Generalized solutions of the Cauchy problem for the Navier-Stokes system and diffusion processes [PDF]
arXiv, 2007 We reduce the construction of a weak solution of the Cauchy problem for the Navier-Stokes system to the construction of a solution to a stochastic problem. Namely, we construct diffusion processes which allow us to obtain a probabilistic representation of a weak (in distributional sense) solution to the Cauchy problem for the Navier- Stokes system.
arxiv
On the Solutions of the Three Dimensional Navier-Stokes Problem [PDF]
arXiv, 2008 The aim of this paper is to solve the three dimensional Navier-Stokes problem with conservative source term. We use convolution methods to construct "well behaved" smooth solutions of the initial boundary value problem for the system of Navier-Stokes with conservative source term.
arxiv
Blow-up rates for the general curve shortening flow [PDF]
arXiv, 2009 The blow-up rates of derivatives of the curvature function will be presented when the closed curves contract to a point in finite time under the general curve shortening flow. In particular, this generalizes a theorem of M.E. Gage and R.S. Hamilton about mean curvature flow in $\mathbb{R}^{2}$.
arxiv
The blow up analysis of the general curve shortening flow [PDF]
arXiv, 2009 It is shown that the curvature function satisfies a nonlinear evolution equation under the general curve shortening flow and a detailed asymptotic behavior of the closed curves is presented when they contract to a point in finite time.
arxiv
Note on quantitative homogenization results for parabolic systems in R d. [PDF]
J Evol Equ, 2021 Meshkova Y.
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Asymptotic uniform boundedness of energy solutions to the Penrose-Fife model [PDF]
arXiv, 2011 We study a Penrose-Fife phase transition model coupled with homogeneous Neumann boundary conditions. Improving previous results, we show that the initial value problem for this model admits a unique solution under weak conditions on the initial data. Moreover, we prove asymptotic regularization properties of weak solutions.
arxiv
A weighted $L_p$-theory for second-order elliptic and parabolic partial differential systems on a half space [PDF]
arXiv, 2012 In this paper we develop a Fefferman-Stein theorem, a Hardy-Littlewood theorem and sharp function estimations in weighted Sobolev spaces. We also provide uniqueness and existence results for second-order elliptic and parabolic partial differential systems in weighed Sobolev spaces.
arxiv
Analysis of a model arising from invasion by precursor and differentiated cells [PDF]
arXiv, 2013 We study the wave solutions for a degenerated reaction diffusion system arising from the invasion of cells. We show that there exists a family of waves for the wave speed larger than or equals a certain number, and below which there is no monotonic wave solutions. We also investigate the monotonicity, uniqueness and asymptotics of the waves.
arxiv
On the metastable behavior of solutions to a class of parabolic systems [PDF]
arXiv, 2014 In this paper we describe the metastable behavior of solutions to a class of parabolic systems. In particular, we improve some results contained in \cite{MS} by using different techniques to describe the slow motion of the internal layers. Numerical simulations illustrate the results.
arxiv
Well-posedness for a class of nonlinear degenerate parabolic equations [PDF]
arXiv, 2015 In this paper we obtain well-posedness for a class of semilinear weakly degenerate reaction-diffusion systems with Robin boundary conditions. This result is obtained through a Gagliardo-Nirenberg interpolation inequality and some embedding results for weighted Sobolev spaces.
arxiv