Lifeng Zhao

University of Science and Technology of China | USTC · School of Mathematical Sciences

This article is devoted to the simplified Ericksen-Leslie's hyperbolic system for incompressible liquid crystal model in two spatial dimensions, which is a nonlinear coupling of incompressible Navier-Stokes equations with wave map to circle $\mathbb{S}^1$. The structure of second-order material derivative is exploited in order to close the energy e...

We study the Ericksen–Leslie hyperbolic system for compressible liquid crystal model in three spatial dimensions. Global regularity and scattering for small and smooth initial data near equilibrium are proved for the case that the system is a nonlinear coupling of compressible Navier–Stokes equations with wave map to [Formula: see text]. The main s...

We consider the three-dimensional incompressible Euler equation \begin{equation*}\left\{\begin{aligned} &\partial_t \Omega+U \cdot \nabla \Omega+\Omega\cdot \nabla U=0 \\ &\Omega(x,0)=\Omega_0(x) \end{aligned}\right. \end{equation*} in the whole space $\mathbb{R}^3$. Under the assumption that the initial velocity is helical and without swirl, we pr...

The present paper considers a homogeneous bubble inside an unbounded polytropic compressible liquid with viscosity. The system is governed by the Navier-Stokes equation with free boundary which is determined by the kinematic and dynamic boundary conditions on the bubble-liquid interface. The global existence of solution is proved, and the $\dot{H}^...

We consider the energy‐critical focusing wave equation ∂t2u(t,x)−Δu(t,x)=u(t,x)u(t,x),t∈ℝ,x∈ℝ6, and we prove the existence of infinite time blowup at the vertices of any regular polyhedron. The blowup rate of each bubble is asymptotic to ckt−1 as t goes to +∞, where the constants ck depend on the distances between the vertices. This result is an ad...

In this paper, we establish the soliton resolution for the energy critical wave equation with inverse square potential in the radial case and in all dimensions $N\geq3$. The structure of the radial linear operator $\mathcal{L}_a :=-\Delta +\frac{a}{|x|^2}=A^*A$, is essential for the channel of energy, where $A$ is a first order differential operato...

We describe completely 2-solitary waves related to the ground state of the nonlinear damped Klein–Gordon equation ∂ttu+2α∂tu-Δu+u-|u|p-1u=0on RN, for 1⩽N⩽5 and energy subcritical exponents p>2. The description is twofold. First, we prove that 2-solitary waves with same sign do not exist. Second, we construct and classify the full family of 2-solita...

We prove shock formation for 3D Euler-Poisson system for electron fluid plasma. The shock solution we construct is of large initial data, also compactly supported during the lifespan. In addition, the blowup time and location can be computed explicitly.

The incompressible Schr\"odinger flow is a Madelung's hydrodynamical form of quantum mechanics, which can simulate classical fluids with particular advantage in its simplicity and its ability of capturing thin vortex dynamics. This model enables robust simulation of intricate phenomena such as vortical wakes and interacting vortex filaments. In thi...

In this paper, we constuct the multi-point blowup solutions of self-similar type for the inviscid Burgers equation. The shape and blowup dynamics are precisely described. Moreover, the solutions we construct are stable under small perturbations on initial data restricted in a compact set.

In this paper, we study the energy critical equivariant Landau-Lifschitz flow with target manifold $\mathbb{S}^2$. We prove that there exists a codimension one smooth well-localized set of initial data which generates finite-time type-II blowup solutions, and then give a precise description of the corresponding singularity formation. In our proof,...

We construct a special $C^{1,\alpha}$ blow up solution to the three dimensional system modeling electro-hydrodynamics, which is strongly coupled with incompressible Euler equation and Nernst-Planck-Poisson equation. Our construction lies on the framework established in Elgindi [11] and relies on a special solution to variant spherical Laplacian.

We prove that any sufficiently differentiable space-like hypersurface of {{\mathbb{R}}^{1+N}} coincides locally around any of its points with the blow-up surface of a finite-energy solution of the focusing nonlinear wave equation {\partial_{tt}u-\Delta u=|u|^{p-1}u} on {{\mathbb{R}}\times{\mathbb{R}}^{N}} , for any {1\leq N\leq 4} and {1<p\leq\frac...

We describe completely 2-solitary waves related to the ground state of the nonlinear damped Klein-Gordon equation \begin{equation*} \partial_{tt}u+2\alpha\partial_{t}u-\Delta u+u-|u|^{p-1}u=0 \end{equation*} on $\bf R^N$, for $1\leq N\leq 5$ and energy subcritical exponents $p>2$. The description is twofold. First, we prove that 2-solitary waves wi...

