Related papers: Global regularity for a logarithmically supercriti…
We prove maximal Schauder regularity for solutions to elliptic systems and Cauchy problems, in the space $C_b(\mathbb{R}^d;\mathbb{R}^m)$ of bounded and continuous functions, associated to a class of nonautonomous weakly coupled…
We consider the Cauchy problem of the higher-order KdV-type equation: \[ \partial_t u + \frac{1}{\mathfrak{m}} |\partial_x|^{\mathfrak{m}-1} \partial_x u = \partial_x (u^{\mathfrak{m}}) \] where $\mathfrak{m} \ge 4$. The nonlinearity is…
We study the pointwise decay properties of solutions to the incompressible Navier-Stokes equations, both in the space and time variables. It is well known that generic global solutions on $\mathbb{R}^n$ do not decay faster at infinity than…
We study the Navier-Stokes equations with an extra eddy viscosity term in the whole space in three dimensions. We introduce a suitable regularized system for which we prove the existence of a regular solution defined for all time. We prove…
In this paper, we study the global well-posedness of classical solution to 2D Cauchy problem of the compressible Navier-Stokes equations with large initial data and vacuum. It is proved that if the shear viscosity $\mu$ is a positive…
In this paper, we investigate Cauchy problem of the two-dimensional full Maxwell-Navier-Stokes system, and prove the global-in-time existence and uniqueness of solution in the borderline space which is very close to $L^2$-energy space by…
We prove global existence of smooth solutions of the 3D loglog energy-supercritical wave equation $\partial_{tt} u - \triangle u = -u^{5} \log^{c} (log(10+u^{2})) $ with $0 < c < {8/225}$ and smooth initial data $(u(0)=u_{0}, \partial_{t}…
In this paper, we prove that the 1D Cauchy problem of the compressible Navier-Stokes equations admits a unique global classical solution $(\rho,\rm u)$ if the viscosity $\mu(\rho)=1+\rho^{\beta}$ with $\beta\geq0$. The initial data can be…
We consider the Hookean dumbbell model, a system of nonlinear PDEs arising in the kinetic theory of homogeneous dilute polymeric fluids. It consists of the unsteady incompressible Navier-Stokes equations in a bounded Lipschitz domain,…
We prove that the solutions to the 3D Navier-Stokes equation with constant rotation exist globally for small axisymmetric initial data, where the smallness is uniform with respect to the viscosity $\nu \in [0,\infty)$. This expands the work…
In this paper we prove Liouville type theorem for the stationary Navier-Stokes equations in $\Bbb R^3$ under the assumptions on the relative decays of velocity, pressure and the head pressure. More precisely, we show that any smooth…
The Gaussian-filtered Navier-Stokes equations are examined theoretically and a generalized theory of their numerical stability is proposed. Using the exact expansion series of subfilter-scale stresses or integration by parts, the terms…
We consider the system of partial differential equations governing two-dimensional flows of a robust class of viscoelastic rate-type fluids with stress diffusion, involving a general objective derivative. The studied system generalizes the…
In this paper we consider the Cauchy problem for the 3D Navier-Stokes equations for incompressible flows. The initial data are assumed to be smooth and rapidly decaying at infinity. A famous open problem is whether classical solutions can…
The current paper is principally motivated by establishing the global well-posedness to the three-dimensional Boussinesq system with zero diffusivity in the setting of axisymmetric flows without swirling with $v_0\in…
Motivated by Kolmogorov's theory of turbulence we present a unified approach to the regularity problems for the 3D Navier-Stokes and Euler equations. We introduce a dissipation wavenumber $\Lambda (t)$ that separates low modes where the…
In 1987, Michael Wiegner in his seminal paper [17] provided an important result regarding the energy decay of Leray solutions $\boldsymbol u(\cdot,t)$ to the incompressible Navier-Stokes in $\mathbb{R}^{n}$: if the associated Stokes flows…
In this paper, we consider the global existence and blowup phenomena of the following Cauchy problem \begin{align*} \left\{\begin{array}{ll}&-i u_t=\Delta u-V(x)u+f(x,|u|^2)u+(W\star|u|^2)u, \quad x\in\mathbb{R}^N, \quad t>0,…
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…
We study the Cauchy problem for a system of semi-linear coupled fractional-diffusion equations with polynomial nonlinearities posed in $% \mathbb{R}_{+}\times \mathbb{R}^{N}$. Under appropriate conditions on the exponents and the orders of…