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Qidi Zhang

Publications and source records attributed to Qidi Zhang.

11 recordsLinked to original sources
arXiv2025

Finite-time singularity formations for the Landau-Lifshitz-Gilbert equation in dimension two

We construct finite time blow-up solutions to the Landau-Lifshitz-Gilbert equation (LLG) from ${\mathbb R}^2$ into $S^2$ \begin{equation*} \begin{cases} u_t= a(Δu+|\nabla u|^2u) -b u\wedge Δu &\ \mbox{ in }\ {\mathbb R}^2\times(0,T), u(\cdot,0) = u_0\in S^2 &\ \mbox{ in }\ {\mathbb R}^2, \end{cases} \end{equation*} where $a^2+b^2=1,~a > 0,~ b\in {\mathbb R}$. Given any prescribed $N$ points in $\mathbb{R}^2$ and small $T>0$, we prove that there exists regular initial data such that the solution blows up precisely at these points at finite time $t=T$, taking around each point the profile of sharply scaled degree 1 harmonic map with the type II blow-up speed \begin{equation*} \| \nabla u\|_{L^\infty } \sim \frac{|\ln(T-t)|^2}{ T-t } \ \mbox{ as } \ t\to T. \end{equation*} The proof is based on the {\em parabolic inner-outer gluing method}, developed in \cite{17HMF} for Harmonic Map Flow (HMF). However, a direct consequence of the presence of dispersion is the {\em lack of maximum principle} for suitable quantities, which makes the analysis more delicate even at the linearized level. To overcome this difficulty, we make use of two key technical ingredients: first, for the inner problem we employ the tool of {\em distorted Fourier transform}, as developed by Krieger, Miao, Schlag and Tataru \cite{Krieger09Duke,KMS20WM}. Second, the linear theory for the outer problem is achieved by means of the sub-Gaussian estimate for the fundamental solution of parabolic system in non-divergence form with coefficients of Dini mean oscillation in space ($\mathsf{DMO_x}$), which was proved by Dong, Kim and Lee \cite{dong22-non-divergence} recently.

math.AP↗
arXiv2024

Seed-TTS: A Family of High-Quality Versatile Speech Generation Models

We introduce Seed-TTS, a family of large-scale autoregressive text-to-speech (TTS) models capable of generating speech that is virtually indistinguishable from human speech. Seed-TTS serves as a foundation model for speech generation and excels in speech in-context learning, achieving performance in speaker similarity and naturalness that matches ground truth human speech in both objective and subjective evaluations. With fine-tuning, we achieve even higher subjective scores across these metrics. Seed-TTS offers superior controllability over various speech attributes such as emotion and is capable of generating highly expressive and diverse speech for speakers in the wild. Furthermore, we propose a self-distillation method for speech factorization, as well as a reinforcement learning approach to enhance model robustness, speaker similarity, and controllability. We additionally present a non-autoregressive (NAR) variant of the Seed-TTS model, named $\text{Seed-TTS}_\text{DiT}$, which utilizes a fully diffusion-based architecture. Unlike previous NAR-based TTS systems, $\text{Seed-TTS}_\text{DiT}$ does not depend on pre-estimated phoneme durations and performs speech generation through end-to-end processing. We demonstrate that this variant achieves comparable performance to the language model-based variant and showcase its effectiveness in speech editing. We encourage readers to listen to demos at \url{https://bytedancespeech.github.io/seedtts_tech_report}.

eess.AS↗
arXiv2024

Co-existence of Type II blow-ups with multiple blow-up rates for five-dimensional heat equation with critical nonlinear boundary conditions

We consider the following five-dimensional heat equation with critical boundary condition \begin{equation*} \partial_t u=Δu \mbox{ \ in \ } \mathbb{R}_+^5\times (0,T) , \quad -\partial_{x_5}u =|u|^\frac{2}{3}u \mbox{ \ on \ } \pp \mathbb{R}^5_+ \times (0,T) . \end{equation*} Given $\mathfrak{o}$ distinct boundary points $q^{[i]} \in \partial \mathbb{R}_+^5$, and $\mathfrak{o}$ integers $l_i\in \mathbb{N}$ (possibly duplicated), $i=1,2,\dots, \mathfrak{o}$, for $T>0$ sufficiently small, we construct a finite-time blow-up solution $u$ with a type II blow-up rate $(T-t)^{-3l_i -3}$ for $x$ near $q^{[i]}$. This seems to be the first result of the co-existence of type II blowups with different blow-up rates. To accommodate highly unstable blowups with different blowup rates, we first develop a unified linear theory for the inner problem with more time decay in the blow-up scheme through restriction on the spatial growth of the right-hand side, and then use vanishing adjustment functions for deriving multiple rates at distinct points. This paper is inspired by [25, 52, 60].

