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Mimi Dai

Publications and source records attributed to Mimi Dai.

At least 19 recordsLinked to original sources
arXiv2026

Instantaneous blowup of incompressible flow with passive tracer

We construct a family of solutions $(u,b)$ of the incompressible flow with a passive tracer for which both $\|u(t)\|_{L^\infty}$ and $\|b(t)\|_{L^\infty}$ blow up at time $T_*$. Away from $T_*$, the solutions remain smooth in both space and time. The argument adapts the inverse cascade mechanism from [CDP25] to the presence of an advected scalar, but the passive component creates a new compatibility constraint: the iteration must propagate the tracer while preserving the same principal velocity profiles from one stage to the next. We resolve it by introducing a simultaneous decomposition lemma for a symmetric tensor and a vector field. We also show the existence of an infinitely family of instantaneous blowup solutions to the 2D MHD system, with critical blowup rate for the velocity component according to the scaling of the system. Moreover, the non-uniqueness is sharp in the sense that it occurs in spaces borderline to $L^2_tL_x^\infty$, the endpoint space of the Ladyzhenskaya--Prodi--Serrin type where uniqueness is known.

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arXiv2026

Determining wavenumbers for Hall and electron magnetohydrodynamics turbulence

In turbulent flows, the Kolmogorov wavenumber characterizes the smallest scales at which viscous effects dominate. A mathematical analogue of this notion first introduced by Foias and Prodi [8] -- a determining wavenumber -- quantifies the minimal set of modes that uniquely determine the long-time behavior of solutions. Extending this framework from the Navier-Stokes equations to magnetized plasma models, we focus on the Hall-MHD and Electron-MHD turbulence in sub-ion and dissipation ranges. We prove existence of time-dependent determining wavenumbers for weak solutions of the Hall- and electron-MHD, improving upon previous results that were not optimal and lacked any comparison with phenomenological dissipation scales. Under explicit scale-localized intermittency assumptions, we show that their time averages are bounded above by Kolmogorov-like dissipation wavenumbers predicted by phenomenological studies of plasma turbulence. For strong electron-MHD solutions, we also establish a uniform bound on the magnetic determining wavenumber from Besov regularity.

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arXiv2026

Instantaneous blowup and non-uniqueness of smooth solutions of MHD

We construct a family of solutions $(u,B)$ of the incompressible magnetohydrodynamic (MHD) system, the $L^\infty$ norm of which blows up instantaneously at the critical rate. The solutions remain smooth except at the blowup time. An inverse energy cascade mechanism and a convex integration scheme along a time sequence are the main ingredients of the construction, inspired by our recent work [CDP25] for the Navier-Stokes equations. The challenge of the construction for the MHD system stems from the coupling and the necessity of preserving the same ansatz of the principal solution at every iterative step while implementing convex integration. Existing convex integration schemes for MHD can treat the coupling but fail to produce the same ansatz of the principal solution recursively. To achieve the goal, we introduce a coupled geometric lemma that decomposes a symmetric tensor and a skew-symmetric tensor simultaneously. We emphasize that such coupled geometric lemma is new and of independent interest.

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arXiv2025

Instantaneous Type I blow-up and non-uniqueness of smooth solutions of the Navier-Stokes equations

For any smooth, divergence-free initial data, we construct a solution of the Navier--Stokes equations that exhibits Type~I blow-up of the $L^\infty$ norm at time $T_*>0$, while remaining smooth in space and time on $\mathbb T^d\times([0,T]\setminus\{T_*\})$. An instantaneous injection of energy from infinite wavenumber initiates a bifurcation from the classical solution, producing an infinite family of spatially smooth solutions with the same data and thereby violating uniqueness of the Cauchy problem. A key ingredient is the first known construction of a complete inverse energy cascade realized by a classical Navier--Stokes flow, which transfers energy from infinitely high to low frequencies. The result holds in all dimensions $d\geq2$.

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arXiv2025

Discretely self-similar solutions for electron MHD

We study discretely self-similar solutions for the electron magnetohydrodynamics (MHD) without resistivity. Under several different decay and non-decay conditions, we show the absence of non-trivial discretely self-similar blowup solutions.

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arXiv2025

Well-posedness of the electron MHD with partial resistivity

Due to the singular nonlinear Hall term, the non-resistive electron magnetohydrodynamics (MHD) is not known to be locally well-posed in general. In this paper we consider the $2\frac12$D electron MHD with either horizontal or vertical resistivity and show local well-posedness in Sobolev spaces.

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arXiv2025

Blowup for the forced electron MHD

The electron magnetohydrodynamics (MHD) contains a highly nonlinear Hall term with an interesting structure. Exploring the Hall nonlinear structure, we investigate possible phenomena of finite time blow up for the electron MHD with a (non-rough) forcing. When the magnetic field has zero horizontal components, the vertical component equation has a mixing feature with the mixer being the current flow. By constructing a magnetic field profile whose current density is approximately a hyperbolic flow near the origin, we show blowup develops in finite time. In another setting when the magnetic field is a shear type, the Hall term vanishes, and finite time blowup can be obtained for the forced electron MHD as well.

