arXiv subjects
Publications and source records attributed to Tadahiro Oh.
We discuss boundedness properties of certain classes of discrete bilinear operators that are similar to those of the continuous bilinear pseudodifferential operators with symbols in the Hörmander classes $BS^ω_{1, 0}$. In particular, we investigate their relation to discrete analogues of the bilinear Calderón-Zygmund singular integral operators and show compactness of their commutators.
We study the Cauchy problem for the nonlinear wave equations (NLW) with random data and/or stochastic forcing on a two-dimensional compact Riemannian manifold without boundary. (i) We first study the defocusing stochastic damped NLW driven by additive space-time white-noise, and with initial data distributed according to the Gibbs measure. By introducing a suitable space-dependent renormalization, we prove local well-posedness of the renormalized equation. Bourgain's invariant measure argument then allows us to establish almost sure global well-posedness and invariance of the Gibbs measure for the renormalized stochastic damped NLW. (ii) Similarly, we study the random data defocusing NLW (without stochastic forcing), and establish the same results as in the previous setting. (iii) Lastly, we study the stochastic NLW without damping. By introducing a space-time dependent renormalization, we prove its local well-posedness with deterministic initial data in all subcritical spaces. These results extend the corresponding recent results on the two-dimensional torus obtained by (i) Gubinelli-Koch-Oh-Tolomeo (2018), (ii) Oh-Thomann (2017), and (iii) Gubinelli-Koch-Oh (2018), to a general class of compact manifolds. The main ingredient is the Green's function estimate for the Laplace-Beltrami operator in this setting to study regularity properties of stochastic terms appearing in each of the problems.
We study the nonlinear Schrödinger equation (NLS) with the quadratic nonlinearity $|u|^2$, posed on the two-dimensional torus $\mathbb{T}^2$. While the relevant $L^3$-Strichartz estimate is known only with a derivative loss, we prove local well-posedness of the quadratic NLS in $L^2(\mathbb{T}^2)$, thus resolving an open problem of thirty years since Bourgain (1993). In view of ill-posedness in negative Sobolev spaces, this result is sharp. We establish a crucial bilinear estimate by separately studying the non-resonant and nearly resonant cases. As a corollary, we obtain a tri-linear version of the $L^3$-Strichartz estimate without any derivative loss.
We prove probabilistic well-posedness for a 2D viscous nonlinear wave equation modeling fluid-structure interaction between a 3D incompressible, viscous Stokes flow and nonlinear elastodynamics of a 2D stretched membrane. The focus is on (rough) data, often arising in real-life problems, for which it is known that the deterministic problem is ill-posed. We show that random perturbations of such data give rise almost surely to the existence of a unique solution. More specifically, we prove almost sure global well-posedness for a viscous nonlinear wave equation with the subcritical initial data in the Sobolev space $\mathcal{H}^s (\mathbb{R}^2)$, $s > - \frac 15$, which are randomly perturbed using Wiener randomization. This result shows "robustness" of nonlinear FSI problems/models, and provides confidence that even for the "rough data" (data in $\mathcal{H}^s$, $s > -\frac 1 5$) random perturbations of such data (due to e.g., randomness in real-life data, numerical discretization, etc.) will almost surely provide a unique solution which depends continuously on the data in the $\mathcal{H}^s$ topology.
We study the stochastic viscous nonlinear wave equations (SvNLW) on $\mathbb T^2$, forced by a fractional derivative of the space-time white noise $ξ$. In particular, we consider SvNLW with the singular additive forcing $D^\frac{1}{2}ξ$ such that solutions are expected to be merely distributions. By introducing an appropriate renormalization, we prove local well-posedness of SvNLW. By establishing an energy bound via a Yudovich-type argument, we also prove global well-posedness of the defocusing cubic SvNLW. Lastly, in the defocusing case, we prove almost sure global well-posedness of SvNLW with respect to certain Gaussian random initial data.
