Nonlinear Differential Equations and Applications NoDEA
1420-9004
1021-9722
Cơ quản chủ quản: Birkhauser Verlag Basel , SPRINGER INT PUBL AG
Lĩnh vực:
AnalysisApplied Mathematics
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Các bài báo tiêu biểu
Semi-linear fractional $$\varvec{\sigma }$$ -evolution equations with mass or power non-linearity
Tập 25 - Trang 1-43 - 2018
In this paper we study the global (in time) existence of small data solutions to semi-linear fractional
$$\sigma $$
-evolution equations with mass or power non-linearity. Our main goal is to explain on the one hand the influence of the mass term and on the other hand the influence of higher regularity of the data on qualitative properties of solutions. Using modified Bessel functions we prove some polynomial decay in
$$L^p-L^q$$
estimates for solutions to the corresponding linear fractional
$$\sigma $$
-evolution equations with vanishing right-hand sides. By a fixed point argument the existence of small data solutions is proved for some admissible range of powers p.
On the behavior in time of solutions to motion of Non-Newtonian fluids
Tập 27 - Trang 1-18 - 2020
We study the behavior on time of weak solutions to the non-stationary motion of an incompressible fluid with shear rate dependent viscosity in bounded domains when the initial velocity
$${u}_0 {\in } {L}^2$$
. Our estimates show the different behavior of the solution as the growth condition of the stress tensor varies. In the “dilatant” or “shear thickening” case we prove that the decay rate does not depend on
$$u_0$$
, then our estimates also apply for irregular initial velocity.
On the global well-posedness of a class of Boussinesq–Navier–Stokes systems
Tập 18 Số 6 - Trang 707-735 - 2011
On C 1+α regularity of solutions of Isaacs parabolic equations with VMO coefficients
Tập 21 - Trang 63-85 - 2013
We prove that boundary value problems for fully nonlinear second-order parabolic equations admit L
p
-viscosity solutions, which are in C
1+α
for an
$${\alpha \in (0, 1)}$$
. The equations have a special structure that the “main” part containing only second-order derivatives is given by a positive homogeneous function of second-order derivatives and as a function of independent variables it is measurable in the time variable and, so to speak, VMO in spatial variables.
A family of parameter-dependent diffeomorphisms acting on function spaces over a Riemannian manifold and applications to geometric flows
Tập 22 - Trang 45-85 - 2014
It is the purpose of this article to establish a technical tool to study regularity of solutions to parabolic equations on manifolds. As applications of this technique, we prove that solutions to the Ricci-DeTurck flow, the surface diffusion flow and the mean curvature flow enjoy joint analyticity in time and space, and solutions to the Ricci flow admit temporal analyticity.
The Rayleigh–Taylor instability for the Verigin problem with and without phase transition
Tập 26 - Trang 1-35 - 2019
Isothermal compressible two-phase flows in a capillary are modeled with and without phase transition in the presence of gravity, employing Darcy’s law for the velocity field. It is shown that the resulting systems are thermodynamically consistent in the sense that the available energy is a strict Lyapunov functional. In both cases, the equilibria with flat interface are identified. It is shown that the problems are well-posed in an
$$L_p$$
-setting and generate local semiflows in the proper state manifolds. The main result concerns the stability of equilibria with flat interface, i.e. the Rayleigh–Taylor instability.
Existence in the nonlinear Schrödinger equation with bounded magnetic field
Tập 29 - Trang 1-14 - 2022
The paper studies existence of ground states for the nonlinear Schrödinger equation
0.1
$$\begin{aligned} -(\nabla + {\mathbf {i}}A(x))^2u+V(x)u=|u|^{p-1}u ,\quad 2
Global center manifolds in singular systems
Tập 3 - Trang 19-34 - 1996
The problem of existence of aglobal center manifold for a system of O.D.E. like
(*)
$$\left\{ {\begin{array}{*{20}c} {\dot x = A(y)x + F(x,y)} \\ {\dot y = G(x,y), (x,y) \in \mathbb{R}^n \times \mathbb{R}^m ,} \\ \end{array} } \right.$$
is considered. We give conditions onA(y), F(x, y), G(x, y) in order that a functionH: ℝ
m
→ℝ
n
, with the same smoothness asA(y), F(x, y), G(x, y), exists and is such that the manifoldC={(x,y)∈ℝ
n
×ℝ
m
∣x=H(y),y∈ℝ
m
} is an invariant manifold for (*), and there exists ρ>0 such that any solution of (*) satisfying sup
t∈ℝ∣x(t)∣ <ρ must belong toC. This is why we callC global center manifold. Applications are given to the problem of existence of heteroclinic orbits in singular systems.
Global inversion of functions: an introduction
Tập 1 - Trang 229-248 - 1994
This is an exposition of some basic ideas in the realm of Global Inverse Function theorems. We address ourselves mainly to readers who are interested in the applications to Differential Equations. But we do not deal with those applications and we give a ‘self-contained” elementary exposition. The first part is devoted to the celebrated Hadamard-Caccioppoli theorem on proper local homeomorphisms treated in the framework of the Hausdorff spaces. In the proof, the concept of ‘ω-limit set’ is used in a crucial way and this is perhaps the novelty of our approach. In the second part we deal with open sets in Banach spaces. The concept of ‘attraction basin’ here is the main tool of our exposition which also shows a few recent results, here extended from finite dimensional to general Banach spaces, together with the classical theorem of Hadamard-Levy which assumes that the operator norm of the inverse of the derivative does not grow too fast (roughly at most linearly).