اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدAn Eulerian-Lagrangian Form for the Euler Equations in Sobolev Spaces
119
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
In 2000 Constantin showed that the incompressible Euler equations can be written in an Eulerian-Lagrangian form which involves the back-to-labels map (the inverse of the trajectory map for each fixed time). In the same paper a local existence result is proved in certain Holder spaces $C^{1,mu}$. We review the Eulerian-Lagrangian formulation of the equations and prove that given initial data in $H^s$ for $ngeq2$ and $s>frac{n}{2}+1$, a unique local-in-time solution exists on the $n$-torus that is continuous into $H^s$ and $C^1$ into $H^{s-1}$. These solutions automatically have $C^1$ trajectories. The proof here is direct and does not appeal to results already known about the classical formulation. Moreover, these solutions are regular enough that the classical and Eulerian-Lagrangian formulations are equivalent, therefore what we present amounts to an alternative approach to some of the standard theory.
قيم البحث
اقرأ أيضاً
In the first part of this paper we establish a uniqueness result for continuity equations with velocity field whose derivative can be represented by a singular integral operator of an $L^1$ function, extending the Lagrangian theory in cite{BouchutCri
ppa13}. The proof is based on a combination of a stability estimate via optimal transport techniques developed in cite{Seis16a} and some tools from harmonic analysis introduced in cite{BouchutCrippa13}. In the second part of the paper, we address a question that arose in cite{FilhoMazzucatoNussenzveig06}, namely whether 2D Euler solutions obtained via vanishing viscosity are renormalized (in the sense of DiPerna and Lions) when the initial data has low integrability. We show that this is the case even when the initial vorticity is only in~$L^1$, extending the proof for the $L^p$ case in cite{CrippaSpirito15}.
This article is concerned with the local well-posedness problem for the compressible Euler equations in gas dynamics. For this system we consider the free boundary problem which corresponds to a physical vacuum. Despite the clear physical interest in
this system, the prior work on this problemis limited to Lagrangian coordinates, in high regularity spaces. Instead, the objective of the present work is to provide a new, fully Eulerian approach to this problem, which provides a complete, Hadamard style well-posedness theory for this problem in low regularity Sobolev spaces. In particular we give new proofs for both existence, uniqueness, and continuous dependence on the data with sharp, scale invariant energy estimates, and continuation criterion.
It is proved that modulation in time and space of periodic wave trains, of the defocussing nonlinear Schrodinger equation, can be approximated by solutions of the Whitham modulation equations, in the hyperbolic case, on a natural time scale. The erro
r estimates are based on existence, uniqueness, and energy arguments, in Sobolev spaces on the real line. An essential part of the proof is the inclusion of higher-order corrections to Whitham theory, and concomitant higher-order energy estimates.
We prove that the Cauchy problem for the dispersion generalized Benjamin-Ono equation [partial_t u+|partial_x|^{1+alpha}partial_x u+uu_x=0, u(x,0)=u_0(x),] is locally well-posed in the Sobolev spaces $H^s$ for $s>1-alpha$ if $0leq alpha leq 1$. The n
ew ingredient is that we develop the methods of Ionescu, Kenig and Tataru cite{IKT} to approach the problem in a less perturbative way, in spite of the ill-posedness results of Molinet, Saut and Tzvetkovin cite{MST}. Moreover, as a bi-product we prove that if $0<alpha leq 1$ the corresponding modified equation (with the nonlinearity $pm uuu_x$) is locally well-posed in $H^s$ for $sgeq 1/2-alpha/4$.
This short note investigates the compact embedding of degenerate matrix weighted Sobolev spaces into weighted Lebesgue spaces. The Sobolev spaces explored are defined as the abstract completion of Lipschitz functions in a bounded domain $Omega$ with
respect to the norm: $$|f|_{QH^{1,p}(v,mu;Omega)} = |f|_{L^p_v(Omega)} + | abla f|_{mathcal{L}^p_Q(mu;Omega)}$$ where the weight $v$ is comparable to a power of the pointwise operator norm of the matrix valued function $Q=Q(x)$ in $Omega$. Following our main theorem, we give an explicit application where degeneracy is controlled through an ellipticity condition of the form $$w(x)|xi|^p leq left(xicdot Q(x)xiright)^{p/2}leq tau(x)|xi|^p$$ for a pair of $p$-admissible weights $wleq tau$ in $Omega$. We also give explicit examples demonstrating the sharpness of our hypotheses.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
