We consider the semi-linear beam equation on the d dimensional irrational torus with smooth nonlinearity of order n -- 1 with n $\ge$ 3 and d $\ge$ 2. If $ε$ $\ll$ 1 is the size of the initial datum, we prove that the lifespan T$ε$ of solutions is O($ε$ --A(n--2) --) where A $\not\equiv$ A(d, n) = 1 + 3 d--1 when n is even and A = 1 + 3 d--1 + max(4--d d--1 , 0) when n is odd. For instance for d = 2 and n = 3 (quadratic nonlinearity) we obtain T$ε$ = O($ε$ --6 --), much better than O($ε$ --1), the time given by the local existence theory. The irrationality of the torus makes the set of differences between two eigenvalues of $\sqrt$ $Δ$ 2 + 1 accumulate to zero, facilitating the exchange between the high Fourier modes and complicating the control of the solutions over long times. Our result is obtained by combining a Birkhoff normal form step and a modified energy step.
We prove a bilinear Strichartz type estimate for irrational tori via a decoupling type argument, \cite{bourgain2014proof}, recovering and generalizing the result of \cite{de2006global}. As a corollary, we derive a global well-posedness result for the cubic defocusing NLS on two dimensional irrational tori with data of infinite energy.
We revisit the work of Bourgain on the invariance of the Gibbs measure for the cubic, defocusing nonlinear Schrödinger equation in 2D on a square torus, and we prove the equivalent result on any tori.
A characteristic of the defocusing cubic nonlinear Schrödinger equation (NLSE), when defined so that the space variable is the multi-dimensional square (hence rational) torus, is that there exist solutions that start with arbitrarily small norms Sobolev norms and evolve to develop arbitrarily large modes at later times; this phenomenon is recognized as a weak energy transfer to high modes for the NLSE. In this paper, we show that when the system is considered on an irrational torus, energy transfer is more difficult to detect.
We consider the quantum hydrodynamic system on a $d$-dimensional irrational torus with $d=2,3$. We discuss the behaviour, over a "non trivial" time interval, of the $H^s$-Sobolev norms of solutions. More precisely we prove that, for generic irrational tori, the solutions, evolving from $\varepsilon$-small initial conditions, remain bounded in $H^s$ for a time scale of order $O(\varepsilon^{-1-1/(d-1)+})$, which is strictly larger with respect to the time-scale provided by local theory. We exploit a Madelung transformation to rewrite the system as a nonlinear Schrödinger equation. We therefore …
We consider the cubic nonlinear Schrödinger equation on $2$-dimensional irrational tori. We construct solutions which undergo growth of Sobolev norms. More concretely, for every $s>0$, $s\neq 1$ and almost every choice of spatial periods we construct solutions whose $H^s$ Sobolev norms grow by any prescribed factor. Moreover, for a set of spatial periods with positive Hausdorff dimension we construct solutions whose Sobolev norms go from arbitrarily small to arbitrarily large. We also provide estimates for the time needed to undergo the norm explosion. Note that the irrationality of the space …
In this paper we prove a result of almost global existence for some abstract nonlinear PDEs on flat tori and apply it to some concrete equations, namely a nonlinear Schrödinger equation with a convolution potential, a beam equation and a quantum hydrodinamical equation. We also apply it to the stability of plane waves in NLS. The main point is that the abstract result is based on a nonresonance condition much weaker than the usual ones, which rely on the celebrated Bourgain's Lemma which provides a partition of the "resonant sites" of the Laplace operator on irrational tori.
