Sign-changing tower of bubbles for a sinh-Poisson equation with asymmetric exponents

Angela Pistoia; Tonia Ricciardi

doi:10.3934/dcds.2017245

Article Contents

2017, Volume 37, Issue 11: 5651-5692. Doi: 10.3934/dcds.2017245

This issue Previous Article The initial-boundary value problems for a class of sixth order nonlinear wave equation Next Article 2-manifolds and inverse limits of set-valued functions on intervals

Sign-changing tower of bubbles for a sinh-Poisson equation with asymmetric exponents

Angela Pistoia^1, and
Tonia Ricciardi^2,

1.
Dipartimento SBAI, Università di Roma "La Sapienza", Via Antonio Scarpa 16, 00161 Rome, Italy
2.
Dipartimento di Matematica e Applicazioni "R. Caccioppoli", Università di Napoli Federico Ⅱ, Via Cintia, 80126 Naples, Italy

Received: February 28, 2017

Revised: April 30, 2017

Published: October 2017

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Abstract

Motivated by the statistical mechanics description of stationary 2D-turbulence, for a sinh-Poisson type equation with asymmetric nonlinearity, we construct a concentrating solution sequence in the form of a tower of singular Liouville bubbles, each of which has a different degeneracy exponent. The asymmetry parameter $γ∈(0, 1]$ corresponds to the ratio between the intensity of the negatively rotating vortices and the intensity of the positively rotating vortices. Our solutions correspond to a superposition of highly concentrated vortex configurations of alternating orientation; they extend in a nontrivial way some known results for $\gamma=1$. Thus, by analyzing the case $\gamma≠1$ we emphasize specific properties of the physically relevant parameter $\gamma$ in the vortex concentration phenomena.

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Mathematics Subject Classification: 35J91, 35A01, 35B44, 35B30.

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	E. Caglioti , P. L. Lions , C. Marchioro and M. Pulvirenti , A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics description, Commun. Math. Phys., 174 (1995) , 229-260. doi: 10.1007/BF02099602.
	M. del Pino , P. Esposito and M. Musso , Nondegeneracy of entire solutions of a singular Liouville equation, Proc. Am. Math. Soc., 140 (2012) , 581-588. doi: 10.1090/S0002-9939-2011-11134-1.
	M. del Pino , M. Kowalczyk and M. Musso , Singular limits in Liouville-type equations, Calc. Var. Partial Differential Equations, 24 (2005) , 47-81. doi: 10.1007/s00526-004-0314-5.
	P. Esposito , M. Grossi and A. Pistoia , On the existence of blowing-up solutions for a mean field equation, Ann. I. H. Poincaré, 22 (2005) , 227-257. doi: 10.1016/j.anihpc.2004.12.001.
	G. L. Eyink and K. R. Sreenivasan , Onsager and the theory of hydrodynamic turbulence, Reviews of Modern Physics, 78 (2006) , 87-135. doi: 10.1103/RevModPhys.78.87.
	M. Grossi , C. Grumiau and F. Pacella , Lane Emden problems with large exponents and singular Liouville equations, J. Math. Pures Appl., 101 (2014) , 735-754. doi: 10.1016/j.matpur.2013.06.011.
	M. Grossi and A. Pistoia , Multiple blow-up phenomena for the sinh-Poisson equation, Arch. Rational Mech. Anal., 209 (2013) , 287-320. doi: 10.1007/s00205-013-0625-9.
	A. Jevnikar , An existence result for the mean-field equation on compact surfaces in a doubly supercritical regime, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013) , 1021-1045. doi: 10.1017/S030821051200042X.
	A. Jevnikar and W. Yang, Analytic aspects of the Tzitzéica equation: Blow-up analysis and existence results, Calc. Var. Partial Differential Equations, 56 (2017), Paper No. 43, 38 pp. doi: 10.1007/s00526-017-1136-6.
	D. D. Joseph and T. S. Lundgren , Quasilinear problems driven by positive sources, Arch. Rat. Mech. Anal., 49 (1973) , 241-269. doi: 10.1007/BF00250508.
	J. Jost , G. Wang , D. Ye and C. Zhou , The blow up analysis of solutions of the elliptic sinh-Gordon equation, Calc. Var. Partial Differential Equations, 31 (2008) , 263-276. doi: 10.1007/s00526-007-0116-7.
	G. Joyce and D. Montgomery , Negative temperature states for the two-dimensional guiding centre plasma, J. Plasma Phys., 10 (1973) , 107-121.
	C. S. Lin , An expository survey on recent development of mean field equations, Discr. Cont. Dynamical Systems, 19 (2007) , 387-410. doi: 10.3934/dcds.2007.19.387.
	A. Malchiodi , Topological methods for an elliptic equation with exponential nonlinearities, Discr. Cont. Dynamical Systems, 21 (2008) , 277-294. doi: 10.3934/dcds.2008.21.277.
	J. Moser , A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J, 20 (1970/71) , 1077-1092. doi: 10.1512/iumj.1971.20.20101.
	C. Neri , Statistical Mechanics of the $N$ -point vortex system with random intesities on a bounded domain, Ann. I. H. Poincaré, 21 (2004) , 381-399. doi: 10.1016/j.anihpc.2003.05.002.
	H. Ohtsuka , T. Ricciardi and T. Suzuki , Blow-up analysis for an elliptic equation describing stationary vortex flows with variable intensities in 2D-turbulence, J. Differential Equations, 249 (2010) , 1436-1465. doi: 10.1016/j.jde.2010.06.006.
	H. Ohtsuka and T. Suzuki , Mean field equation for the equilibrium turbulence and a related functional inequality, Adv. Differential Equations, 11 (2006) , 281-304.
	L. Onsager , Statistical hydrodynamics, Nuovo Cimento Suppl, 6 (1949) , 279-287. doi: 10.1007/BF02780991.
	A. Pistoia and T. Ricciardi , Concentrating solutions for a Liouville type equation with variable intensities in 2D-turbulence, Nonlinearity, 29 (2016) , 271-297. doi: 10.1088/0951-7715/29/2/271.
	Y. B. Pointin and T. S. Lundgren , Statistical mechanics of two-dimensional vortices in a bounded container, Phys. Fluids, 19 (1976) , 1459-1470.
	J. Prajapat and G. Tarantello , On a class of elliptic problems in $\mathbb R^2$ : Symmetry and uniqueness results, Proc. R. Soc. Edinb. Sect. A, 131 (2001) , 967-985. doi: 10.1017/S0308210500001219.
	T. Ricciardi , Mountain-pass solutions for a mean field equation from two-dimensional turbulence, Differential and Integral Equations, 20 (2007) , 561-575.
	T. Ricciardi and G. Zecca, Minimal blow-up masses and existence of solutions for an asymmetric sinh-Poisson equation, arXiv: 1605.05895
	K. Sawada and T. Suzuki , Derivation of the equilibrium mean field equations of point vortex and vortex filament system, Theoret. Appl. Mech. Japan, 56 (2008) , 285-290.
	R. Takahashi, Analysis Seminar, Naples Federico Ⅱ University, March 2016.
	N. S. Trudinger , On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967) , 473-483.
	D. Ye , Une remarque sur le comportement asymptotique des solutions de $-Δ u=λ f(u)$, C.R. Acad. Sci. Paris, 325 (1997) , 1279-1282. doi: 10.1016/S0764-4442(97)82353-1.
	C. Zhou , Existence of solution for mean field equation for the equilibrium turbulence, Nonlinear Anal., 69 (2008) , 2541-2552. doi: 10.1016/j.na.2007.08.029.