math.AP↗
arXiv2023

Long-time dynamics for the energy critical heat equation in $R^5$

We investigate the long-time behavior of global solutions to the energy critical heat equation in $R^5$ \begin{equation*} \begin{cases} \pp_t u=Δu+|u|^{\frac{4}{3}} u ~&\mbox{ in }~ R^5 \times (t_0,\infty), u(\cdot,t_0)=u_0~&\mbox{ in }~ R^5. \end{cases} \end{equation*} For $t_0$ sufficiently large, we show the existence of positive solutions for a class of initial value $u_0(x)\sim |x|^{-γ}$ as $|x|\rightarrow \infty$ with $γ>\frac32$ such that the global solutions behave asymptotically \begin{equation*} \| u(\cdot,t) \|_{L^\infty (\R^5)} \sim \begin{cases} t^{-\frac{3(2-γ)}{2}} ~&\mbox{ if }~ \frac32<γ<2 (\ln t)^{-3} ~&\mbox{ if }~ γ=2 1 ~&\mbox{ if }~ γ>2 \end{cases} \mbox{ \ for \ } t >t_0, \end{equation*} which is slower than the self-similar time decay $t^{-\frac{3}{4}}$. These rates are inspired by Fila-King \cite[Conjecture 1.1]{FilaKing12}.

math.AP↗
arXiv2023

Trichotomy dynamics of the 1-equivariant harmonic map flow

For the 1-equivariant harmonic map flow from $ R^2$ into $S^2$ \begin{equation*} \left\{ \begin{aligned} &v_t=v_{rr}+\frac{v_r}{r} - \frac{\sin(2v)}{2r^2} , ~\quad(r,t)\in R_+\times (t_0,+\infty),\\ &v(r,t_0)=v_0, \qquad\qquad\qquad\quad r\in R_+, \end{aligned} \right. \end{equation*} we construct global growing, bounded and decaying solutions with the initial data $v_0(r)$ satisfying $$v_0(0)=π~\mbox{ and }~ v_0(r)\sim r^{1-γ} ~\mbox{ as }~ r\to+\infty, \quad γ>1.$$ These global solutions exhibit the following trichotomy long-time asymptotic behavior \begin{equation*} \| v_r(\cdot,t) \|_{L^\infty ([0,\infty))} \sim \begin{cases} t^{\frac{γ-2}{2}}\ln t ~&\mbox{ if }~ 1<γ<2,\\ 1 ~&\mbox{ if }~ γ=2,\\ \ln t ~&\mbox{ if }~ γ>2,\\ \end{cases} ~\mbox{ as }~ t\to +\infty. \end{equation*}

math.AP↗
arXiv2022

On Fila-King Conjecture in Dimension Four

We consider the following Cauchy problem for the four-dimensional energy critical heat equation \begin{equation*} \begin{cases} u_t=Δu+u^{3} ~&\mbox{ in }~ {\mathbb R}^4 \times (0,\infty),\\ u(x,0)=u_0(x) ~&\mbox{ in }~ {\mathbb R}^4. \end{cases} \end{equation*} We construct a positive infinite time blow-up solution $u(x,t)$ with the blow-up rate $ \| u(\cdot,t)\|_{L^\infty({\mathbb R}^4)} \sim \ln t$ as $t\to \infty$ and show the stability of the infinite time blow-up. This gives a rigorous proof of a conjecture by Fila and King \cite[Conjecture 1.1]{filaking12}.

math.AP↗
arXiv2020

Sub-solutions and a point-wise Hopf's Lemma for Fractional p-Laplacian

We prove a Hopf's lemma in the point-wise sense. The essential technique is to prove $(-Δ)^s_p u(x)$ is uniformly bounded in the unit ball $B_1\subset\mathbb{R}^n$, where $u(x)=(1-|x|^2)^s_{+}$. Also we study the global Hölder continuity of bounded positive solutions for $(-Δ)^s_p u(x)=f(x,u).$

math.AP↗
arXiv2020

Type II Finite time blow-up for the three dimensional energy critical heat equation

We consider the following Cauchy problem for three dimensional energy critical heat equation \begin{equation*} \begin{cases} u_t=Δu+u^{5},~&\mbox{ in } \ {\mathbb R}^3 \times (0,T),\\ u(x,0)=u_0(x),~&\mbox{ in } \ {\mathbb R}^3. \end{cases} \end{equation*} We construct type II finite time blow-up solution $u(x,t)$ with the blow-up rates $ \| u\|_{L^\infty} \sim (T-t)^{-k}$, where $ k=1,2,... $. This gives a rigorous proof of the formal computations by Filippas, Herrero and Velazquez \cite{fhv}. This is the first instance of type II finite time blow-up for three dimensional energy critical heat equation.

math.AP↗
arXiv2015

Global existence of small equivariant wave maps on rotationally symmetric manifolds

We introduce a class of rotationally invariant manifolds, which we call \emph{admissible}, on which the wave flow satisfies smoothing and Strichartz estimates. We deduce the global existence of equivariant wave maps from admissible manifolds to general targets, for small initial data of critical regularity $H^{\frac n2}$. The class of admissible manifolds includes in particular asymptotically flat manifolds and perturbations of real hyperbolic spaces $\mathbb{H}^{n}$ for $n\ge3$.

math.AP↗
arXiv2008

Long-time existence for semi-linear Klein-Gordon equations with quadratic potential

We prove that small smooth solutions of semi-linear Klein-Gordon equations with quadratic potential exist over a longer interval than the one given by local existence theory, for almost every value of mass. We use normal form for the Sobolev energy. The difficulty in comparison with some similar results on the sphere comes from the fact that two successive eigenvalues $λ, λ'$ of $\sqrt{-Δ+|x|^2}$ may be separated by a distance as small as $\frac{1}{\sqrtλ}$.

math.AP↗