Continuous dependence of very weak solutions for the stationary Navier-Stokes equations

In this work we show the continuous dependence of the very weak solutions for the stationary Navier-Stokes system with respect to boundary data belonging to space L2 (Γ)

Self-similarity And Asymptotic Stability For Coupled Nonlinear Schrödinger Equations In High Dimensions

This paper is concerned with systems of coupled Schrödinger equations with polynomial nonlinearities and dimension n<1. We show the existence of global self-similar solutions and prove that they are asymptotically stable in a framework based on weak- Lp spaces, whose elements have local finite L2-mass. The radial symmetry of the solutions is also addressed. © 2011 Elsevier B.V. All rights reserved.2415534542Bronski, J.C., Carr, L.D., Deconink, B., Kutz, J.N., Bose Einstein condensates in standing waves (2001) Phys. Rev. Lett., 86, pp. 1402-1405Spatschek, K.H., Coupled localized electron-plasma waves and oscillatory ion-acoustic perturbations (1978) Phys. Fluids, 21, pp. 1032-1035Yew, A.C., Multipulses of Nonlinearly Coupled Schrödinger Equations (2001) Journal of Differential Equations, 173 (1), pp. 92-137. , DOI 10.1006/jdeq.2000.3922, PII S0022039600939226Menyuk, C.R., Schiek, R., Torner, L., Solitary waves due to X ( 2): X ( 2) cascading (1994) J. Opt. Soc. Amer. B, 11, pp. 2434-2443Sammut, A.R., Buryak, A.V., Kivshar, Y.S., Bright and dark solitary waves in the presence of the third-harmonic generation (1998) J. Opt. Soc. Am. B, 15, pp. 1488-1496Ambrosetti, A., Colorado, E., Bound and ground states of coupled nonlinear Schrödinger equations (2006) Comptes Rendus Mathematique, 342 (7), pp. 453-458. , DOI 10.1016/j.crma.2006.01.024, PII S1631073X06000525De Figueiredo, D.G., Lopes, O., Solitary waves for some nonlinear Schrödinger systems (2008) Ann. Inst. H. Poincaré Anal. Non Linéaire, 25, pp. 149-161Lin, T.C., Wei, J., Ground states of N coupled nonlinear Schrödinger equations in Rn, n &lt3 (2008) Comm. Math. Phys., 277, pp. 573-576. , (Erratum)Yew, A.C., Stability analysis of multipulses in nonlinearly-coupled Schrödinger equations (2000) Indiana Univ. Math. J., 49 (3), pp. 1079-1124Ohta, M., Stability of solitary waves for coupled nonlinear Schrödinger equations (1996) Nonlinear Analysis, Theory, Methods and Applications, 26 (5), pp. 933-939. , DOI 10.1016/0362-546X(94)00340-8Pastor, A., Orbital stability of periodic travelling waves for coupled nonlinear Schrödinger equations (2010) Electron. J. Differential Equations, 7, pp. 1-19Pastor, A., Nonlinear and spectral stability of periodic travelling wave solutions for a nonlinear Schrödinger system (2010) Differential Integral Equations, 23, pp. 125-154Pelinovsky, D.E., Kivshar, Y.S., Stability criterion for multicomponent solitary waves (2000) Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, 62 (6 B), pp. 8668-8676Pelinovsky, D.E., Inertia law for spectral stability of solitary waves in coupled nonlinear Schrödinger equations (2005) Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 461 (2055), pp. 783-812. , DOI 10.1098/rspa.2004.1345Lopes, O., Stability of solitary waves of some coupled systems (2006) Nonlinearity, 19 (1), pp. 95-113. , DOI 10.1088/0951-7715/19/1/006, PII S0951771506057124Braz Silva, E.P., Ferreira, L.C.F., Villamizar-Roa, E.J., On the existence of infinite energy solutions for nonlinear Schrödinger equations (2009) Proc. Amer. Math. Soc., 137, pp. 1977-1987Cazenave, T., Weissler, F.B., Asymptotically self-similar global solutions of the nonlinear Schrödinger and heat equations (1998) Math. Z., 228 (1), pp. 83-120Giga, Y., Miyakawa, T., NavierStokes flow in R 3 with measures as initial vorticity and Morrey spaces (1989) Commn. Partial Differential Equations, 14, pp. 577-618Dudley, J.M., Finot, C., Richardson, D.J., Millot, G., (2007) Nature Phys., 3 (9), pp. 597-603Fermann, M.E., Kruglov, V.I., Thomsen, B.C., Dudley, J.M., Harvey, J.D., Self-similar propagation and amplification of parabolic pulses in optical fibers (2000) Phys. Rev. Lett., 84, pp. 6010-6013Soffer, A., Weinstein, M.I., Multichannel nonlinear scattering for nonintegrable equations. II, the case of anisotropic potentials and data (1992) J. Differential Equations, 98 (2), pp. 376-390Kirr, E., Zarnescu, A., On the asymptotic stability of bound states in 2D cubic Schrödinger equation (2007) Comm. Math. Phys., 272 (2), pp. 443-468Kuznetsov, E.A., (2009) Wave Collapse in Nonlinear Optics, 114 VOL., pp. 175-190. , Topics in Applied Physics SpringerPérez-García, V.M., Self-similar solutions and collective coordinate methods for nonlinear Schrödinger equations (2004) Physica D, 191 (34), pp. 211-218Lin, T.-C., Wei, J., Solitary and self-similar solutions of two-component system of nonlinear Schrödinger equations (2006) Physica D: Nonlinear Phenomena, 220 (2), pp. 99-115. , DOI 10.1016/j.physd.2006.07.009, PII S0167278906002363Bergh, J., Lofstrom, J., (1976) Interpolation Spaces, , Springer-Verlag BerlinNew YorkFerreira, L.C.F., Mateus, E., Self-similarity and uniqueness of solutions for semilinear reactiondiffusion systems (2010) Adv. Difference Equ., 15 (12), pp. 73-9

