 10.3934/proc.1998.1998.124 [5] Qian Liu, Shuang Liu, King-Yeung Lam. Asymptotic spreading of interacting species with multiple fronts Ⅰ: A geometric optics approach. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3683-3714. doi: 10.3934/dcds.2020050 [6] Daomin Cao, Ezzat S. Noussair, Shusen Yan. On the profile of solutions for an elliptic problem arising in nonlinear optics. Discrete & Continuous Dynamical Systems, 2004, 11 (2&3) : 649-666. doi: 10.3934/dcds.2004.11.649 [7] Tianhong Li. Some special solutions of the multidimensional Euler equations in $R^N$. Communications on Pure & Applied Analysis, 2005, 4 (4) : 757-762. doi: 10.3934/cpaa.2005.4.757 [8] Danny Calegari. Geometry and topology of $\mathbb{R}$-covered foliations. Electronic Research Announcements, 2000, 6: 31-39. [9] Abbas Moameni. Soliton solutions for quasilinear Schrödinger equations involving supercritical exponent in $\mathbb R^N$. Communications on Pure & Applied Analysis, 2008, 7 (1) : 89-105. doi: 10.3934/cpaa.2008.7.89 [10] Li-Bin Liu, Ying Liang, Jian Zhang, Xiaobing Bao. A robust adaptive grid method for singularly perturbed Burger-Huxley equations. Electronic Research Archive, 2020, 28 (4) : 1439-1457. doi: 10.3934/era.2020076 [11] Aibin Zang. Kato's type theorems for the convergence of Euler-Voigt equations to Euler equations with Drichlet boundary conditions. Discrete & Continuous Dynamical Systems, 2019, 39 (9) : 4945-4953. doi: 10.3934/dcds.2019202 [12] Mapundi K. Banda, Michael Herty, Axel Klar. Coupling conditions for gas networks governed by the isothermal Euler equations. Networks & Heterogeneous Media, 2006, 1 (2) : 295-314. doi: 10.3934/nhm.2006.1.295 [13] Maria Do Rosario Grossinho, Rogério Martins. Subharmonic oscillations for some second-order differential equations without Landesman-Lazer conditions. Conference Publications, 2001, 2001 (Special) : 174-181. doi: 10.3934/proc.2001.2001.174 [14] Jaume Llibre, Yuzhou Tian. Meromorphic integrability of the Hamiltonian systems with homogeneous potentials of degree -4. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021228 [15] Gennaro Infante. Positive solutions of differential equations with nonlinear boundary conditions. Conference Publications, 2003, 2003 (Special) : 432-438. doi: 10.3934/proc.2003.2003.432 [16] Carlos Gutierrez, Víctor Guíñez. Simple umbilic points on surfaces immersed in $\R^4$. Discrete & Continuous Dynamical Systems, 2003, 9 (4) : 877-900. doi: 10.3934/dcds.2003.9.877 [17] Jonathan E. Rubin, Justyna Signerska-Rynkowska, Jonathan D. Touboul, Alexandre Vidal. Wild oscillations in a nonlinear neuron model with resets: (Ⅱ) Mixed-mode oscillations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 4003-4039. doi: 10.3934/dcdsb.2017205 [18] Marcel Oliver. The Lagrangian averaged Euler equations as the short-time inviscid limit of the Navier–Stokes equations with Besov class data in $\mathbb{R}^2$. Communications on Pure & Applied Analysis, 2002, 1 (2) : 221-235. doi: 10.3934/cpaa.2002.1.221 [19] Yuming Qin, Lan Huang, Zhiyong Ma. Global existence and exponential stability in $H^4$ for the nonlinear compressible Navier-Stokes equations. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1991-2012. doi: 10.3934/cpaa.2009.8.1991 [20] E. N. Dancer, Shusen Yan. On the existence of multipeak solutions for nonlinear field equations on $R^N$. Discrete & Continuous Dynamical Systems, 2000, 6 (1) : 39-50. doi: 10.3934/dcds.2000.6.39

2020 Impact Factor: 1.916