We consider the Cauchy problem for the nonlinear Schrödinger equations (NLS) with non-algebraic nonlinearities on the Euclidean space. In particular, we study the energy-critical NLS on R d , d = 5, 6, and energy-critical NLS without gauge invariance and prove that they are almost surely locally well-posed with respect to randomized initial data below the energy space. We also study the long time behavior of solutions to these equations: (i) we prove almost sure global well-posedness of the (standard) energy-critical NLS on R d , d = 5, 6, in the defocusing case, and (ii) we present a probabilistic construction of finite time blowup solutions to the energy-critical NLS without gauge invariance below the energy space. Contents 2010 Mathematics Subject Classification. 35Q55. Key words and phrases. nonlinear Schrödinger equation; almost sure local well-posedness; almost sure global well-posedness; finite time blowup. c T γ φ 2 H s such that for each ω ∈ Ω T , there exists a unique solution, where S(t) = e it∆ and X 1 T is defined in Section 3 below. Almost sure local well-posedness with respect to the Wiener randomization has been studied in the context of the cubic NLS and the quintic NLS on R d [2, 3, 8] which are energy-critical in dimensions 4 and 3, respectively. Note that when d = 5, 6, the energycritical nonlinearity |u| 4 d−2 u is no longer algebraic, presenting a new difficulty in applying the argument in [2,3,8].

We study the two-dimensional stochastic nonlinear wave equation (SNLW) and stochastic nonlinear heat equation (SNLH) with a quadratic nonlinearity, forced by a fractional derivative (of order α > 0) of a space-time white noise. In particular, we show that the well-posedness theory breaks at α = 1 2 for SNLW and at α = 1 for SNLH. This provides a first example showing that SNLW behaves less favorably than SNLH. (i) As for SNLW, Deya (2020) essentially proved its local well-posedness for 0 < α < 1 2 . We first revisit this argument and establish multilinear smoothing of order 1 4 on the second order stochastic term in the spirit of a recent work by Gubinelli, Koch, and Oh (2018). This allows us to simplify the local well-posedness argument for some range of α. On the other hand, when α ≥ 1 2 , we show that SNLW is ill-posed in the sense that the second order stochastic term is not a continuous function of time with values in spatial distributions. This shows that a standard method such as the Da Prato-Debussche trick or its variant, based on a higher order expansion, breaks down for α ≥ 1 2 . (ii) As for SNLH, we establish analogous results with a threshold given byThese examples show that in the case of rough noises, the existing well-posedness theory for singular stochastic PDEs breaks down before reaching the critical values (α = 3 4 in the wave case and α = 2 in the heat case) predicted by the scaling analysis (due to Deng, Nahmod, and Yue (2019) in the wave case and due to Hairer (2014) in the heat case).

In the present paper, we consider the Cauchy problem of fourth order nonlinear Schrödinger type equations with a derivative nonlinearity. In one dimensional case, we prove that the fourth order nonlinear Schrödinger equation with the derivative quartic nonlinearity ∂ x (u 4 ) is the small data global in time well-posed and scattering to a free solution. Furthermore, we show that the same result holds for the d ≥ 2 and derivative polynomial type nonlinearity, for example |∇|(u m ) with (m − 1)d ≥ 4.

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Abstract. We consider ill-posedness of the Cauchy problem for the generalized Boussinesq and Kawahara equations. We prove norm inflation with general initial data, an improvement over the ill-posedness results by Geba et al., Nonlinear Anal. 95 (2014), 404-413 for the generalized Boussinesq equations and by Kato, Adv.

In this note, we study the ill-posedness of nonlinear wave equations (NLW). Namely, we show that NLW experiences norm inflation at every initial data in negative Sobolev spaces. This result covers a gap left open in a paper of Christ, Colliander, and Tao (2003) and extends the result by Oh, Tzvetkov, and the second author (2019) to non-cubic integer nonlinearities. In particular, for some low dimensional cases, we obtain norm inflation above the scaling critical regularity. We also prove ill-posedness for NLW, via norm inflation at general initial data, in negative regularity Fourier-Lebesgue and Fourieramalgam spaces.

We study the Cauchy problem of the damped wave equation ∂ 2 t u − ∆u + ∂tu = 0 and give sharp L p -L q estimates of the solution for 1 ≤ q ≤ p < ∞ (p = 1) with derivative loss. This is an improvement of the so-called Matsumura estimates. Moreover, as its application, we consider the nonlinear problem with initial, s ≥ 0, and β = (n − 1)| 1 2 − 1 r |, and prove the local and global existence of solutions. In particular, we prove the existence of the global solution with small initial data for the critical nonlinearity with the power 1 + 2r n , while it is known that the critical power 1+ 2 n belongs to the blow-up region when r = 1. We also discuss the asymptotic behavior of the global solution in supercritical cases. Moreover, we present blow-up results in subcritical cases. We give estimates of lifespan by an ODE argument.

We consider the Cauchy problem associated with the Chern–Simons–Dirac system in ℝ1+2. Using gauge invariance, we reduce the Chern–Simons–Dirac system to a Dirac equation and we uncover the null structure of this Dirac equation. Next, relying on null structure estimates, we establish that the Cauchy problem associated with this Dirac equation is locally-in-time well-posed in the Sobolev space Hs for all s > 1/4. Our proof uses modified L4-type estimates.