We consider the focusing energy subcritical nonlinear wave equation ∂ttu−Δu=|u|p−1u in RN, N≥1. Given any compact set K⊂RN, we construct finite energy solutions which blow up at t=0 exactly on K. The construction is based on an appropriate ansatz. The initial ansatz is simply U0(t,x)=κ(t+A(x))−2p−1, where A≥0 vanishes exactly on K, which is a solut...

In this article, we consider the Ericksen-Leslie's hyperbolic system for compressible liquid crystal model in three spatial dimensions. Global regularity for small and smooth initial data near equilibrium is proved for the case that the system is a nonlinear coupling of compressible Navier-Stokes equations with wave map to $\mathbb{S}^2$. Our argum...

In this article, we consider the Ericksen-Leslie's hyperbolic system for compressible liquid crystal model in three spatial dimensions. Global regularity for small and smooth initial data near equilibrium is proved for the case that the system is a nonlinear coupling of compressible Navier-Stokes equations with wave map to S 2. Our argument is a co...

We prove that any sufficiently differentiable space-like hypersurface of ${\mathbb R}^{1+N} $ coincides locally around any of its points with the blow-up surface of a finite-energy solution of the focusing nonlinear wave equation $\partial_{tt} u - \Delta u=|u|^{p-1} u$ on ${\mathbb R} \times {\mathbb R} ^N$, for any $1\leq N\leq 4$ and $1 < p \le...

We consider the nonlinear Schrödinger equation (Equation Presented) for H¹ -subcritical or critical nonlinearities: (N - 2)α ≤ 4. Under the additional technical assumptions α ≥ 2 (and thus N ≤ 4), we construct H solutions that blow up in finite time with explicit blow-up profiles and blow-up rates. In particular, blowup can occur at any given finit...

In this article, we consider the Ericksen-Leslie's hyperbolic system for incompressible liquid crystal model without kinematic transport in three spatial dimensions. Global regularity for small and smooth initial data near an equilibrium is proved for the case that the system is a nonlinear coupling of incompressible Navier-Stokes equations with wa...

In this article, we consider the Ericksen-Leslie's hyperbolic system for incompressible liquid crystal model without kinematic transport in three spatial dimensions. Global regularity for small and smooth initial data near an equilibrium is proved for the case that the system is a nonlinear coupling of incompressible Navier-Stokes equations with wa...

We consider the focusing energy subcritical nonlinear wave equation $\partial_{tt} u - \Delta u= |u|^{p-1} u$ in ${\mathbb R}^N$, $N\ge 1$. Given any compact set $ E \subset {\mathbb R}^N $, we construct finite energy solutions which blow up at $t=0$ exactly on $ E$. The construction is based on an appropriate ansatz. The initial ansatz is simply $...

We consider the nonlinear Schr\"odinger equation \[ u_t = i \Delta u + | u |^\alpha u \quad \mbox{on ${\mathbb R}^N $, $\alpha>0$,} \] for $H^1$-subcritical or critical nonlinearities: $(N-2) \alpha \le 4$. Under the additional technical assumptions $\alpha\geq 2$ (and thus $N\leq 4$), we construct $H^1$ solutions that blow up in finite time with e...

This article addresses third grade nanofluidic flow instigated by riga plate and Cattaneo-Christov theory is adopted to investigate thermal and mass diffusions with the incorporation of newly eminent zero nanoparticles mass flux condition. The governing system of equations is nondimensionalized through relevant similarity transformations and signif...

In this paper, we prove that the small energy harmonic maps from H ² to H ² are asymptotically stable under the wave map equation in the sub-critical perturbation class. This result may be seen as an example supporting the soliton resolution conjecture for geometric wave equations without equivariant assumptions on the initial data. In this paper,...

We consider the Cauchy problem for the energy-critical nonlinear Schrödinger equation with fractional Laplacian (fNLS) in the radial case. We obtain global well-posedness and scattering in the energy space in the defo-cusing case, and in the focusing case with energy below the ground state. The main feature of the present work is the nonlocality of...

We analyze the capillary rise dynamics for magnetohydrodynamics(MHD) fluid flow through deformable porous material in the presence of gravity effects. The modeling is performed using mixture theory approach and mathematical manipulation yields a nonlinear free boundary problem. Due to the capillary rise action, the pressure gradient in the liquid g...

In this paper, we study the asymptotic behaviors of finite energy solutions to the Landau–Lifshitz flows from R² into Kähler manifolds. First, we prove that the solution with initial data below the critical energy converges to a constant map in the energy space as t→ ∞ for the compact Riemannian surface targets. In particular, when the target is a...