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arXiv2025

Well-posedness and blowup of 1D electron magnetohydrodynamics

The one-dimensional toy models proposed for the three-dimensional electron magnetohydrodynamics in our previous work share some similarities with the original dynamics under certain symmetry. We continue to study the well-posedness issue and explore the potential singularity formation scenario for these models.

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arXiv2024

Ill-posedness of $2\frac12$D electron MHD

We consider the electron magnetohydrodynamics (MHD) in the context where the 3D magnetic field depends only on the two horizontal plane variables. In particular, the magnetic field takes the form $B=\nabla\times (a\vec e_z)+b\vec e_z$ with $a=a(x,y)$ and $b=b(x,y)$. Initial data $(a_0,b_0)$ is constructed in the Sobolev space $H^\beta \times H^{\beta-1}$ with $1<\beta<4$ such that the solution to this electron MHD system either escapes the space or develops norm inflation in $\dot H^\beta \times \dot H^{\beta-1}$.

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arXiv2024

Beale--Kato--Majda-type continuation criteria for Hall- and electron-magnetohydrodynamics

We show that regular solutions to electron-MHD with resistivity can be continued as long as the time integral of the supremum of the current gradient remains finite. This dimensionless continuation criterion is analogous to the celebrated result of Beale--Kato--Majda for the incompressible Euler and Navier--Stokes equations. A similar continuation criterion, formulated in terms of the time integral of the supremum of the vorticity, velocity gradient and current gradient, is established for the Hall-MHD with resistivity as well.

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arXiv2024

An Onsager-type theorem for SQG

We construct non-trivial weak solutions $\theta\in C_t^0C_x^{0-}$ to the surface quasi-geostrophic (SQG) equations, which have compact support in time and, thus, violate the conservation of the Hamiltonian. The result is sharp in view of the fact that such a conservation law holds for all weak solutions in the class $C_{t,x}^0 \subset L_{t,x}^3$ (Isett-Vicol, 2015) and resolves the Onsager conjecture for SQG. The construction is achieved by means of a Nash iteration together with the linear decoupling method recently introduced in Giri-Radu (2023).

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arXiv2024

Non-unique solutions for electron MHD

We consider the electron magnetohydrodynamics (MHD) equation on the 3D torus $\mathbb T^3$. For a given smooth vector field $H$ with zero mean and zero divergence, we can construct a weak solution $B$ to the electron MHD in the space $L^\gamma_tW^{1,p}_x$ for appropriate $(\gamma, p)$ such that $B$ is arbitrarily close to $H$ in this space. The parameters $\gamma$ and $p$ depend on the resistivity. As a consequence, non-uniqueness of weak solutions is obtained for the electron MHD with hyper-resistivity. In particular, non-Leray-Hopf solutions can be constructed. As a byproduct, we also show the existence of weak solutions to the electron MHD without resistivity.

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arXiv2024

Self-similar singularities for electron MHD

We study several types of self-similar solutions for the electron magnetohydrodynamics (MHD) without resistivity, including locally self-similar solutions and pseudo-self-similar solutions. We show that under certain conditions, these types of self-similar blowup solutions can be excluded.

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arXiv2023

Non-uniqueness of forced active scalar equations with even drift operators

We consider forced active scalar equations with even and homogeneous degree 0 drift operator on $\mathbb T^d$. Inspired by the non-uniqueness construction for dyadic fluid models, by implementing a sum-difference convex integration scheme we obtain non-unique weak solutions for the active scalar equation in space $C_t^0C_x^\alpha$ with $\alpha<\frac{1}{2d+1}$. We note that in 1D, the regularity $\alpha<\frac13$ is sharp as the energy identity is satisfied for solutions in $C^\alpha$ with $\alpha>\frac13$. Without external forcing, Isett and Vicol constructed non-unique weak solutions for such active scalar equations with spatial regularity $C_x^\alpha$ for $\alpha<\frac{1}{4d+1}$.

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arXiv2023

Non-unique weak solutions of forced SQG

We construct non-unique weak solutions $\theta\in C_t^0C_x^{0-}$ for forced surface quasi-geostrophic (SQG) equation. This is achieved through a convex integration scheme adapted to the sum-difference system of two distinct solutions. Without external forcing, non-unique weak solutions $\theta$ in space $C_t^0C_x^{\alpha}$ with $\alpha<-\frac15$ were constructed by Buckmaster, Shkoller and Vicol, and Isett and Ma.

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arXiv2023

Global existence of 2D electron MHD near a steady state

We study the electron magnetohydrodynamics (MHD) in two dimensional geometry, which has a rich family of steady states. In an anisotropic resistivity context, we show global in time existence of small smooth solution near a shear type steady state. Convergence rate of the solution to the steady state is also obtained.

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arXiv2023

Uniqueness for a stochastic ideal dyadic MHD model

We study a stochastic dyadic model with both forward and backward energy cascade mechanisms for the inviscid and non-resistive magnetohydrodynamics. For a particular class of stochastic forcing, we show weak uniqueness for the stochastic system. However the solution dissipates the energy which is formally an invariant quantity for the system.

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