On The Heat Equation With Concave-convex Nonlinearity And Initial Data In Weak-lp Spaces

We analyze the local well-posedness of the initial-boundary value problem for the heat equation with nonlinearity presenting a combined concave- convex structure and taking the initial data in weak-Lp spaces. Moreover we give a new uniqueness class and obtain some results about the behavior of the solutions near t = 0+.10617151732Aguirre, J., Escobedo, M., A Cauchy problem for ut-Δu = u p with 0 &ltp &lt1. Asymptotic behaviour of solutions (1986) Ann. Fac. Sci. Toulouse Math., 8, pp. 175-203Ambrosetti, A., Brezis, H., Cerami, G., Combined effects of concave and convex nonlinea-rities in some elliptic problems (1994) J. Funct. Anal., 122, pp. 519-543Bergh, J., Lofstrom, J., (1976) Interpolation Spaces, , Springer-Verlag, New YorkBoccardo, L., Peral, I., Escobedo, M., A Dirichlet problem involving critical exponent (1995) Journal of Nonlinear Analysis T.M.A., 24, pp. 1639-1648Brezis, H., Cazenave, T., A nonlinear heat equation with singular initial data (1996) J. Anal. Math., 68, pp. 277-304Cannone, M., Planchon, F., Self-similar solutions for Navier-Stokes equations in ℝ 3 (1996) Comm. Partial Differential Equations, 21, pp. 179-193Cazenave, T., Dickstein, F., Escobedo, M., A semilinear heat equation with concave-convex nonlinearity (1999) Rend. Mat. Appl., 19, pp. 211-242Ferreira, L.C.F., Villamizar-Roa, E.J., Self-similar solutions, uniqueness and long-time asymptotic behavior for semilinear heat equations (2006) Differential and Integral Equations, 19, pp. 1349-1370Ferreira, L.C.F., Villamizar-Roa, E.J., On the existence of solutions for the Navier-Stokes system in a sum of weak-L p spaces (2010) Discrete Contin. Dyn. Syst., 27, pp. 171-183Giga, Y., Solutions for semilinear parabolic equations in L p and regularity of weak solutions of the Navier-Stokes system (1986) J. Differential Equations, 62, pp. 186-212Kozono, H., Yamazaki, Y., Semilinear heat equations and the Navier-Stokes equations with distributions in new function spaces as initial data (1994) Comm. P.D.E., 19, pp. 959-1014Loayza, M., The heat equation with singular nonlinearity and singular initial data (2006) J. Differ-ential Equations, 229, pp. 509-528Maekawa, Y., Terasawa, T., The Navier-Stokes equations with initial data in uniformly local L p spaces (2006) Differential Integral Equations, 19, pp. 369-400Weissler, F.B., Local existence and nonexistence for semilinear parabolic equations in L p (1980) Indiana Univ. Math. J., 29, pp. 79-102Weissler, F.B., Existence and nonexistence of global solutions for a semilinear heat equation (1981) Israel J. Math., 38, pp. 29-40Yamazaki, M., The Navier-Stokes equations in the weak-Ln spaces with time-dependent ex-ternal force (2000) Math. Ann., 317, pp. 635-67

