اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدGlobal solutions for the Muskat problem in the scaling invariant Besov space $dot B^1_{infty, 1}$
109
0
0.0
(
0
)
تأليف
Huy Q. Nguyen
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
The one-phase and two-phase Muskat problems with arbitrary viscosity contrast are studied in all dimensions. They are quasilinear parabolic equations for the graph free boundary. We prove that small data in the scaling invariant homogeneous Besov space $dot B^1_{infty, 1}$ lead to unique global solutions. The proof exploits a new structure of the Dirichlet-Neumann operator which allows us to implement a robust fixed-point argument. As a consequence of this method, the initial data is only assumed to be in $dot B^1_{infty, 1}$ and the solution map is Lipschitz continuous in the same topology. For the general Muskat problem, the only known scaling invariant result was obtained in the Wiener algebra (plus an $L^2$ assumption) which is strictly contained in $dot B^1_{infty, 1}$.
قيم البحث
اقرأ أيضاً
The free boundary problem for a two-dimensional fluid filtered in porous media is studied. This is known as the one-phase Muskat problem and is mathematically equivalent to the vertical Hele-Shaw problem driven by gravity force. We prove that if the
initial free boundary is the graph of a periodic Lipschitz function, then there exists a global-in-time Lipschitz solution in the strong $L^infty_t L^2_x$ sense and it is the unique viscosity solution. The proof requires quantitative estimates for layer potentials and pointwise elliptic regularity in Lipschitz domains. This is the first construction of unique global strong solutions for the Muskat problem with initial data of arbitrary size.
We show the existence of self-similar solutions for the Muskat equation. These solutions are parameterized by $0<s ll 1$; they are exact corners of slope $s$ at $t=0$ and become smooth in $x$ for $t>0$.
The singular limit of the thin film Muskat problem is performed when the density (and possibly the viscosity) of the lighter fluid vanishes and the porous medium equation is identified as the limit problem. In particular, the height of the denser flu
id is shown to converge towards the solution to the porous medium equation and an explicit rate for this convergence is provided in space dimension d $le$ 4. Moreover, the limit of the height of the lighter fluid is determined in a certain regime and is given by the corresponding initial condition.
For the thin obstacle problem in 3d, we show that half-space solutions form an isolated family in the space of $7/2$-homogeneous solutions. For a general solution with one blow-up profile in this family, we establish the rate of convergence to this p
rofile. As a consequence, we obtain regularity of the free boundary near such contact points.
We prove that the viscous Burgers equation has a globally defined smooth solution in all dimensions provided the initial condition and the forcing term are smooth and bounded together with their derivatives. Such solutions may have infinite energy. T
he proof does not rely on energy estimates, but on a combination of the maximum principle and quantitative Schauder estimates. We obtain precise bounds on the sup norm of the solution and its derivatives, making it plain that there is no exponential increase in time. In particular, these bounds are time-independent if the forcing term is zero. To get a classical solution, it suffices to assume that the initial condition and the forcing term have bounded derivatives up to order two.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
