arXiv subjects
Publications and source records attributed to Yuzhao Wang.
In a seminal paper (1996), Bourgain proved invariance of the Gibbs measure for the defocusing cubic nonlinear Schrödinger equation on the two-dimensional torus by constructing local-in-time solutions in a probabilistic manner. In this note, we revisit and streamline his argument, using the random tensor estimate developed by Deng, Nahmod, and Yue (2022).
Music generation aims to create music segments that align with human aesthetics based on diverse conditional information. Despite advancements in generating music from specific textual descriptions (e.g., style, genre, instruments), the practical application is still hindered by ordinary users' limited expertise or time to write accurate prompts. To bridge this application gap, this paper introduces MusFlow, a novel multimodal music generation model using Conditional Flow Matching. We employ multiple Multi-Layer Perceptrons (MLPs) to align multimodal conditional information into the audio's CLAP embedding space. Conditional flow matching is trained to reconstruct the compressed Mel-spectrogram in the pretrained VAE latent space guided by aligned feature embedding. MusFlow can generate music from images, story texts, and music captions. To collect data for model training, inspired by multi-agent collaboration, we construct an intelligent data annotation workflow centered around a fine-tuned Qwen2-VL model. Using this workflow, we build a new multimodal music dataset, MMusSet, with each sample containing a quadruple of image, story text, music caption, and music piece. We conduct four sets of experiments: image-to-music, story-to-music, caption-to-music, and multimodal music generation. Experimental results demonstrate that MusFlow can generate high-quality music pieces whether the input conditions are unimodal or multimodal. We hope this work can advance the application of music generation in multimedia field, making music creation more accessible. Our generated samples, code and dataset are available at musflow.github.io.
We study the fractional $Φ^4_3$-measure (with order $α> 1$) and the dynamical problem of its canonical stochastic quantization: the three-dimensional stochastic damped fractional nonlinear wave equation with a cubic nonlinearity, also called the fractional hyperbolic $Φ^4_3$-model. We first construct the fractional $Φ^4_3$-measure via the variational approach by Barashkov-Gubinelli (2020). When $α\leq \frac{9}{8}$, this fractional $Φ^4_3$-measure turns out to be singular with respect to the base Gaussian measure. We then prove almost sure global well-posedness of the fractional hyperbolic $Φ^4_3$-model and invariance of the fractional $Φ^4_3$-measure for all $α> 1$ by further developing the globalization framework due to Oh-Okamoto-Tolomeo (2024) on the hyperbolic $Φ^3_3$-model. Furthermore, when $α> \frac{9}{8}$ , we prove weak universality of the fractional hyperbolic $Φ^4_3$-model by utilizing the convergence of Gibbs measures.
We consider the ill-posedness issue for the cubic nonlinear heat equation and prove norm inflation with infinite loss of regularity in the Hölder-Besov space $\mathcal C^s = B^{s}_{\infty, \infty}$ for $ s \le -\frac 23$. In particular, our result includes the subcritical range $-1< s \le -\frac 23$, which is above the scaling critical regularity $s = -1$ with respect to the Hölder-Besov scale. In view of the well-posedness result in $\mathcal C^s$, $s > -\frac 23$, our ill-posedness result is sharp.
We study the Boltzmann equation with the constant collision kernel in the case of spatially periodic domain $\mathbb{T}^d$, $d\geq 2$. Using the existing techniques from nonlinear dispersive PDEs, we prove the local well-posedness result in $L^{2,r}_vH^s_x$ for $s>\frac{d}{2}-\frac{1}{4}$ and $r>\frac{d}{2}$. To reach the result, the main tool we establish is the $L^4$ Strichartz estimate for solutions to the corresponding linear equation.
We consider the Cauchy problem for the one-dimensional periodic cubic nonlinear fractional Schr{ö}dinger equation (FNLS) with initial data distributed via its associated Gibbs measure. We construct global strong solutions with the flow property for the FNLS on the support of the Gibbs measure in the full dispersive range, thus resolving a question proposed by Sun-Tzvetkov (2021). As a byproduct, we prove the invariance of the Gibbs measure and almost sure global well-posedness for FNLS.
We study the construction of the Gibbs measures for the {\it focusing} mass-critical fractional nonlinear Schrödinger equation on the multi-dimensional torus. We identify the sharp mass threshold for normalizability and non-normalizability of the focusing Gibbs measures, which generalizes the influential works of Lebowitz-Rose-Speer (1988), Bourgain (1994), and Oh-Sosoe-Tolomeo (2021) on the one-dimensional nonlinear Schrödinger equations. To this purpose, we establish an almost sharp fractional Gagliardo-Nirenberg-Sobolev inequality on the torus, which is of independent interest.