Strong Solutions And Inviscid Limit For Boussinesq: System With Partial Viscosity

We consider the convection problem of a fluid with viscosity depending on tempera-ture in either a bounded or an exterior domain Ω⊂ RN,N =2,3. It is assumed that the temperature is transported without thermal conductance (dissipation) by the velocity field which is described by the Navier-Stokes flow. This model is commonly called the Boussinesq system with partial viscosity. In this paper we prove the existence and uniqueness of strong solutions for the Boussinesq system with partial viscosity with initial data in W2-2/p,p(Ω)×W1,q(Ω). For a bounded domain Ω, we also analyze the inviscid limit problem when the conductivity in the fully viscous Boussinesq system goes to zero. © 2013 International Press.112421439de Almeida, M., Ferreira, L.C.F., On the well posedness and large-time behavior for Boussi-nesq equations in Morrey spaces (2011) Diff. Integral Eqs., 24, pp. 719-742Abels, H., Nonstationary Stokes system with variable viscosity in bounded and unbounded do-mains (2010) Discrete Contin. Dyn. Syst. Ser. S, 3, pp. 141-157Abidi, H., Sur l'unicit́e pour le syst'eme de Boussinesq avec diffusion non lińeaire (2009) J. Math. Pures et. Appl., 91, pp. 80-99Boldrini, J.L., Duŕan, M., Rojas-Medar, M.A., Existence and uniqueness of strong solution for the incompressible micropolar fluid equations in domains of R3 (2010) Ann. Univ. Ferrara, 56, pp. 37-51Brandolese, L., Schonbek, M.E., Large time decay and growth for solutions of a viscous Boussinesq system (2012) Trans. Amer. Math. Soc., 364, pp. 5057-5090Cannon, J.R., DiBenedetto, E., The initial value problem for the Boussinesq equations with data in Lp, in ApproximationMethods for Navier-Stokes Problems (1980) Lecture Notes inMath, 771, pp. 129-144Chae, D., Global regularity for the 2D Boussinesq equations with partial viscosity terms (2006) Adv. in Math., 203, pp. 497-513Chae, D., Imanuvilov, O.Y., Generic solvability of the axisymmetric 3-D Euler equations and the 2-D Boussinesq equations (1999) J. Diff. Eqs., 156, pp. 1-17Danchin, R., Density-dependent incompressible fluids in bounded domains (2006) J. Math. Fluid Mech., 8, pp. 333-381Danchin, R., Paicu, M., Existence and uniqueness results for the Boussinesq system with data in Lorenz spaces (2008) Physica D, 237, pp. 1444-1460Danchin, R., Paicu, M., Le théoréme de Leray et le théoréme de Fujita-Kato pour le systéme de Boussinesq partiellement visqueux (2008) Bull. Soc. Math. France, 136, pp. 261-309DiPerna, R.J., Lions, P.L., Ordinary differential equations, transport theory and Sobolev