Local well-posedness and blow-up in the energy space for a class of \(L^2\) critical dispersion generalized Benjamin-Ono equations. (English. French summary) Zbl 1230.35102
Summary: We consider a family of dispersion generalized Benjamin-Ono equations (dgBO)
\[
u_t - \partial_x |D|^\alpha u + |u|^{2\alpha} \partial_x u = 0, \qquad (t,x) \in {\mathbb R} \times {\mathbb R},
\]
where \(\widehat{|D|^\alpha u} = |\xi|^\alpha\widehat{u}\) and \(1\leq\alpha \leq 2\). These equations are critical with respect to the \(L^{2}\) norm and global existence and interpolate between the modified BO equation (\(\alpha =1\)) and the critical gKdV equation (\(\alpha =2\)).
First, we prove local well-posedness in the energy space for \(1<\alpha <2\), extending results in [C.E. Kenig, {G. Ponce} and L. Vega, J. Am. Math. Soc. 4, No. 2, 323–347 (1991; Zbl 0737.35102); Commun. Pure Appl. Math. 46, No. 4, 527–620 (1993; Zbl 0808.35128)] and for the generalized KdV equations.
Second, we address the blow-up problem in the spirit of Y. Martel and F. Merle [J. Math. Pures Appl. (9) 79, No. 4, 339–425 (2000; Zbl 0963.37058)] and F. Merle [J. Am. Math. Soc. 14, No. 3, 555–578 (2001; Zbl 0970.35128)] concerning the critical gKdV equation, by studying rigidity properties of the dgBO flow in a neighborhood of the solitons. We prove that for \(\alpha\) close to 2, solutions of negative energy close to solitons blow up in finite or infinite time in the energy space \(H^{\alpha/2}\).
The blow-up proof requires both extensions to dgBO of monotonicity results for local \(L^{2}\) norms by pseudo-differential operator tools and perturbative arguments close to the gKdV case to obtain structural properties of the linearized flow around solitons.
First, we prove local well-posedness in the energy space for \(1<\alpha <2\), extending results in [C.E. Kenig, {G. Ponce} and L. Vega, J. Am. Math. Soc. 4, No. 2, 323–347 (1991; Zbl 0737.35102); Commun. Pure Appl. Math. 46, No. 4, 527–620 (1993; Zbl 0808.35128)] and for the generalized KdV equations.
Second, we address the blow-up problem in the spirit of Y. Martel and F. Merle [J. Math. Pures Appl. (9) 79, No. 4, 339–425 (2000; Zbl 0963.37058)] and F. Merle [J. Am. Math. Soc. 14, No. 3, 555–578 (2001; Zbl 0970.35128)] concerning the critical gKdV equation, by studying rigidity properties of the dgBO flow in a neighborhood of the solitons. We prove that for \(\alpha\) close to 2, solutions of negative energy close to solitons blow up in finite or infinite time in the energy space \(H^{\alpha/2}\).
The blow-up proof requires both extensions to dgBO of monotonicity results for local \(L^{2}\) norms by pseudo-differential operator tools and perturbative arguments close to the gKdV case to obtain structural properties of the linearized flow around solitons.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 35B44 | Blow-up in context of PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
References:
| [1] | Amick, C. J.; Toland, J. F., Uniqueness and related analytic properties for the Benjamin-Ono equation—a nonlinear Neumann problem in the plane, Acta Math., 167, 107-126 (1991) · Zbl 0755.35108 |
| [2] | Amick, C. J.; Toland, J. F., Uniqueness of Benjaminʼs solitary-wave solution of the Benjamin-Ono equation, IMA J. Appl. Math., 46, 21-28 (1991) · Zbl 0735.35105 |
| [3] | Caffarelli, L.; Silvestre, L., An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32, 1245-1260 (2007) · Zbl 1143.26002 |
| [4] | Cui, S.; Kenig, C. E., Weak continuity of the flow map for the Benjamin-Ono equation on the line, J. Fourier Anal. Appl., 16, 1021-1052 (2010) · Zbl 1223.35272 |
| [5] | Frank, R. L.; Lenzmann, E., Uniqueness and nondegeneracy of ground states for \((- \Delta)^s Q + Q - Q^{\alpha + 1} = 0\) in \(R\), preprint |
| [6] | Gnedenko, B. V.; Kolmogorov, A. N., Limit Distributions for Sums of Independent Random Variables (1954), Addison-Wesley Publishing Company · Zbl 0056.36001 |
| [7] | Goubet, O.; Molinet, L., Global weak attractor for weakly damped nonlinear Schrödinger equations in \(L^2(R)\), Nonlinear Anal., 71, 317-320 (2009) · Zbl 1170.35534 |
| [8] | Halmos, P. R.; Sunder, V. S., Bounded Integral Operators on \(L^2\) Spaces, Ergeb. Math. Grenzgeb., vol. 96 (1978), Springer-Verlag: Springer-Verlag Berlin · Zbl 0389.47001 |