In this paper, we study the Gibbs measures associated to the focusing nonlinear Schrödinger equation with harmonic potential on Euclidean spaces. We establish a dichotomy for normalizability vs non-normalizability in the one dimensional case, and under radial assumption in the higher dimensional cases. In particular, we complete the programs of constructing Gibbs measures in the presence of a harmonic potential initiated by Burq-Thomann-Tzvetkov (2005) in dimension one and Deng (2013) in dimension two with radial assumption.
In this note, we study the ill-posedness problem for the derivative nonlinear Schrödinger equation (DNLS) in the one-dimensional setting. More precisely, by using a ternary-quinary tree expansion of the Duhamel formula we prove norm inflation in Sobolev spaces below the (scaling) critical regularity for the gauged DNLS. This ill-posedness result is sharp since DNLS is known to be globally well-posed in $L^2(\mathbb{R})$. The main novelty of our approach is to control the derivative loss from the cubic nonlinearity by the quintic nonlinearity with carefully chosen 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.
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 $1\leq p \leq \frac32$.
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 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.
In this paper, we first introduce a new function space $MH^{θ, p}$ whose norm is given by the $\ell^p$-sum of modulated $H^θ$-norms of a given function. In particular, when $θ< -\frac 12$, we show that the space $MH^{θ, p}$ agrees with the modulation space $M^{2, p}(\mathbb R)$ on the real line and the Fourier-Lebesgue space $\mathcal F L^{p}(\mathbb T)$ on the circle. We use this equivalence of the norms and the Galilean symmetry to adapt the conserved quantities constructed by Killip-Vişan-Zhang to the modulation space setting. By applying the scaling symmetry, we then prove global well-posedness of the one-dimensional cubic nonlinear Schrödinger equation (NLS) in almost critical spaces. More precisely, we show that the cubic NLS on $\mathbb R$ is globally well-posed in $M^{2, p}(\mathbb R)$ for any $p < \infty$, while the renormalized cubic NLS on $\mathbb T$ is globally well-posed in $\mathcal FL^p(\mathbb T)$ for any $p < \infty$. In Appendix, we also establish analogous global-in-time bounds for the modified KdV equation (mKdV) in the modulation spaces on the real line and in the Fourier-Lebesgue spaces on the circle. An additional key ingredient of the proof in this case is a Galilean transform which converts the mKdV to the mKdV-NLS equation.
We study the two-dimensional stochastic sine-Gordon equation (SSG) in the hyperbolic setting. In particular, by introducing a suitable time-dependent renormalization for the relevant imaginary multiplicative Gaussian chaos, we prove local well-posedness of SSG for any value of a parameter $β^2 > 0$ in the nonlinearity. This exhibits sharp contrast with the parabolic case studied by Hairer and Shen (2016) and Chandra, Hairer, and Shen (2018), where the parameter is restricted to the subcritical range: $0 < β^2 < 8 π$. We also present a triviality result for the unrenormalized SSG.
In this note, we study the hyperbolic stochastic damped sine-Gordon equation (SdSG), with a parameter $β^2 > 0$, and its associated Gibbs dynamics on the two-dimensional torus. After introducing a suitable renormalization, we first construct the Gibbs measure in the range $0<β^2<4π$ via the variational approach due to Barashkov-Gubinelli (2018). We then prove almost sure global well-posedness and invariance of the Gibbs measure under the hyperbolic SdSG dynamics in the range $0<β^2<2π$. Our construction of the Gibbs measure also yields almost sure global well-posedness and invariance of the Gibbs measure for the parabolic sine-Gordon model in the range $0<β^2<4π$.
We study the stochastic cubic nonlinear Schrödinger equation (SNLS) with an additive noise on the one-dimensional torus. In particular, we prove local well-posedness of the (renormalized) SNLS when the noise is almost space-time white noise. We also discuss a notion of criticality in this stochastic context, comparing the situation with the stochastic cubic heat equation (also known as the stochastic quantization equation).
We construct global-in-time singular dynamics for the (renormalized) cubic fourth order nonlinear Schrödinger equation on the circle, having the white noise measure as an invariant measure. For this purpose, we introduce the "random-resonant / nonlinear decomposition", which allows us to single out the singular component of the solution. Unlike the classical McKean, Bourgain, Da Prato-Debussche type argument, this singular component is nonlinear, consisting of arbitrarily high powers of the random initial data. We also employ a random gauge transform, leading to random Fourier restriction norm spaces. For this problem, a contraction argument does not work and we instead establish convergence of smooth approximating solutions by studying the partially iterated Duhamel formulation under the random gauge transform. We reduce the crucial nonlinear estimates to boundedness properties of certain random multilinear functionals of the white noise.