On a stochastic Hardy–Littlewood–Sobolev inequality with application to Strichartz estimates for a noisy dispersion
[Sur une inégalité de Hardy–Littlewood–Sobolev stochastique avec une application aux inégalités de Strichartz pour des dispersions bruitées]
Duboscq, Romain1 ; Réveillac, Anthony1
1 INSA Toulouse & Institut de Mathématiques de Toulouse, IMT / UMR CNRS 5219, 135 avenue de Rangueil, 31077 Toulouse Cedex 4 (France)
Annales Henri Lebesgue, Tome 5 (2022), pp. 263-274.

Dans ce papier, nous nous intéressons á une inégalité de Hardy–Littlewood–Sobolev stochastique. Étant donné la nature non-homogène du le potentiel dans l’inégalité, nous montrons qu’une constante proportionnelle á la longueur de l’intervalle considéré apparaît dans le membre de droite. Une application directe de ce résultat est d’obtenir des inégalités de Strichartz locales pour des dispersions modulées par un bruit et, ainsi, résoudre le problème de Cauchy pour des équations de Schrödinger non-linéaires critiques.

In this paper, we investigate a stochastic Hardy–Littlewood–Sobolev inequality. Due to the non-homogenous nature of the potential in the inequality, we show that a constant proportional to the length of the interval appears on the right-hand-side. As a direct application, we derive local Strichartz estimates for randomly modulated dispersions and solve the Cauchy problem of the critical nonlinear Schrödinger equation.

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DOI : 10.5802/ahl.122
Classification : 35Q55, 60H15
Mots clés : Stochastic regularization, Stochastic Partial Differential Equations, Nonlinear Schrödinger equation, Hardy–Littlewood–Sobolev inequality
Duboscq, Romain 1 ; Réveillac, Anthony 1

1 INSA Toulouse & Institut de Mathématiques de Toulouse, IMT / UMR CNRS 5219, 135 avenue de Rangueil, 31077 Toulouse Cedex 4 (France) 
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Duboscq, Romain; Réveillac, Anthony. On a stochastic Hardy–Littlewood–Sobolev inequality with application to Strichartz estimates for a noisy dispersion. Annales Henri Lebesgue, Tome 5 (2022), pp. 263-274. doi : 10.5802/ahl.122. http://www.numdam.org/articles/10.5802/ahl.122/

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