| [9] | Hörmander, Lars, The Analysis of Linear Partial Differential Operators. III. Pseudodifferential Operators, Grundlehren Math. Wiss., vol. 274 (1985), Springer-Verlag: Springer-Verlag Berlin · Zbl 0601.35001 |
| [10] | Kato, T., On the Cauchy problem for the (generalized) Korteweg-de Vries equation, (Studies in Applied Mathematics. Studies in Applied Mathematics, Adv. Math. Suppl. Stud., vol. 8 (1983), Academic Press: Academic Press New York), 93-128 · Zbl 0549.34001 |
| [11] | Kenig, C. E.; Martel, Y., Asymptotic stability of solitons for the Benjamin-Ono equation, Rev. Mat. Iberoam., 25, 909-970 (2009) · Zbl 1247.35133 |
| [12] | Kenig, C. E.; Ponce, G.; Vega, L., Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J., 40, 33-69 (1991) · Zbl 0738.35022 |
| [13] | Kenig, C. E.; Ponce, G.; Vega, L., Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc., 4, 323-347 (1991) · Zbl 0737.35102 |
| [14] | Kenig, C. E.; Ponce, G.; Vega, L., Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46, 527-620 (1993) · Zbl 0808.35128 |
| [15] | Kenig, C. E.; Takaoka, H., Global wellposedness of the modified Benjamin-Ono equation with initial data in \(H^{1 / 2}\), Int. Math. Res. Not. (2006), Art. ID 95702 · Zbl 1140.35386 |
| [16] | Lions, P.-L., The concentration-compactness principle in the calculus of variations: the locally compact case. Parts 1 and 2, Ann. Inst. H. Poincaré Non Linéaire, 1, 109-145 (1984), 223-283 · Zbl 0541.49009 |
| [17] | Martel, Y., Linear problems related to asymptotic stability of solitons of the generalized KdV equations, SIAM J. Math. Anal., 38, 759-781 (2006) · Zbl 1126.35055 |
| [18] | Martel, Y.; Merle, F., Instability of solitons for the critical generalized Korteweg-de Vries equation, Geom. Funct. Anal., 11, 74-123 (2001) · Zbl 0985.35071 |
| [19] | Martel, Y.; Merle, F., A Liouville theorem for the critical generalized Korteweg-de Vries equation, J. Math. Pures Appl., 79, 339-425 (2000) · Zbl 0963.37058 |
| [20] | Martel, Y.; Merle, F., Stability of the blow up profile and lower bounds on the blow up rate for the critical generalized KdV equation, Ann. of Math., 155, 235-280 (2002) · Zbl 1005.35081 |
| [21] | Martel, Y.; Merle, F., Blow up in finite time and dynamics of blow up solutions for the \(L^2\)-critical generalized KdV equation, J. Amer. Math. Soc., 15, 617-664 (2002) · Zbl 0996.35064 |
| [22] | Merle, F., Existence of blow-up solutions in the energy space for the critical generalized Korteweg-de Vries equation, J. Amer. Math. Soc., 14, 555-578 (2001) · Zbl 0970.35128 |
| [23] | Merle, F.; Raphaël, P., On universality of blow up profile for \(L^2\) critical nonlinear Schrödinger equation, Invent. Math., 156, 565-672 (2004) · Zbl 1067.35110 |
| [24] | Merle, F.; Raphaël, P., Sharp lower bound on the blow up rate for critical nonlinear Schrödinger equation, J. Amer. Math. Soc., 19, 37-90 (2006) · Zbl 1075.35077 |
| [25] | Merle, F.; Raphaël, P., Profiles and quantization of the blow up mass for critical nonlinear Schrödinger equation, Comm. Math. Phys., 253, 675-704 (2005) · Zbl 1062.35137 |
| [26] | Molinet, L.; Ribaud, F., Well-posedness results for the generalized Benjamin-Ono equation with small initial data, J. Math. Pures Appl., 83, 277-311 (2004) · Zbl 1084.35094 |
| [27] | Reed, M.; Simon, B., Methods of Modern Mathematical Physics IV. Analysis of Operators (1978), Academic Press: Academic Press New York, San Francisco, London · Zbl 0401.47001 |
| [28] | Weinstein, M. I., Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87, 567-576 (1983) · Zbl 0527.35023 |
| [29] | Weinstein, M. I., Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16, 472-491 (1985) · Zbl 0583.35028 |
| [30] | Weinstein, M. I., Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math., 39, 51-68 (1986) · Zbl 0594.35005 |
| [31] | Weinstein, M. I., Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation, Comm. Partial Differential Equations, 12, 1133-1173 (1987) · Zbl 0657.73040 |
| [32] | Weinstein, M. I., Solitary waves of nonlinear dispersive evolution equations with critical power nonlinearities, J. Differential Equations, 69, 192-203 (1987) · Zbl 0636.35015 |
| [33] | Yamazato, M., Unimodality of infinitely divisible distribution functions of class \(L\), Ann. Probab., 6, 523-531 (1978) · Zbl 0394.60017 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