spaces (1989) Invent. Math., 98, pp. 511-547Feireisl, E., Schonbek, M.E., On the Oberbeck-Boussinesq approximation on unbounded do-mains, Nonlin. Par (2012) Diff. Equ, Abel Symposia, 7, pp. 131-168Ferreira, L.C.F., Villamizar-Roa, E.J., On the stability problem for the Boussinesq equations in weak-Lp spaces (2010) Commun. Pure Appl. Anal., 9, pp. 667-684Ferreira, L.C.F., Villamizar-Roa, E.J., Existence of solutions to the convection problem in a pseudomeasure-type space (2008) Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 464, pp. 1983-1999Ferreira, L.C.F., Villamizar-Roa, E.J., Well-posedness and asymptotic behaviour for the convection problem in Rn (2006) Nonlinearity, 19, pp. 2169-2191Graham, A., Shear patterns in a unstable layer of air (1933) Philos. Trans. Roy. Soc. London Ser. A, 232, pp. 285-296Hishida, T., On a class of stable steady flows to the exterior convection problem (1997) J. Diff. Eqs., 141, pp. 54-85Hmidi, T., Keraani, S., On the global well-possedness of the two-dimensional Boussinesq system with a zero diffusivity (2007) Adv. Diff. Eqs., 12, pp. 461-480Hou, T.Y., Li, C., Global well-posedness of the viscous Boussinesq equations (2005) Discrete Cont. Dyn. Syst. A., 12, pp. 1-12Ladyszhenskaya, O., Solonnikov, V.A., Unique solvability of an initial and boundary value problem for viscous incompressible nonhomogeneous fluids (1976) Zap. Naûcn Sem. -Leningrado Otdel Math. Inst. Steklov., 52, pp. 52-109. , (English Translation, J. Soviet Math., 9, 697-749, 1978)Lai, M., Pan, R., Zhao, K., Initial boundary value problem for 2D viscous Boussinesq equa-tions (2011) Arch. Rational Mech. Anal., 199, pp. 739-760Lorca, S., Boldrini, J.L., The initial value problem for a generalized Boussinesq model (1999) Nonlinear Anal, 36, pp. 457-480Majda, A., Introduction to PDEs and Waves for the Atmosphere and Ocean, Courant Lect (2003) Notes Math, 9. , American Mathematical Society/CIMAMorimoto, H., Nonstationary Boussinesq equations (1992) J. Fac. Sci. Univ. Tokyo Sect. IA Math., 39, pp. 61-75Pedloski, J., (1978) Geophysical Fluid Dynamics, , Springer-Verlag, New YorkTriebel, H., (1978) Interpolation Theory, Function Spaces, Differential Operators, , North HollandTurcotte, D.L., Schubert, G., (1982) Geodynamics: Applications of Continuum Physics to Geological Problems, , JohnWiley and SonsVillamizar-Roa, E.J., Ortega-Torres, E., On a generalized Boussinesq model around a rotat-ing obstacle Existence of strong solutions (2011) Discrete Contin. Dyn. Syst. Ser. B, 15, pp. 825-84

On The Existence Of Solutions For The Navier-stokes System In A Sum Of Weak-lp Spaces

