Shuolin Zhang, Zhaonan Luo, Zhaoyang Yin
In this paper, we study the well-posedness of Fractional Rough Burgers equation driven by space-time noise in $H^s(\mathbb T)$ space. For the higher dissipation $γ\in(\frac{4}{3},2]$, we establish local well-posedness. Global well-posedness is further obtained when $γ$ is restricted to the interval $(\frac{5}{3}, 2]$. For the lower dissipation $γ\in(\frac{5}{4},\frac{4}{3}]$, we use the regularity analysis derivation the para-controlled solution.
Zipeng Chen, Song Liu, Zhaoyang Yin
Recently, Coiculescu and Palasek \cite{Coiculescu2025} shows the non-uniqueness of solutions for the 3D incompressible Navier-Stokes equations with initial data in $BMO^{-1}$. Inspired by their breakthrough work, we develop their schemes for the incompressible magnetohydrodynamic equations and obtain a similar result in 5 dimensional case. More precisely, we construct two distinct global solutions with a initial data, which has nonvanishing velocity and magnetic fields in $BMO^{-1}(\mathbb{T}^5)$.
Yingying Guo, Zhaoyang Yin
This paper is devoted to the study of the existence and uniqueness of global admissible conservative weak solutions for the periodic single-cycle pulse equation. We first transform the equation into an equivalent semilinear system by introducing a new set of variables. Using the standard ordinary differential equation theory, we then obtain the global solution to the semilinear system. Next, returning to the original coordinates, we get the global admissible conservative weak solution for the periodic single-cycle pulse equation. Finally, given an admissible conservative weak solution, we find a equation to single out a unique characteristic curve through each initial point and prove the uniqueness of global admissible conservative weak solution without any additional assumptions.
Zhaonan Luo, Wei Luo, Zhaoyang Yin
In this paper we mainly investigate the inviscid limit for the strong solutions of the finite extensible nonlinear elastic (FENE) dumbbell model. By virtue of the Littlewood-Paley theory, we first obtain a uniform estimate for the solution to the FENE dumbbell model with viscosity in Besov spaces. Moreover, we show that the data-to-solution map is continuous. Finally, we prove that the strong solution of the FENE dumbbell model converges to a Euler system couple with a Fokker-Planck equation. Furthermore, convergence rates in Lebesgue spaces are obtained also.
Zihua Guo, Xingxing Liu, Luc Molinet, Zhaoyang Yin
We prove norm inflation and hence ill-posedness for a class of shallow water wave equations, such as the Camassa-Holm equation, Degasperis-Procesi equation and Novikov equation etc., in the critical Sobolev space $H^{3/2}$ and even in the Besov space $B^{1+1/p}_{p,r}$ for $p\in [1,\infty], r\in (1,\infty]$. Our results cover both real-line and torus cases (only real-line case for Novikov), solving an open problem left in the previous works (\cite{Danchin2,Byers,HHK}).
Wei Luo, Zhaoyang Yin
In this paper we mainly investigate the traveling wave solution of the two dimensional Euler equations with gravity at the free surface over a flat bed. We assume that the free surface is almost periodic in the horizontal direction. Using conformal mappings, one can change the free boundary problem into a fixed boundary problem with some unknown functions in the boundary condition. By virtue of the Hilbert transform, the problem is equivalent to a quasilinear pseudodifferential equation for a almost periodic function of one variable. The bifurcation theory ensures us to obtain a existence result. Our existence result generalizes and covers the recent result in \cite{Constantin2011v}. Moreover, our result implies a non-uniqueness result at the same bifurcation point.
Jun Wang, Zhaoyang Yin
In this paper, we study the following Schrödinger equations with potentials and general nonlinearities \begin{equation*} \left\{\begin{aligned} & -Δu+V(x)u+λu=|u|^{q-2}u+βf(u), \\ & \int |u|^2dx=Θ, \end{aligned} \right. \end{equation*} both on $\mathbb{R}^N$ as well as on domains $r Ω$ where $Ω\subset \mathbb{R}^N$ is an open bounded convex domain and $r>0$ is large. The exponent satisfies $2+\frac{4}{N}\leq q\leq2^*=\frac{2 N}{N-2}$ and $f:\mathbb{R}\rightarrow \mathbb{R}$ satisfies $L^2$-subcritical or $L^2$-critical growth. This paper generalizes the conclusion of Bartsch et al. in \cite{TBAQ2023}(2023, arXiv preprint). Moreover, we consider the Sobolev critical case and $L^2$-critical case of the above problem.