We study the stochastic cubic nonlinear wave equation (SNLW) with an additive noise on the three-dimensional torus $\mathbb{T}^3$. In particular, we prove local well-posedness of the (renormalized) SNLW when the noise is almost a space-time white noise. In recent years, the paracontrolled calculus has played a crucial role in the well-posedness study of singular SNLW on $\mathbb{T}^3$ by Gubinelli, Koch, and the first author (2018), Okamoto, Tolomeo, and the first author (2020), and Bringmann (2020). Our approach, however, does not rely on the paracontrolled calculus. We instead proceed with the second order expansion and study the resulting equation for the residual term, using multilinear dispersive smoothing.
We study an optimal mass threshold for normalizability of the Gibbs measures associated with the focusing mass-critical nonlinear Schrödinger equation on the one-dimensional torus. In an influential paper, Lebowitz, Rose, and Speer (1988) proposed a critical mass threshold given by the mass of the ground state on the real line. We provide a proof for the optimality of this critical mass threshold. The proof also applies to the two-dimensional radial problem posed on the unit disc. In this case, we answer a question posed by Bourgain and Bulut (2014) on the optimal mass threshold. Furthermore, in the one-dimensional case, we show that the Gibbs measure is indeed normalizable at the optimal mass threshold, thus answering an open question posed by Lebowitz, Rose, and Speer (1988). This normalizability at the optimal mass threshold is rather striking in view of the minimal mass blowup solution for the focusing quintic nonlinear Schrödinger equation on the one-dimensional torus.
We continue the study on the transport properties of the Gaussian measures on Sobolev spaces under the dynamics of the cubic fourth order nonlinear Schrödinger equation. By considering the renormalized equation, we extend the quasi-invariance results in [30, 27] to Sobolev spaces of negative regularity. Our proof combines the approach introduced by Planchon, Tzvetkov, and Visciglia [35] with the normal form approach in [30, 27].
We prove quasi-invariance of Gaussian measures supported on Sobolev spaces under the dynamics of the three-dimensional defocusing cubic nonlinear wave equation. As in the previous work on the two-dimensional case, we employ a simultaneous renormalization on the energy functional and its time derivative. Two new ingredients in the three-dimensional case are (i) the construction of the weighted Gaussian measures, based on a variational formula for the partition function inspired by Barashkov and Gubinelli (2018), and (ii) an improved argument in controlling the growth of the truncated weighted Gaussian measures, where we combine a deterministic growth bound of solutions with stochastic estimates on random distributions.
In this note, we consider the ill-posedness issue for the cubic nonlinear Schrödinger equation. In particular, we prove norm inflation based at every initial condition in negative Sobolev spaces below or at the scaling critical regularity.
In this paper, we study the one-dimensional cubic nonlinear Schrödinger equation (NLS) on the circle. In particular, we develop a normal form approach to study NLS in almost critical Fourier-Lebesgue spaces. By applying an infinite iteration of normal form reductions introduced by the first author with Z. Guo and S. Kwon (2013), we derive a normal form equation which is equivalent to the renormalized cubic NLS for regular solutions. For rough functions, the normal form equation behaves better than the renormalized cubic NLS, thus providing a further renormalization of the cubic NLS. We then prove that this normal form equation is unconditionally globally well-posed in the Fourier-Lebesgue spaces $\mathcal{F} L^p(\mathbb{T})$, $1 \leq p < \infty$. By inverting the transformation, we conclude global well-posedness of the renormalized cubic NLS in almost critical Fourier-Lebesgue spaces in a suitable sense. This approach also allows us to prove unconditional uniqueness of the (renormalized) cubic NLS in $\mathcal{F} L^p(\mathbb{T})$ for $1\leq p \leq \frac32$.
Using ideas from paracontrolled calculus, we prove local well-posedness of a renormalized version of the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity forced by an additive space-time white noise on a periodic domain. There are two new ingredients as compared to the parabolic setting. (i) In constructing stochastic objects, we have to carefully exploit dispersion at a multilinear level. (ii) We introduce novel random operators and leverage their regularity to overcome the lack of smoothing of usual paradifferential commutators.