We study the Navier-Stokes system with initial data belonging to sum of two weak-Lp spaces, which contains the sum of homogeneous function with different degrees. The domain Ω can be either an exterior domain, the half-space, the whole space or a bounded domain with dimension n ≥ 2. We obtain the existence of local mild solutions in the same class of initial data and moreover we show results about uniqueness, regularity and continuous dependence of solutions with respect to the initial data. To obtain our results we prove a new Hölder-type inequality on the sum of Lorentz spaces.271171183Barraza, O., Self similar solutions in weak Lp-spaces of the Navier-Stokes equations (1996) Rev. Mat. Iber., 12, pp. 411-439Bennett, C., Sharpley, R., Interpolation of operators (1988) Pure and Applied Mathematics, 129. , Academic PressBorchers, W., Miyakawa, T., On stability of exterior stationary Navier-Stokes flows (1995) Acta Math., 174, pp. 311-382Borchers, W., Varnhorn, W., On the boundedness of the Stokes semigroup in two-dimensional exterior domains (1993) Math. Z., 213, pp. 275-299Cannone, M., (1995) Ondelettes, Paraproduits et Navier-Stokes, , Diderot Editeur, ParisDan, W., Shibata, Y., On the Lq-Lr estimates of the Stokes semigroup in a twodimensional exterior domain (1999) J. Math. Soc. Japan, 51, pp. 181-207Dickstein, F., Blow up stability of solutions of the nonlinear heat equation with a large life span (2006) J. Diff. Eq., 223, pp. 303-328Ferreira, L.C.F., Villamizar-Roa, E.J., Well-posedness and asymptotic behaviour for the convection problem in Rn (2006) Nonlinearity, 19, pp. 2169-2191Giga, Y., Solutions in Lrof the Navier-Stokes initial value problem (1985) Arch. Rational Mech. Anal., 89, pp. 267-281Giga, Y., Miyakawa, T., Navier-Stokes flow in R3with measures as initial vorticity and Morrey spaces (1989) Comm. P.D.E., 14, pp. 577-618Hishida, T., Shibata, Y., Lp -Lq estimate of the Stokes operator and Navier-Stokes flows in the exterior of a rotating obstacle (2009) Arch. Rational Mech. Anal., 193, pp. 339-421Iwashita, H., Lq -Lr estimates for solutions of the nonstationary Stokes equations in an exterior domain and the Navier-Stokes initial value problems in Lq spaces (1989) Math. Ann., 285, pp. 265-288Kato, T., Strong Lp-solutions of the Navier-Stokes Equation in R m, with applications to weak solutions (1984) Math. Z., 187, pp. 471-480Kato, T., Strong solutions of the Navier-Stokes equations in Morrey spaces (1992) Bol. Soc. Brasil Math., 22, pp. 127-155Koch, H., Tataru, D., Well-posedness for the Navier-Stokes equations (2001) Advanced in Math., 157, pp. 22-35Kozono, H., Yamazaki, Y., Semilinear heat equations and the Navier-Stokes equations with distributions in new function spaces as initial data (1994) Comm. P.D.E., 19, pp. 959-1014Lemarié-Rieusset, P., (2002) Recent Developments in the Navier-Stokes Problem, , Chapman & Hall/ CRC Press, Boca RatonMaekawa, Y., Terasawa, T., The Navier-Stokes equations with initial data in uniformly local L p spaces (2006) Differential Integral Equations, 19, pp. 369-400Taniuchi, Y., Uniformly local W estimate for 2 -D vorticity equation and its applications to Euler equations with initial vorticity in bmo (2004) Comm. Math. Phys., 248, pp. 169-186Taylor, M.E., Analysis on Morrey spaces and applications to Navier-Stokes equations and other evolution equations (1992) Comm. P.D.E., 17, pp. 1407-1456Ukai, S., A solution formula for the Stokes equation in Rn + (1987) Comm. Pure Appl. Math., 40, pp. 611-621Weissler, F., The Navier-Stokes initial value problem in Lp (1980) Arch. Rational Mech. Anal., 74, pp. 219-230Yamazaki, M., The Navier-Stokes equations in the weak-Ln spaces with time-dependent external force (2000) Math. Ann., 317, pp. 635-67

