Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userWell-posedness for the diffusive 3D Burgers equations with initial data in $H^{1/2}$
150
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this note we discuss the diffusive, vector-valued Burgers equations in a three-dimensional domain with periodic boundary conditions. We prove that given initial data in $H^{1/2}$ these equations admit a unique global solution that becomes classical immediately after the initial time. To prove local existence, we follow as closely as possible an argument giving local existence for the Navier--Stokes equations. The existence of global classical solutions is then a consequence of the maximum principle for the Burgers equations due to Kiselev and Ladyzhenskaya (1957). In several places we encounter difficulties that are not present in the corresponding analysis of the Navier--Stokes equations. These are essentially due to the absence of any of the cancellations afforded by incompressibility, and the lack of conservation of mass. Indeed, standard means of obtaining estimates in $L^2$ fail and we are forced to start with more regular data. Furthermore, we must control the total momentum and carefully check how it impacts on various standard estimates.
rate research
Read More
The Cauchy problem for a scalar conservation laws admits a unique entropy solution when the data $u_0$ is a bounded measurable function (Kruzhkov). The semi-group $(S_t)_{tge0}$ is contracting in the $L^1$-distance. For the multi-dimensional Burgers equation, we show that $(S_t)_{tge0}$ extends uniquely as a continuous semi-group over $L^p(mathbb{R}^n)$ whenever $1le p<infty$, and $u(t):=S_tu_0$ is actually an entropy solution to the Cauchy problem. When $ple qle infty$ and $t>0$, $S_t$ actually maps $L^p(mathbb{R}^n)$ into $L^q(mathbb{R}^n)$. These results are based upon new dispersive estimates. The ingredients are on the one hand Compensated Integrability, and on the other hand a De Giorgi-type iteration.
This paper concerns the local well-posedness for the good Boussinesq equation subject to quasi-periodic initial conditions. By constructing a delicately and subtly iterative process together with an explicit combinatorial analysis, we show that there exists a unique solution for such a model in a small region of time. The size of this region depends on both the given data and the frequency vector involved. Moreover the local solution has an expansion with exponentially decaying Fourier coefficients.
In this paper, we consider the almost sure well-posedness of the Cauchy problem to the Cahn-Hilliard-Navier-Stokes equation with a randomization initial data on a torus $mathbb{T}^3$. First, we prove the local existence and uniqueness of solution. Furthermore, we prove the global existence and uniqueness of solution and give the relative probability estimate under the condition of small initial data.
In this paper, we first prove the local well-posedness of strong solutions to the incompressible Hall-MHD system with initial data $(u_0,B_0)in H^{frac{1}{2}+sigma}(mathbb{R}^3)times H^{frac{3}{2}}(mathbb{R}^3)$ and $sigmain (0,2)$. In particular, if the viscosity coefficient is equal to the resistivity coefficient, we can reduce $sigma$ to $0$ with the aid of the new formulation of the Hall-MHD system observed by cite{MR4193644}. Moreover, we establish the global well-posedness in $H^{frac{1}{2}+sigma}(mathbb{R}^3)times H^{frac{3}{2}}(mathbb{R}^3)$ with $sigmain (0,2)$ for small initial data and get the optimal time-decay rates of solutions. Our results improves some previous works.
We prove global well-posedness of the fifth-order Korteweg-de Vries equation on the real line for initial data in $H^{-1}(mathbb{R})$. By comparison, the optimal regularity for well-posedness on the torus is known to be $L^2(mathbb{R}/mathbb{Z})$. In order to prove this result, we develop a strategy for integrating the local smoothing effect into the method of commuting flows introduced previously in the context of KdV. It is this synthesis that allows us to go beyond the known threshold on the torus.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
