Related papers: Singular and regular solutions of a non-linear par…
In this paper, we prove the global existence and singularity formation for a wave system from modelling nematic liquid crystals in one space dimension. In our model, although the viscous damping term is included, the solution with smooth…
We consider the NLS system of the third-harmonic generation, which was introduced by Sammut. Our interest is in solitary wave solutions and their stability properties. The recent work of Oliveira and Pastor, discussed global well-posedness…
We prove the nonexistence of local self-similar solutions of the three dimensional incompressible Navier-Stokes equations. The local self-similar solutions we consider here are different from the global self-similar solutions. The…
We prove existence of global regular axially-symmetric solutions to the Navier-Stokes equations in a cylindrical domain. We assume the periodic boundary conditions on the top and the bottom of the cylinder, but on the lateral part we assume…
We study the non-uniqueness of weak solutions for the two-dimensional hyper-dissipative Navier-Stokes equations in the super-critical spaces $L_{t}^{\gamma}L_{x}^{p}$ when $\alpha\in[1,\frac{3}{2})$, and obtain the conclusion that the…
Using some nonlinear domain decomposition method, we prove the existence of singular limits for solution of semilinear elliptic problems with exponential nonlinearity.
In the paper, we have introduced the notion of mild bounded ancient solutions to the Navier-Stokes equations in a half space. They play a certain role in understanding whether or not solutions to the initial boundary value problem for the…
We consider the singularity formation of strong solutions to the two-dimensional full compressible Navier-Stokes equations with zero heat conduction in a bounded domain. It is shown that for the initial density allowing vacuum, the strong…
We consider the following parabolic system whose nonlinearity has no gradient structure: $$\left\{\begin{array}{ll} \partial_t u = \Delta u + |v|^{p-1}v, \quad & \partial_t v = \mu \Delta v + |u|^{q - 1}u,\\ u(\cdot, 0) = u_0, \quad &…
We study the uniqueness, the continuity in $L^2$ and the large time decay for the Leray solutions of the $3D$ incompressible Navier-Stokes equations with the nonlinear exponential damping term $a (e^{b |u|^{\bf 2}}-1)u$, ($a,b>0$) studied…
We consider the Cauchy problem for the incompressible Navier-Stokes equations in $\mathbb{R}^3$ for a one-parameter family of explicit scale-invariant axi-symmetric initial data, which is smooth away from the origin and invariant under the…
Consider an infinite system \[\partial_tu_t(x)=(\mathscr{L}u_t)(x)+ \sigma\bigl(u_t(x)\bigr)\partial_tB_t(x)\] of interacting It\^{o} diffusions, started at a nonnegative deterministic bounded initial profile. We study local and global…
Direct Numerical Simulation is performed of the forced Navier-Stokes equation in four spatial dimensions. Well equilibrated, long time runs at sufficient resolution were obtained to reliably measure spectral quantities, the velocity…
This article presents an innovative extension of the Smagorinsky model incorporating dynamic boundary conditions and advanced regularity methods. We formulate the modified Navier-Stokes equations with the Smagorinsky term to model…
In studies of superlinear parabolic equations \begin{equation*} u_t=\Delta u+u^p,\quad x\in {\mathbb R}^N,\ t>0, \end{equation*} where $p>1$, backward self-similar solutions play an important role. These are solutions of the form $ u(x,t) =…
First, the solution uniqueness and existence of a stationary anisotropic (linear) Stokes system with constant viscosity coefficients in a compressible framework on $n$-dimensional flat torus are analysed in a range of periodic Sobolev…
We investigate weak Serrin-type blowup criterion of the three-dimensional full compressible Navier-Stokes equations for the Cauchy problem, Dirichlet problem and Navier-slip boundary condition. It is shown that the strong or smooth solution…
In this paper we construct smooth, non-radial solutions of the compressible Euler and Navier-Stokes equation that develop an imploding finite time singularity. Our construction is motivated by the works [Merle, Rapha\"{e}l, Rodnianski, and…
We study the uniqueness of singular radial (forward and backward) self-similar positive solutions of the equation $u_t-\Delta u = u^p, \quad x\in{\mathbb R}^n,\ t>0,$ where $p\geq(n+2)/(n-2)_+$.
We study the existence of global-in-time solutions for a nonlinear heat equation with nonlocal diffusion, power nonlinearity and suitably small data (either compared pointwisely to the singular solution or in the norm of a critical Morrey…