On The Stability Problem For The Boussinesq Equations In Weak-lp Spaces

We consider the Boussinesq equations in either an exterior domain in ℝn, the whole space ℝn, the half space ℝn + or a bounded domain in ℝn, where the dimension n satisfies n ≥ 3. We give a class of stable steady solutions, which improves and complements the previous stability results. Our results give a complete answer to the stability problem for the Boussinesq equations in weak-Lp spaces, in the sense that we only assume that the stable steady solution belongs to scaling invariant class L σ (n,∞) x L(n,∞). Moreover, some considerations about the exponential decay (in bounded domains) and the uniqueness of the disturbance are done.93667684Bergh, J., Löfström, J., (1976) Interpolation Spaces, , Springer-Verlag, BerlinBarraza, O., Regularity and stability for the solutions of the Navier-Stokes equations in Lorentz spaces (1999) Nonlinear Analysis, 35, pp. 747-764Borchers, W., Miyakawa, T., On stability of exterior stationary Navier-Stokes flows (1995) Acta Math, 174, pp. 311-382Biler, P., Cannone, M., Karch, G., Asymptotic stability of Navier-Stokes flow past an ob-stacle (2004) Nonlocal elliptic and parabolic problems, Banach Center Publ, 66, pp. 47-59Cannone, M., Karch, G., About the regularized Navier-Stokes equations (2005) J. Math. Fluid Mechanics, 7, pp. 1-28Karch, G., Prioux, N., Self-similarity in viscous Boussinesq equations (2008) Proc. Amer. Math. Soc, 136, pp. 879-888Cannone, M., Planchon, F., Self-similar solutions for Navier-Stokes equations in ℝ3 (1996) Comm. Partial Differential Equations, 21, pp. 179-193Chandrasekhar, S., (1981) Hydrodynamic and Hydromagnetic Stability, , Dover, New YorkChen, Z., Kagei, Y., Miyakawa, T., Remarks on stability of purely conductive steady states to the exterior Boussinesq problem (1992) Adv. Math. Sci. Appl, 1, pp. 411-430Galdi, G., Padula, M., A new approach to energy theory in the stability of fluid motion (1990) Arch. Rational Mech. Anal, 110, pp. 187-286D. Fujiwara and H. Morimoto, An Lr-Theorem of the Helmholtz decomposition of vector fields, Fac. Sci. Univ. Tokio, sec. IA, 24 (1977), 685-700Hishida, T., Asymptotic behavior and stability of solutions to the exterior convection problem (1994) Nonlinear Anal, 22, pp. 895-925Hishida, T., Global existence and exponential stability of convection (1995) J. Math. Anal. Appl, 196, pp. 699-721Hishida, T., On a class of stable steady flow to the exterior convection problem (1997) J. Diff. Eq, 141, pp. 54-85Joseph, D., (1976) Stability of Fluid Motion, , Springer-Verlag, BerlinKozono, H., Yamazaki, M., On a larger class of stable solutions to the Navier-Stokes equations in exterior domains (1998) Math. Z, 228, pp. 751-785Landau, L., Lifshitz, E., Theorical Physics: Fluid Mechanics (1987) 2nd edition, , Pergamon Press, OxfordFerreira, L.C.F., Villamizar-Roa, E.J., Well-posedness and asymptotic behaviour for the convection problem in ℝ n (2006) Nonlinearity, 19, pp. 2169-2191Ferreira, L.C.F., Villamizar-Roa, E.J., Existence of solutions to the convection problem in a pseudomeasure-type space (2008) Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci, 464, pp. 1983-1999Lemarié-Rieusset, P., (2002) Recent Developments in the Navier-Stokes Problem, , Chapman & Hall/ CRC Press, Boca RatonMeyer, Y., (1997) Wavelets, paraproducts and Navier-Stokes equations, current developments in Mathematics 1996, pp. 105-212. , International Press, CambridgeO'Neil, R., Convolution operators and L(p, q) spaces (1963) Duke Math. J, 30, pp. 129-142Yamazaki, M., The Navier-Stokes equations in the weak-Ln spaces with time-dependent ex-ternal force (2000) Math. Ann, 317, pp. 635-67

On Existence And Scattering Theory For The Klein-gordon-schrödinger System In An Infinite l2l^{2}<math Xmlns:xlink=http://www.w3.org/1999/xlink><msup><mi>l</mi><mn>2</mn></msup></math>-norm Setting

