اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدMulti-dimensional Burgers equation with unbounded initial data: well-posedness and dispersive estimates
125
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
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.
قيم البحث
اقرأ أيضاً
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 classica
l 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.
We discuss the minimal integrability needed for the initial data, in order that the Cauchy problem for a multi-dimensional conservation law admit an entropy solution. In particular we allow unbounded initial data. We investigate also the decay of the
solution as time increases, in relation with the nonlinearity. The main ingredient is our recent theory of divergence-free positive symmetric tensor. We apply in particular the so-called compensated integrability to a tensor which generalizes the one that L. Tartar used in one space dimension. It allows us to establish a Strichartz-like inequality, in a quasilinear context. This program is carried out in details for a multi-dimensional version of the Burgers equation.
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. Fu
rthermore, 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 address the local well-posedness of the spatially inhomogeneous non-cutoff Boltzmann equation when the initial data decays polynomially in the velocity variable. We consider the case of very soft potentials $gamma + 2s < 0$. Our mai
n result completes the picture for local well-posedness in this decay class by removing the restriction $gamma + 2s > -3/2$ of previous works. Our approach is entirely based on the Carleman decomposition of the collision operator into a lower order term and an integro-differential operator similar to the fractional Laplacian. Interestingly, this yields a very short proof of local well-posedness when $gamma in (-3,0]$ and $s in (0,1/2)$ in a weighted $C^1$ space that we include as well.
We provide a new analysis of the Boltzmann equation with constant collision kernel in two space dimensions. The scaling-critical Lebesgue space is $L^2_{x,v}$; we prove global well-posedness and a version of scattering, assuming that the data $f_0$ i
s sufficiently smooth and localized, and the $L^2_{x,v}$ norm of $f_0$ is sufficiently small. The proof relies upon a new scaling-critical bilinear spacetime estimate for the collision gain term in Boltzmanns equation, combined with a novel application of the Kaniel-Shinbrot iteration.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
