Local well-posedness and blow-up in the energy space for a class of \( L^{2} \) critical dispersion generalized Benjamin–Ono equations

Abstract

We consider a family of dispersion generalized Benjamin-Ono equations (dgBO)where |D| α u = |ξ| α u and 1 ≤ α ≤ 2. These equations are critical with respect to the L 2 norm and global existence and interpolate between the modified BO equation (α = 1) and the critical gKdV equation (α = 2).First, we prove local well-posedness in the energy space for 1 < α < 2, extending results in [19]-[20] for the generalized KdV equations.Second, we address the blow up problem in the spirit of [25,30] concerning the critical gKdV equation, by studying rigidity properties of the (dgBO) flow in a neighborhood of the solitons. We prove that for α close to 2, solutions of negative energy close to solitons blow up in finite or infinite time in the energy space H Ensuite, nousétudions le phénomène d'explosion dans l'esprit de [25,30] concernant l'équation de gKdV critique, enétudiant les propriétés de rigidité du flot de (dgBO) dans un voisinage des solitons. Nous montrons que pour α proche de 2, les solutions d'énergie négative proches des solitons explosent en temps fini ou infini dans l'espace d'énergie H α 2 . La preuve de ce résultat d'explosion nécessite d'une part l'adaptationà (dgBO) de résultats de monotonie de normes L 2 locales par des méthodes d'opérateurs pseudodifferentiels et d'autre part des arguments de perturbation proche du cas (gKdV) pour obtenir des propriétés structurelles du flot linéarisé autour des solitons.2 Local well-posedness in the energy space Proof of Theorem 1We denote the Fourier transform by F(f )(ξ) =f (ξ) = e −ixξ f (x)dx.We introduce the group W α (t) defined by F(W α (t)f )(ξ) = e it(|ξ| α ξ) f (ξ), 1 < α < 2.Then, we claim (or recall) the following linear estimates (we use classical notation from [20]).

Modulational Instability and Variational Structure

“…To proceed, differentiating (101) and (100) with respect to x∂ x + T ∂ T leads, with help of (51) and (37), to that Furthermore, a complex eigenvalue of D implies modulational instability.…”

“…K ) − ( U ) − T 0 u f (u)(xu x + T u T )dx + 2c( P) + a( M) = 0 and T (u)(xu x + T u T )dx − c( M) = 0,respectively. We then use(35) and(51),(37) to calculate thatw 1 L 1 L −1 0 L 1 v 2 = L 1 w 1 , xu x + T u T − G −1 w 1 , xu x + T u T L 1 w 1 , v 1 = M c L 1 u, xu x + T u T − P c L 1 1, xu x + T u T −G −1 (M c u, xu x + T u T − P c 1, xu x + T u T ) L 1 w 1 , v M c (α(α − 1)K − ( (α K − E)) ) (105a) −G −1 (M c ( P) − P c ( M) )(M c (α K a − E a ) − P c T ). Similarly, L 1 L −1 0 L 1 v 2 = L 1 w 3 , xu x + T u T − G −1 w 1 , xu x + T u T L 1 w 3 , v 1 = −M a (α(α − 1)K − ( (α K − E)) ) (105b) + G −1 (M c ( P) − P c ( M) )(M a (α K a − E a ) − P a T ).Note from (99c) that,w 1 , L 2 v 2 = −2α(α + 1)M c K and w 3 , L 2 v 2 = 2α(α + 1)M a K .…”

On the fractional Schrödinger problem with non‐symmetric potential

“…when s is close to 1 in the subcritical range of p. In [38] and [39], this idea was implemented to obtain the uniqueness and nondegeneracy of the ground state solutions for the nonlinear problem (1.10) with s 2 .0, 1/, which settled a conjecture by [10] and generalized a classical result by Amick and Toland [40] on the uniqueness of solitary waves for the Benjamin-Ono equation. Moreover, the formulation (1.8) in terms of local differential operators plays a central role when deriving bounds on the number of sign changes for eigenfunctions of fractional Schrödinger operator.…”

“…After the excellent work of Caffarelli and Silvestre , applying the s ‐harmonic extension to define the fractional Schrödinger operator, Fall and Valdinoci proved the uniqueness and non‐degeneracy of positive solutions of when s is close to 1 in the subcritical range of p . In and , this idea was implemented to obtain the uniqueness and non‐degeneracy of the ground state solutions for the nonlinear problem with s ∈(0,1), which settled a conjecture by and generalized a classical result by Amick and Toland on the uniqueness of solitary waves for the Benjamin–Ono equation. Moreover, the formulation in terms of local differential operators plays a central role when deriving bounds on the number of sign changes for eigenfunctions of fractional Schrödinger operator.…”

“…We will look for solutions to that decay at infinity is the space of all functions such that The fractional Laplacian (−Δ) s u of a function is defined in terms of its Fourier transform by the relation Fractional powers of the Laplacian (−Δ) s is the infinitesimal generators of Lévy stable diffusion processes (see ). It arises in a numerous variety of equations in mathematical physics and related fields, such as phase transitions, flames propagation, chemical reaction in liquids, population dynamics, American options in finance, and crystal dislocation, see, for example, and refe‐ rences therein.…”