Xi Tu, Zhaoyang Yin, Yingying Guo
This paper is devoted to studying the local well-posedness (existence,uniqueness and continuous dependence) for the generalized Camassa-Holm equations in critial Besov spaces $B^{\frac{1}{p}}_{p,1}$ with $1\leq p<+\infty$, which improves the previous index $s> \max\{\frac{1}{2},\frac{1}{p}\}$ or $s=\frac{1}{p},\ p\in[1,2],\ r=1$ in \cite{linb,tu-yin4}. The main difficulty is to prove the uniqueness, which need to use the Moser-type inequality. To overcome the difficulty, we use the Lagrange coordinate transformation to obtain the uniqueness.
Zhen He, Zhaoyang Yin
In this paper, we first establish the local well-posednesss for the Cauchy problem of a modified Camassa-Holm (MOCH) equation in critical Besov spaces $B^{\frac 1 p}_{p,1}$ with $1\leq p<+\infty.$ The obtained results improve considerably the recent result in \cite{Luo1}. Then we show the persiscence property of MOCH.
Weikui Ye, Zhaoyang Yin, Wei Luo
In this paper, we mainly investigate the Cauchy problem for the incompressible Navier-Stokes equations in homogeneous Besov spaces $\dot{B}^{\frac{d}{p}-1}_{p,r}$ with $1\leq p<\infty,\ 1\leq r\leq \infty, \ d\geq 2$. Firstly, we prove the local existence of the solution and give a lower bound of the lifespan $T$ of the solution. The lifespan depends on the Littlewood-Paley decomposition of the initial data, that is $\dotΔ_j u_0$. Secondly, if the initial data $u^n_0\rightarrow u_0$ in $\dot{B}^{\frac{d}{p}-1}_{p,r}$, then the corresponding lifespan $T_n\rightarrow T$. Thirdly, we prove that the data-to-solutions map is continuous in $\dot{B}^{\frac{d}{p}-1}_{p,r}$. Therefore, the Cauchy problem of the Navier-Stokes equations is locally well-posed in the critical Besov spaces in the Hadamard sense. Moreover, we also obtain well-posedness and weak-strong uniqueness results in $L^{\infty}L^2\cap L^{2}\dot{H}^1$.
Jinlu Li, Yanghai Yu, Weipeng Zhu, Zhaoyang Yin
In this paper, we construct a class of global large solution to the compressible Navier-Stokes equations in the whole space $\R^d$. Precisely speaking, our choice of special initial data whose $\dot{B}^{-1}_{\infty,\infty}$ norm can be arbitrarily large, namely, $||u_0||_{\dot{B}^{-1}_{\infty,\infty}}\gg 1$, allows to give rise to global-in-time solution to the compressible Navier-Stokes equations.
Teng-Fei Zhang, Zhaoyang Yin
In this paper we consider the non-cutoff Boltzmann equation in spatially inhomogeneous case. We prove the propagation of Gevrey regularity for the so-called smooth Maxwellian decay solutions to the Cauchy problem of spatially inhomogeneous Boltzmann equation, and obtain Gevrey regularity of order $1/(2s)$ in the velocity variable $v$ and order 1 in the space variable $x$. The strategy relies on our recent results for spatially homogeneous case (J. Diff. Equ. 253(4) (2012), 1172-1190. DOI: 10.1016/j.jde.2012.04.023). Rather, we need much more intricate analysis additionally in order to handle with the coupling of the double variables. Combining with the previous result mentioned above, it gives a whole characterization of the Gevrey regularity of the particular kind of solutions to the non-cutoff Boltzmann.
Jingjing Liu, Zhaoyang Yin
This paper is concerned with global existence and blow-up phenomena for two-component Degasperis-Procesi system and two-component b-family system. The strategy relies on our observation on new conservative quantities of these systems. Several new global existence results and a new blowup result of strong solutions to the two-component Degasperis- Procesi system and the two-component b-family system are presented by using these new conservative quantities.
Jinlu Li, Xiaoping Zhai, Zhaoyang Yin
In this paper, we mainly study the Cauchy problem for the full compressible Navier-Stokes equations in Sobolev spaces. We establish the global well-posedness of the equations with small initial data by using Friedrich's method and compactness arguments.
Huijun He, Zhaoyang Yin
In this paper, we mainly consider the Gevrey regularity and analyticity of the solution to a generalized two-component shallow water wave system with higher-order inertia operators, namely, $m=(1-\partial_x^2)^su$ with $s>1$. Firstly, we obtain the Gevrey regularity and analyticity for a short time. Secondly, we show the continuity of the data-to-solution map. Finally, we prove the global Gevrey regularity and analyticity in time.
