More Like this

1. Well-Posedness for Stochastic Fractional Navier–Stokes Equation in the Critical Fourier–Besov Space

   https://doi.org/10.1007/s10959-021-01152-y  

   Yin, Xiuwei ; Wu, Jiang-Lun ; Shen, Guangjun ( January 2022 , Journal of Theoretical Probability)

   The well-posedness of stochastic Navier–Stokes equations with various noises is a hot topic in the area of stochastic partial differential equations. Recently, the consideration of stochastic Navier–Stokes equations involving fractional Laplacian has received more and more attention. Due to the scaling-invariant property of the fractional stochastic equations concerned, it is natural and also very important to study the well-posedness of stochastic fractional Navier–Stokes equations in the associated critical Fourier–Besov spaces. In this paper, we are concerned with the three-dimensional stochastic fractional Navier–Stokes equation driven by multiplicative noise. We aim to establish the well-posedness of solutions of the concerned equation. To this end, by utilising the Fourier localisation technique, we first establish the local existence and uniqueness of the solutions in the critical Fourier–Besov space$$\dot{\mathcal {B}}^{4-2\alpha -\frac{3}{p}}_{p,r}$$${\dot{B}}_{p,r}^{4-2\alpha -\frac{3}{p}}$. Then, under the condition that the initial date is sufficiently small, we show the global existence of the solutions in the probabilistic sense.

    

   more » « less
2. The Littlewood–Paley decomposition for periodic functions and applications to the Boussinesq equations

   https://doi.org/10.1142/s0219530519500234  

   Dai, Yichen ; Hu, Weiwei ; Wu, Jiahong ; Xiao, Bei ( July 2020 , Analysis and Applications)

   The Littlewood–Paley decomposition for functions defined on the whole space [Formula: see text] and related Besov space techniques have become indispensable tools in the study of many partial differential equations (PDEs) with [Formula: see text] as the spatial domain. This paper intends to develop parallel tools for the periodic domain [Formula: see text]. Taking advantage of the boundedness and convergence theory on the square-cutoff Fourier partial sum, we define the Littlewood–Paley decomposition for periodic functions via the square cutoff. We remark that the Littlewood–Paley projections defined via the circular cutoff in [Formula: see text] with [Formula: see text] in the literature do not behave well on the Lebesgue space [Formula: see text] except for [Formula: see text]. We develop a complete set of tools associated with this decomposition, which would be very useful in the study of PDEs defined on [Formula: see text]. As an application of the tools developed here, we study the periodic weak solutions of the [Formula: see text]-dimensional Boussinesq equations with the fractional dissipation [Formula: see text] and without thermal diffusion. We obtain two main results. The first assesses the global existence of [Formula: see text]-weak solutions for any [Formula: see text] and the existence and uniqueness of the [Formula: see text]-weak solutions when [Formula: see text] for [Formula: see text]. The second establishes the zero thermal diffusion limit with an explicit convergence rate. 

   more » « less
3. Global well-posedness and asymptotic behavior in critical spaces for the compressible Euler system with velocity alignment

   https://doi.org/10.1088/1361-6544/ad140b  

   Bai, Xiang ; Miao, Qianyun ; Tan, Changhui ; Xue, Liutang ( January 2024 , Nonlinearity)

   In this paper, we study the Cauchy problem of the compressible Euler system with strongly singular velocity alignment. We prove the existence and uniqueness of global solutions in critical Besov spaces to the considered system with small initial data. The local-in-time solvability is also addressed. Moreover, we show the large-time asymptotic behaviour and optimal decay estimates of the solutions as$t\to \mathrm{\infty }$.

    

   more » « less
4. Sparse optimization problems in fractional order Sobolev spaces

   https://doi.org/10.1088/1361-6420/acbe5e  

   Antil, Harbir ; Wachsmuth, Daniel ( March 2023 , Inverse Problems)

   We consider optimization problems in the fractional order Sobolev spaces with sparsity promoting objective functionals containingL^p-pseudonorms,$p\in (0,1)$. Existence of solutions is proven. By means of a smoothing scheme, we obtain first-order optimality conditions, which contain an equation with the fractional Laplace operator. An algorithm based on this smoothing scheme is developed. Weak limit points of iterates are shown to satisfy a stationarity system that is slightly weaker than that given by the necessary condition.

    

   more » « less
5. Local martingale solutions and pathwise uniqueness for the three-dimensional stochastic inviscid primitive equations

   https://doi.org/10.1007/s40072-022-00266-6  

   Hu, Ruimeng ; Lin, Quyuan ( July 2022 , Stochastics and Partial Differential Equations: Analysis and Computations)

   We study the stochastic effect on the three-dimensional inviscid primitive equations (PEs, also called the hydrostatic Euler equations). Specifically, we consider a larger class of noises than multiplicative noises, and work in the analytic function space due to the ill-posedness in Sobolev spaces of PEs without horizontal viscosity. Under proper conditions, we prove the local existence of martingale solutions and pathwise uniqueness. By adding vertical viscosity,i.e., considering the hydrostatic Navier-Stokes equations, we can relax the restriction on initial conditions to be only analytic in the horizontal variables with Sobolev regularity in the vertical variable, and allow the transport noise in the vertical direction. We establish the local existence of martingale solutions and pathwise uniqueness, and show that the solutions become analytic in the vertical variable instantaneously as$$t>0$$$t>0$and the vertical analytic radius increases as long as the solutions exist.

    

   more » « less