Mimi Dai
We consider the Cauchy problem for the electron magnetohydrodynamics model in the supercritical regime. For rough initial data in $\mathcal H^{-s}(\mathbb T^n)$ with $s>0$, we obtain global in time weak solutions almost surely via an appropriate randomization of the initial data.
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We prove the global regularity of the solution pair to the N-dimensional logarithmically supercritical magnetohydrodynamics system with zero diffusivity. This is the endpoint case omitted in the work of [24]; it also improves some previous results logarithmically.
We study the cubic defocusing nonlinear Schrödinger equation on $\mathbb{R}^4$ with supercritical initial data. For randomized initial data in $H^s(\mathbb{R}^4)$, we prove almost sure local wellposedness for $\frac{1}{7} < s < 1$ and almost sure scattering for $\frac{5}{7} < s < 1$. The randomization is based on a unit-scale decomposition in frequency space, a decomposition in the angular variable, and - for the almost sure scattering result - an additional unit-scale decomposition in physical space. We employ new probabilistic estimates for the linear Schrödinger flow with randomized data, w…
In this article we study the defocusing energy-critical nonlinear wave equation on $\mathbb{R}^4$ with scaling supercritical data. We prove almost sure scattering for randomized initial data in $H^s(\mathbb{R}^4) \times H^{s-1}(\mathbb{R}^4)$ with $\frac{5}{6} < s < 1$. The proof relies on new probabilistic estimates for the linear flow of the wave equation with randomized data, where the randomization is based on a unit-scale decomposition in frequency space, a decomposition in the angular variable, and a unit-scale decomposition of physical space. In particular, we show that the solution to …
This paper studies the quintic nonlinear Schrödinger equation on $\mathbb{R}^d$ with randomized initial data below the critical regularity $H^{\frac{d-1}{2}}$. The main result is a proof of almost sure local well-posedness given a Wiener Randomization of the data in $H^s$ for $s \in (\frac{d-2}{2}, \frac{d-1}{2})$. The argument further develops the techniques introduced in the work of Á. Bényi, T. Oh and O. Pocovnicu on the cubic problem. The paper concludes with a condition for almost sure global well-posedness.
Strong ill-posedness for the MHD system in the supercritical regime: inviscid and viscous
该研究针对三维不可压缩MHD方程组，在超临界Sobolev空间中证明了其强不适定性，引入“单磁ansatz”揭示了磁场驱动超临界不适定性的机制，填补了相关理论空白。
We consider the magnetohydrodynamics system with Hall effect accompanied with initial data in supercritical Sobolev space. Via an appropriate randomization of the supercritical initial data, both local and small data global well-posedness for the system are obtained almost surely in critical Sobolev space.
We consider the Schrödinger equations with arbitrary (large) power non-linearity on the three-dimensional torus. We construct non-trivial probability measures supported on Sobolev spaces and show that the equations are globally well-posed on the supports of these measures, respectively. Moreover, these measures are invariant under the flows that are constructed. Therefore, the constructed solutions are recurrent in time.\\ Also, we show \textit{slow growth} control on the time evolution of the solutions. A generalization to any dimension is given. Our proof relies on a new approach combining t…
We prove that for almost every initial data $(u_0,u_1) \in H^s \times H^{s-1}$ with $s > \frac{p-3}{p-1}$ there exists a global weak solution to the supercritical semilinear wave equation $\partial _t^2u - Δu +|u|^{p-1}u=0$ where $p>5$, in both $\mathbb{R}^3$ and $\mathbb{T}^3$. This improves in a probabilistic framework the classical result of Strauss who proved global existence of weak solutions associated to $H^1 \times L^2$ initial data. The proof relies on techniques introduced by T. Oh and O. Pocovnicu based on the pioneer work of N. Burq and N. Tzvetkov. We also improve the global well-…
In this paper, we present two almost sure global well-posedness (GWP) results for the energy supercritical nonlinear Schrödinger equations (NLS) on the unit ball of $\Bbb R^3$ using two different approaches. First, for the NLS with algebraic nonlinearities with the subcritical initial data, we show the almost sure global well-posedness and the invariance of the underlying measures, and establish controls on the growth of Sobolev norms of the solutions. This global result is based on a deterministic local theory and a probabilistic globalization. Second, for the NLS with generic power nonlinear…
In this paper we prove an almost sure local well-posedness result for the periodic 3D quintic nonlinear Schrödinger equation in the supercritical regime, that is below the critical space $H^1(\mathbb T^3)$.
This paper considers the supercritical Navier-Stokes equations posed in the whole space $\R^d$, with suitably randomized initial data, in the weak solution setting. The global weak solutions are constructed for a large set of initial data in $H^{-s}(\R^d)$ for some $s>0$ via a probabilistic argument, and this in turn implies the almost sure existence.