This paper is concerned with the initial value problem for the nonlinear Klein-Gordon-Schrödinger (KGS) system in (Formula presented.). We consider general polynomial nonlinearities that include in particular the classical Yukawa-KGS model. We show existence of local and global mild solutions for the KGS system with initial data in weak (Formula presented.)-spaces, which is an infinite (Formula presented.)-norm setting. Moreover, we obtain a persistence result in (Formula presented.) when the initial data belong to this class, which shows that the constructed data-solution map in weak-(Formula presented.) recovers (Formula presented.)-regularity. We also prove results of scattering and wave operators in that singular framework. © 2014 Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg

On The Schrödinger-boussinesq System With Singular Initial Data

We study the existence of local and global solutions for coupled Schrödinger-Boussinesq systems with initial data in weak-Lr spaces. These spaces contain singular functions with infinite L2-mass such as homogeneous functions of negative degree. Moreover, we analyze the self-similarity and radial symmetry of solutions by considering initial data with the right homogeneity and radially symmetric, respectively. Since functions in weak-Lr with r&gt;2 have local finite L2-mass, the solutions obtained can be physically realized. Moreover, for initial data in Hs, local solutions belong to Hs which shows that the constructed data-solution map in weak-Lr recovers Hs-regularity. © 2012 Elsevier Ltd.4002487496Makhankov, V., On stationary solutions of Schrödinger equation with a self-consistence potential satisfying Boussinesq's equations (1974) Phys. Lett. A, 50, pp. 42-44Yajima, N., Satsuma, J., Soliton solutions in a diatomic lattice system (1979) Progr. Theoret. Phys., 62, pp. 370-378Farah, L., Local and global solutions for the nonlinear Schrödinger-Boussinesq system (2008) Differential Integral Equations, 21, pp. 743-770Farah, L., Pastor, A., On the periodic Schrödinger-Boussinesq system (2010) J. Math. Anal. Appl., 368, pp. 330-349Linares, F., Navas, A., On Schrödinger-Bousinesq equations (2004) Adv. Differential Equations, 9, pp. 159-176Han, Y., The Cauchy problem of nonlinear Schrödinger-Bousinesq equations in Hs(Rd) (2005) J. Partial Differ. Equ., 18, pp. 1-20Cazenave, T., Weissler, F., Asymptotically self-similar global solutions of the nonlinear Schrödinger and heat equations (1998) Math. Z., 228, pp. 83-120Ferreira, L.C.F., Villamizar-Roa, E.J., Braz e Silva, P., On the existence of infinite energy solutions for nonlinear Schrödinger equations (2009) Proc. Amer. Math. Soc., 137, pp. 1977-1987Ferreira, L.C.F., Villamizar-Roa, E.J., Self-similarity and asymptotic stability for coupled nonlinear Schrodinger equations in high dimensions (2012) Physica D, 241, pp. 534-542Cannone, M., Planchon, F., Self-similar solutions for Navier-Stokes equations in R3 (1996) Comm. Partial Differential Equations, 21, pp. 179-193Ferreira, L.C.F., Villamizar-Roa, E.J., Self-similar solutions, uniqueness and long-time asymptotic behavior for semilinear heat equations (2006) Differential Integral Equations, 19, pp. 1349-1370Ferreira, L.C.F., Villamizar-Roa, E.J., On the stability problem for the Boussinesq equations in weak-Lp spaces (2010) Commun. Pure Appl. Anal., 9, pp. 667-684Ferreira, L.C.F., Villamizar-Roa, E.J., Existence of solutions to the convection problem in a pseudomeasure-type space (2008) Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 464, pp. 1983-1999Ferreira, L.C.F., Villamizar-Roa, E.J., Well-posedness and asymptotic behaviour for the convection problem in Rn (2006) Nonlinearity, 19, pp. 2169-2191Giga, Y., Miyakawa, T., Navier-Stokes flow in R3 with measures as initial vorticity and Morrey spaces (1989) Comm. Partial Differential Equations, 14, pp. 577-618Ferreira, L.C.F., Existence and scattering theory for Boussinesq type equations with singular data (2011) J. Differential Equations, 250, pp. 2372-2388Bergh, J., Löfström, J., (1976) Interpolation Spaces, , Springer-Verlag, Berlin, New Yor