Wei Luo, Zhaoyang Yin
In this paper we mainly study the long time behaviour of solutions to the finite extensible nonlinear elastic (FENE) dumbbell model with dimension two in the co-rotation case. Firstly, we obtain the $L^2$ decay rate of the velocity of the 2D co-rotation FENE model is $(1+t)^{-\frac{1}{2}}$ with small data. Then, by virtue of the Littlewood-Paley theory, we can remove the small condition. Our obtained sharp result improves considerably the recent results in \cite{Luo-Yin,Schonbek}.
Zhaoyang Yin
We establish the local well-posedness for a new nonlinearly dispersive wave equation and we show that the equation has solutions that exist for indefinite times as well as solutions which blowup in finite times. Furthermore, we derive an explosion criterion for the equation and we give a sharp estimate from below for the existence time of solutions with smooth initial data.
Zihua Guo, Jinlu Li, Zhaoyang Yin
We prove the inviscid limit of the incompressible Navier-Stokes equations in the same topology of Besov spaces as the initial data. The proof is based on proving the continuous dependence of the Navier-Stokes equations uniformly with respect to the viscosity. To show the latter, we rely on some Bona-Smith type method in the $L^p$ setting. Our obtained result implies a new result that the Cauchy problem of the Euler equations is locally well-posed in the borderline Besov space $B^{\frac dp+1}_{p,1}(\mathbb{R}^d)$, $1\leq p\leq \infty$, $d\geq 2$, in the sense of Hadmard, which is an open problem left in recent works by Bourgain and Li in \cite{BL,BL1} and by Misiołek and Yoneda in \cite{MY,MY2, MY3}.
Hailiang Liu, Zhaoyang Yin
This paper is concerned with a class of nonlocal dispersive models -- the $θ$-equation proposed by H. Liu [ On discreteness of the Hopf equation, {\it Acta Math. Appl. Sin.} Engl. Ser. {\bf 24}(3)(2008)423--440]: $$ (1-\partial_x^2)u_t+(1-θ\partial_x^2)(\frac{u^2}{2})_x =(1-4θ)(\frac{u_x^2}{2})_x, $$ including integrable equations such as the Camassa-Holm equation, $θ=1/3$, and the Degasperis-Procesi equation, $θ=1/4$, as special models. We investigate both global regularity of solutions and wave breaking phenomena for $θ\in \mathbb{R}$. It is shown that as $θ$ increases regularity of solutions improves: (i) $0 <θ< 1/4$, the solution will blow up when the momentum of initial data satisfies certain sign conditions; (ii) $1/4 \leq θ< 1/2$, the solution will blow up when the slope of initial data is negative at one point; (iii) ${1/2} \leq θ\leq 1$ and $θ=\frac{2n}{2n-1}, n\in \mathbb{N}$, global existence of strong solutions is ensured. Moreover, if the momentum of initial data has a definite sign, then for any $θ\in \mathbb{R}$ global smoothness of the corresponding solution is proved. Proofs are either based on the use of some global invariants or based on exploration of favorable sign conditions of quantities involving solution derivatives. Existence and uniqueness results of global weak solutions for any $θ\in \mathbb{R}$ are also presented. For some restricted range of parameters results here are equivalent to those known for the $b-$equations [e.g. J. Escher and Z. Yin, Well-posedness, blow-up phenomena, and global solutions for the b-equation, {\it J. reine angew. Math.}, {\bf 624} (2008)51--80.]
Jun Wang, Zhaoyang Yin
In this paper, we study the well-posedness theory and the scattering asymptotics for the energy-critical, Schrödinger equation with indefinite potential \begin{equation*} \left\{\begin{array}{l} i \partial_t u+Δu-V(x)u +|u|^{\frac{4}{N-2}}u=0,\ (x, t) \in \mathbb{R}^N \times \mathbb{R}, \\ \left.u\right|_{t=0}=u_0 \in H ^1(\mathbb{R}^N), \end{array}\right. \end{equation*} where $V(x):\mathbb{R}^N\rightarrow \mathbb{R}$ is indefinite and satisfies appropriate conditions. Using contraction mapping method and concentration compactness argument, we obtain the well-posedness theory in proper function spaces and scattering asymptotics. Moreover, we get a positive ground state solution which is radially symmetric by using variational methods. This paper extends the results of \cite{KCEMF2006}(Invent. Math) to the potential equation and develops the recent conclusions.