The present work is focused on behavioral characteristics of gyrotactic microorganisms to describe their role in heat and mass transfer in the presence of magnetohydrodynamic (MHD) forces in Powell-Eyring nanofluids. Implications concerning stretching sheet with respect to velocity, temperature, nanoparticle concentration and motile microorganism d...

In this article, we will show the global wellposedness and scattering of the cubic defocusing nonlinear Schr\"odinger equation on waveguide $\mathbb{R}^2\times \mathbb{T}$ in $H^1$. We first establish the linear profile decomposition in $H^{ 1}(\mathbb{R}^2 \times \mathbb{T})$ motivated by the linear profile decomposition of the mass-critical Schr\...

In this article, we consider the equivariant Schr\"odinger map from $\Bbb H^2$ to $\Bbb S^2$ which converges to the north pole of $\Bbb S^2$ at the origin and spatial infinity of the hyperbolic space. If the energy of the data is less than $4\pi$, we show that the local existence of Schr\"odinger map. Furthermore, if the energy of the data sufficie...

In this article, we consider the infinite dimensional vector-valued resonant nonlinear Schr\"odinger system, which arises from the study of the asymptotic behavior of the defocusing nonlinear Schr\"{o}dinger equation on "wave guide" manifolds like $\mathbb{R}^2\times \mathbb{T}$ in [7]. We show global well-posedness and scattering for this system b...

In this paper, we prove that the small energy harmonic maps from $\Bbb H^2$ to $\Bbb H^2$ are asymptotically stable under the wave map equation in the subcritical perturbation class. This result may be seen as an example supporting the soliton resolution conjecture for geometric wave equations without equivariant assumptions on the initial data. In...

In this paper, we prove the scattering of radial solutions to high dimensional energy-critical nonlinear Schrödinger equations with regular potentials in the defocusing case.

In this paper, we prove that the solution of the Landau-Lifshitz flow $u(t,x)$ from $\mathbb{H}^2$ to $\mathbb{H}^2$ converges to some harmonic map as $t\to\infty$. The essential observation is that although there exist infinite numbers of harmonic maps from $\Bbb H^2$ to $\Bbb H^2$, the heat flow initiated from $u(t,x)$ for any given $t>0$ converg...

In this paper, we study the asymptotic behaviors of finite energy solutions to the Landau-Lifshitz flows from $\Bbb R^2$ into K\"ahler manifolds. First, we prove that the solution with initial data below the critical energy converges to a constant map in the energy space as $t\to \infty$ for the compact Riemannian surface targets. In particular, wh...

In this paper, we proved that if the solution to damped focusing Klein-Gordon equations in inhomogeneous medium is global forward in time, then it will converge to an equilibrium. The core ingredients of our proof are the existence of the "concentration-compact attractor" and the gradient system theory. The existence of "concentration-compact attra...

In this paper, we proved that if the solution to damped focusing Klein-Gordon equations is global forward in time, then it will decouple into a finite number of equilibrium points with different shifts from the origin. The core ingredient of our proof is the existence of the "concentration-compact attractor" which yields a finite number of profiles...

In this paper, we prove the asymptotic stability of nonlinear Schr\"odiger equation on graphs, which partially solves an open problem in D. Noja \cite{DN}. The essential ingredient of our proof is the dispersive estimate for the linearized around the soliton with Kirchhoff boundary condition. In order to obtain the dispersive estimates, we use the...

In this paper, we prove the uniqueness of weak solution to Vlasov-Poisson-Fokker-Planck system in $C([0,T]; L^p)$, by assuming the solution has local bounded density which trends to infinite with a "reasonable" rate as $t$ trends zero. And particularly as a corollary, we get uniqueness of weak solution with initial data $f_0$ satisfying $f_0|v|^2\i...

We prove the decay and scattering of solutions to three dimensional nonlinear Schr\"odinger with a Schawtz potential. For Rollnik potentials, we obtain time decay and scattering in energy space for small initial data for NLS with pure power nonlinearity $\frac{5}{3}<p<5$, which is the sharp exponent for scattering. For radial monotone Rollnik poten...

We consider the Cauchy problem for the nonlinear Schr\"odinger equation with combined nonlinearities, one of which is defocusing mass-critical and the other is focusing energy-critical or energy-subcritical. The threshold is given by means of variational argument. We establish the profile decomposition in $H^1(\Bbb R^d)$ and then utilize the concen...

We consider the Cauchy problem for the energy-critical nonlinear Schr\"odinger equation with fractional Laplacian (fNLS) in the radial case. We obtain global well-posedness and scattering in the energy space in the defocusing case, and in the focusing case with energy below the ground state.