Probabilistic global-wellposedness for the energy-supercritical Schrödinger equations on compact manifolds
针对所有维度下带一般非线性幂次的能量超临界薛定谔方程，在一般紧黎曼流形（含环面、佐尔流形）上构造奇异阶Sobolev空间不变测度，证明解的几乎必然全局适定性及时间增长界。
Time asymptotics, time regularity and separation rates for Navier-Stokes flows in supercritical solution classes
该研究将三维Navier-Stokes方程弱解理论拓展至超临界情形，得到含先验能量界与弱星收敛稳定性的结论，并利用先验界推导出解的局部分离速率上界与高阶时间正则性结果。
Half Strong Ill-Posedness of $2 \frac{1}{2}$D Electron Magnetohydrodynamics with Fractional Resistivity
研究2.5维电子磁流体动力学系统，证明在超临界Sobolev空间中，存在初始数据导致解的范数膨胀。
We consider the Cauchy problem for a generalized KdV equation \begin{eqnarray*} u_{t}+\partial_{x}^{3}u+u^{7}u_{x}=0, \end{eqnarray*} with random data on \R. Kenig, Ponce, Vega(Comm. Pure Appl. Math.46(1993), 527-620)proved that the problem is globally well-posed in H^{s}(\R)$ with s> s_{crit}=\frac{3}{14}, which is the scaling critical regularity indices. Birnir, Kenig, Ponce, Svanstedt, Vega(J. London Math. Soc. 53 (1996), 551-559.) proved that the problem is ill-posed in the sense that the time of existence T and the continuous dependence cannot be expressed in terms of the size of the data…
Thanks to an approach inspired from Burq-Lebeau \cite{bule}, we prove stochastic versions of Strichartz estimates for Schrödinger with harmonic potential. As a consequence, we show that the nonlinear Schrödinger equation with quadratic potential and any polynomial non-linearity is almost surely locally well-posed in $L^{2}(\R^{d})$ for any $d\geq 2$. Then, we show that we can combine this result with the high-low frequency decomposition method of Bourgain to prove a.s. global well-posedness results for the cubic equation: when $d=2$, we prove global well-posedness in $\H^{s}(\R^{2})$ for any $…
Bilinear Strichartz estimates and almost sure global solutions for the nonlinear Schr{ö}dinger equation
本文通过谐振子的 Bourgain 型双线性估计、lens 变换及 Kato 型光滑估计，构造超临界非线性 Schrödinger 方程在随机初值下的几乎必然全局解。
We prove the non-uniqueness of weak solutions to 3D hyper viscous and resistive MHD in the class $L^γ_tW^{s,p}_x$, where the exponents $(s,γ,p)$ lie in two supercritical regimes. The result reveals that the scaling-invariant Ladyženskaja-Prodi-Serrin (LPS) condition is the right criterion to detect non-uniqueness, even in the highly viscous and resistive regime beyond the Lions exponent. In particular, for the classical viscous and resistive MHD, the non-uniqueness is sharp near the endpoint $(0,2,\infty)$ of the LPS condition. Moreover, the constructed weak solutions admit the partial regular…
We extend the convergence method introduced in our works [8]-[10] for almost sure global well-posedness of Gibbs measure evolutions of the nonlinear Schrödinger (NLS) and nonlinear wave (NLW) equations on the unit ball in R^d to the case of the three dimensional NLS. This is the first probabilistic global well-posedness result for NLS with supercritical data on the unit ball in R^3. The initial data is taken as a Gaussian random process lying in the support of the Gibbs measure associated to the equation, and results are obtained almost surely with respect to this probability measure. The key …
Global Well-posedness and Scattering for Stochastic generalized KdV Equations with additive noise
本文研究加性噪声驱动的散焦随机广义KdV方程，在质量临界和超临界情形下证明小初值的整体适定性与散射，并建立一般色散关系的振荡积分估计以控制随机卷积尾部。
We obtain almost-sure scattering for the cubic defocusing Schr{ö}dinger equation in the Euclidean space {$\mathbb{R}^3$}, with randomized radially-symmetric initial data at some supercritical regularity scales. Since we make no smallness assumption, our result generalizes the work of B{é}nyi, Oh and Pocovnicu. It also extends the results of Dodson, L{ü}hrmann and Mendelson on the energy-critical equation in {$\mathbb{R}^4$}, to the energy-subcritical equation in {$\mathbb{R}^3$}. In this latter setting, even if the nonlinear Duhamel term enjoys a stochastic smoothing effect that makes it subcr…
Almost sure local well-posedness for the nonlinear Schrödinger equations on $\Bbb T^d$ with non-algebraic nonlinearity
该研究针对带随机初值的非代数幂型非线性薛定谔方程，通过概率改进的伽利略双线性估计等方法，在任意维度及质量超临界范围建立几乎必然局部适定性，还将能量临界情形的低维代数理论推广到高维非代数模型。
Existence and non-uniqueness of probabilistically strong solutions to 3D stochastic magnetohydrodynamic equations
本文针对三维随机MHD方程，在超临界LPS区域构造无穷多个概率强解，证明其非唯一性，并揭示噪声消失时解的聚点包含所有确定性MHD解。
We consider a degenerate chemotaxis model with two-species and two-stimuli in dimension $d\geq 3$ and find two critical curves intersecting at one same point which separate the global existence and blow up of weak solutions to the problem. More precisely, above these curves (i.e. subcritical case), the problem admits a global weak solution obtained by the limits of strong solutions to an approximated system. Based on the second moment of solutions, initial data are constructed to make sure blow up occurs in finite time below these curves (i.e. critical and supercritical cases). In addition, th…
We prove the existence of global in time, finite energy, weak solutions to a quantum magnetohydrodynamic system (QMHD) with large data, modeling a charged quantum fluid interacting with a self-generated electromagnetic field. The analysis of QMHD relies upon the use of Madelung transformations. The rigorous derivation requires non-trivial smoothing estimates, which are obtained by assuming slightly higher regularity for the electromagnetic potential. These assumptions are motivated by the nonlinear dependence of the hydrodynamic system in terms of the underlying wave function dynamics, which i…
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