We study the two-dimensional stochastic nonlinear heat equation (SNLH) and stochastic damped nonlinear wave equation (SdNLW) with an exponential nonlinearity $λβe^{βu }$, forced by an additive space-time white noise. We prove local and global well-posedness of these equations, depending on the sign of $λ$ and the size of $β^2 > 0$, and invariance of the associated Gibbs measures. See the abstract of the paper for a more precise abstract. (Due to the limit on the number of characters for an abstract set by arXiv, the full abstract can not be displayed here.)
We study global-in-time dynamics of the stochastic nonlinear wave equations (SNLW) with an additive space-time white noise forcing, posed on the two-dimensional torus. Our goal in this paper is two-fold. (i) By introducing a hybrid argument, combining the $I$-method in the stochastic setting with a Gronwall-type argument, we first prove global well-posedness of the (renormalized) cubic SNLW in the defocusing case. Our argument yields a double exponential growth bound on the Sobolev norm of a solution. (ii) We then study the stochastic damped nonlinear wave equations (SdNLW) in the defocusing case. In particular, by applying Bourgain's invariant measure argument, we prove almost sure global well-posedness of the (renormalized) defocusing SdNLW with respect to the Gibbs measure and invariance of the Gibbs measure.
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 $α= \frac 12$ 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 $0 < α< \frac 12$. We first revisit this argument and establish multilinear smoothing of order $\frac 14$ 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 $α\geq \frac 12$, 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 $α\ge \frac 12$. (ii) As for SNLH, we establish analogous results with a threshold given by $α= 1$. These 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 ($α= \frac 34$ 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).
We study the three-dimensional cubic nonlinear wave equation (NLW) with random initial data below $L^2(\mathbb{T}^3)$. By considering the second order expansion in terms of the random linear solution, we prove almost sure local well-posedness of the renormalized NLW in negative Sobolev spaces. We also prove a new instability result for the defocusing cubic NLW without renormalization in negative Sobolev spaces, which is in the spirit of the so-called triviality in the study of stochastic partial differential equations. More precisely, by studying (un-renormalized) NLW with given smooth deterministic initial data plus a certain truncated random initial data, we show that, as the truncation is removed, the solutions converge to $0$ in the distributional sense for any deterministic initial data.
In this paper, we present a globalization argument for stochastic nonlinear dispersive PDEs with additive noises by adapting the $I$-method (= the method of almost conservation laws) to the stochastic setting. As a model example, we consider the defocusing stochastic cubic nonlinear Schrödinger equation (SNLS) on $\mathbb R^3$ with additive stochastic forcing, white in time and correlated in space, such that the noise lies below the energy space. By combining the $I$-method with Ito's lemma and a stopping time argument, we construct global-in-time dynamics for SNLS below the energy space.
We study a nonlinear stochastic heat equation forced by a space-time white noise on closed surfaces, with nonlinearity $e^{βu}$. This equation corresponds to the stochastic quantization of the Liouville quantum gravity (LQG) measure. (i) We first revisit the construction of the LQG measure in Liouville conformal field theory (LCFT) in the $L^2$ regime $0<β<\sqrt{2}$. This uniformizes in this regime the approaches of David-Kupiainen-Rhodes-Vargas (2016), David-Rhodes-Vargas (2016) and Guillarmou-Rhodes-Vargas (2019) which treated the case of a closed surface with genus 0, 1 and $> 1$ respectively. Moreover, our argument shows that this measure is independent of the approximation procedure for a large class of smooth approximations. (ii) We prove almost sure global well-posedness of the parabolic stochastic dynamics, and invariance of the measure under this stochastic flow. In particular, our results improve previous results obtained by Garban (2020) in the cases of the sphere and the torus with their canonical metric, and are new in the case of closed surfaces with higher genus.