In this paper, we continue the study in \cite{MiaoWZ:NLS:3d Combined} to show the scattering and blow-up result of the solution for the nonlinear Schr\"{o}dinger equation with the energy below the threshold $m$ in the energy space $H^1(\R^d)$, iu_t + \Delta u = -|u|^{4/(d-2)}u + |u|^{4/(d-1)}u, \; d\geq 5. \tag{CNLS} The threshold is given by the g...

We consider the defocusing energy-critical nonlinear Schrödinger equation of fourth order iut+Δ2u=−|u|8d−4u. We prove that any finite energy solution is global and scatters both forward and backward in time in dimensions d⩾9.

In this paper, we show the scattering and blow-up result of the radial solution with the energy below the threshold for the nonlinear Schrödinger equation (NLS) with the combined terms $$iu_{t} + \Delta{u} = -|u|^{4}u + |u|^{2}u \qquad \qquad \qquad \qquad {\rm (CNLS)}$$in the energy space \({{H^{1}(\mathbb{R}^{3})}}\) . The threshold is given by t...

Using the same induction on energy argument in both the frequency space and the spatial space simultaneously as in [6, 33, 38], we obtain the global well-posedness and scattering of energy solutions of the defocusing energy-critical nonlinear Hartree equation in ℝ × ℝ (n ≥ 5), which removes the radial assumption on the data in [25]. The new ingredi...

We characterize the dynamics of the finite time blow up solutions with minimal mass for the focusing mass critical Hartree equation with H1(R4) data and L2(R4) data, where we make use of the refined Gagliardo-Nirenberg inequality of convolution type and the profile decomposition. Moreover, we also analyze the mass concentration phenomenon of such b...

We prove the global well-posedness and scattering for the defocusing H12-subcritical (that is, 2γ3) Hartree equation with low regularity data in Rd, d⩾3. Precisely, we show that a unique and global solution exists for initial data in the Sobolev space Hs(Rd) with s>4(γ−2)/(3γ−4), which also scatters in both time directions. This improves the result...

We consider the focusing energy-critical nonlinear Schrodinger equation Of fourth order iu(t) + Delta(2)u = vertical bar u vertical bar(8/d-4)u, d >= 5, We prove that if a maximal-lifespan radial solution u: I x R d, C obeys sup(t is an element of l) parallel to Delta u(t)parallel to(2) < parallel to Delta W parallel to(2), then it is global and sc...

We establish global well-posedness and scattering for solutions to the mass-critical nonlinear Hartree equation iut+Δu=±(−2|x|∗2|u|)u for large spherically symmetric initial data; in the focusing case we require, of course, that the mass is strictly less than that of the ground state.RésuméNous établissons l'existence globale et la diffusion des so...

We consider the focusing energy-critical nonlinear Schr\"odinger equation of fourth order $iu_t+\Delta^2 u=|u|^\frac{8}{d-4}u$. We prove that if a maximal-lifespan radial solution $u: I\times\Bbb R^d\to\mathbb{C}$ obeys $\displaystyle\sup_{t\in I}\|\Delta u(t)\|_{2}<\|\Delta W\|_{2}$, then it is global and scatters both forward and backward in time...

We establish global existence, scattering for radial solutions to the energy-critical focusing Hartree equation with energy and $\dot{H}^1$ norm less than those of the ground state in $\mathbb{R}\times \mathbb{R}^d$, $d\geq 5$.

In this paper, we systematically study the wellposedness, illposedness of the Hartree equation, and obtain the sharp local wellposedness, the global existence in H s , s ≥ 1 and the small scattering result in H s for 2 < γ < n and s ≥ γ 2 − 1. In addition, we study the nonexistence of nontrivial asymptotically free solutions of the Hartree equation...

We consider the defocusing, H-1 -critical Hartree equation for the radial data in all dimensions (n >= 5). We show the global well-posedness and scattering results in the energy space. The new ingredient in this paper is that we first take advantage of the term - integral I integral vertical bar x vertical bar <= A vertical bar I vertical bar 1/2 v...

By using the isometric decomposition to the frequency spaces, we will introduce a new class of function spaces , which is a subspace of Gevrey 1-class G1(Rn)⊂C∞(Rn) for λ>0, and we will study the Cauchy problem for the nonlinear Schrödinger equation, the complex Ginzburg–Landau equation and the Navier–Stokes equation. Some well-posed results are ob...

The global well-posedness for the Cauchy problem of the derivative complex Ginzburg–Landau equation is shown in the H1-critical and H1-subcritical cases. A spatial decaying estimate of solutions in is also obtained and the decaying rate is independent of time t∈[0,∞).