Recursos de colección
Project Euclid (Hosted at Cornell University Library) (192.320 recursos)
Advances in Differential Equations
Advances in Differential Equations
Cabada, Alberto; Saavedra, Lorena
The aim of this paper consists on the study of the following fourth-order
operator:
\begin{equation*}
T[M]\,u(t)\equiv u^{(4)}(t)+p_1(t)u'''(t)+p_2(t)u''(t)+Mu(t),\ t\in I \equiv
[a,b] ,
\end{equation*}
coupled with the two point boundary conditions:
\begin{equation*}
u(a)=u(b)=u''(a)=u''(b)=0 .
\end{equation*}
So, we define the following space:
\begin{equation*}
X=\left\lbrace u\in C^4(I) : u(a)=u(b)=u''(a)=u''(b)=0 \right\rbrace .
\end{equation*}
Here, $p_1\in C^3(I)$ and $p_2\in C^2(I)$.
By assuming that the second order linear differential equation
\begin{equation*}
L_2\, u(t)\equiv u''(t)+p_1(t)\,u'(t)+p_2(t)\,u(t)=0\,,\quad t\in I,
\end{equation*}
is disconjugate on $I$, we characterize the parameter's set
where the Green's function related to operator $T[M]$ in $X$
is of constant sign on $I \times I$. Such a characterization
is equivalent to the strongly inverse positive (negative)
character of operator $T[M]$ on $X$ and comes from the
first eigenvalues of operator $T[0]$...
Yan, Wei; Li, Yongsheng; Zhai, Xiaoping; Zhang, Yimin
In this paper, we investigate the Cauchy problem
for the shallow water type equation
\begin{align*}
u_{t}+\partial_{x}^{3}u
+ \tfrac{1}{2}\partial_{x}(u^{2})+\partial_{x}
(1-\partial_{x}^{2})^{-1}\left[u^{2}+\tfrac{1}{2}
u_{x}^{2}\right]=0, \ \ x\in {\mathbf T}={\mathbf R}/2\pi
\lambda,
\end{align*}
with low regularity data and $\lambda\geq1$. By applying the bilinear
estimate in $W^{s}$, Himonas and Misiołek (Commun. Partial Diff. Eqns.,
23 (1998), 123-139) proved that the problem is locally
well-posed in $H^{s}([0, 2\pi))$ with $s\geq {1}/{2}$
for small initial data. In this paper, we show that, when $s < {1}/{2}$, the bilinear
estimate in $W^{s}$ is invalid. We also demonstrate that the bilinear
estimate in $Z^{s}$ is indeed valid for ${1}/{6} < s < {1}/{2}$.
This enables us to prove that the
problem is locally well-posed in $H^{s}(\mathbf{T})$ with
${1}/{6}...
Holmes, John; Thompson, Ryan C.
In this paper, well-posedness in $C^1(\mathbb R)$ (a.k.a. classical solutions)
for a generalized Camassa-Holm equation (g-$k$$b$CH) having
$(k+1)$-degree nonlinearities is shown. This result holds for the
Camassa-Holm, the Degasperis-Procesi and the Novikov equations,
which improves upon earlier results in Sobolev and Besov spaces.
Wolf, Jörg
In the study of local regularity of weak solutions to systems
related to incompressible viscous fluids local energy estimates
serve as important ingredients. However, this requires certain
information on the pressure. This fact has been used by
V. Scheffer in the notion of a suitable weak solution
to the Navier-Stokes equations, and in the proof of
the partial regularity due to Caffarelli, Kohn and Nirenberg.
In general domains, or in case of complex viscous fluid
models a global pressure does not necessarily exist.
To overcome this problem, in the present paper we construct
a local pressure distribution by showing that every
distribution $ \partial _t \boldsymbol u +\boldsymbol F $, which...
Zagatti, Sandro
We study the Dirichlet problem for stationary
Hamilton-Jacobi equations
$$
\begin{cases}
H(x, u(x), \nabla u(x)) = 0 & \ \textrm{in} \ \Omega \\
u(x)=\varphi(x) & \ \textrm{on} \ \partial \Omega.
\end{cases}
$$
We consider a Caratheodory hamiltonian $H=H(x,u,p)$, with a
Sobolev-type
(but not continuous) regularity
with respect to the space variable $x$, and prove existence and
uniqueness
of a Lipschitz continuous maximal generalized solution which, in the
continuous
case, turns out to be the classical viscosity solution.
In addition, we prove a continuous dependence property of the solution
with respect to the boundary datum $\varphi$, completing
in such a way a well posedness theory.
Carreño, N.
In this paper, we prove the existence of controls insensitizing the
$L^2$-norm of the solution of the Boussinesq system. The novelty here
is that no control is used on the temperature equation. Furthermore,
the control acting on the fluid equation can be chosen to have one
vanishing component. It is well known that the insensitizing control
problem is equivalent to a null controllability result for a cascade
system, which is obtained thanks to a suitable Carleman estimate for
the adjoint of the linearized system and an inverse mapping theorem.
The particular form of the adjoint equation will allow us to obtain
the null controllability of the linearized system.
Bui, The Anh; Duong, Xuan
Let $(X, d, \mu)$ be a space of homogeneous type equipped with a
distance $d$ and a measure $\mu$. Assume that $L$ is a closed linear
operator which
generates an analytic semigroup $e^{-tL}, t > 0$. Also assume that
$L$ has a bounded $H_\infty$-calculus on $L^2(X)$ and satisfies the
$L^p-L^q$
semigroup estimates (which is weaker than the pointwise Gaussian or Poisson heat kernel
bounds). The aim of this paper is to establish a theory of
inhomogeneous Besov spaces associated to such an operator $L$. We
prove the molecular decompositions for the new Besov spaces
and obtain the boundedness of the fractional powers $(I+L)^{-\gamma},
\gamma > 0$ on these Besov spaces.
Finally, we...
Gallarati, Chiara; Veraar, Mark
In this paper, we prove maximal $L^p$-regularity for
a system of parabolic PDEs, where the elliptic operator
$A$ has coefficients which depend on time in a measurable
way and are continuous in the space variable. The proof
is based on operator-theoretic methods and one of the
main ingredients in the proof is the construction of
an evolution family on weighted $L^q$-spaces.
Auscher, Pascal; Egert, Moritz
We study second-order divergence-form systems on
half-infinite cylindrical domains with a bounded
and possibly rough base, subject to homogeneous
mixed boundary conditions on the lateral boundary
and square integrable Dirichlet, Neumann, or
regularity data on the cylinder base. Assuming that
the coefficients $A$ are close to coefficients $A_0$
that are independent of the unbounded direction with
respect to the modified Carleson norm of Dahlberg,
we prove a priori estimates and establish
well-posedness if $A_0$ has a special structure.
We obtain a complete characterization of weak
solutions whose gradient either has an $L^2$-bounded
non-tangential maximal function or satisfies a
Lusin area bound. To this end, we combine the
first-order approach to elliptic systems with
the Kato square...
Galise, Giulio; Vitolo, Antonio
In this paper, we introduce some fully nonlinear
second order operators defined as weighted partial sums
of the eigenvalues of the Hessian matrix, arising in
geometrical contexts, with the aim to extend maximum
principles and removable singularities results to
cases of highly degenerate ellipticity.
Hu, Liang-Gen
In this paper, we are concerned with the weighted elliptic system
\begin{equation*}
\begin{cases}
-\Delta u=|x|^{\beta} v^{\vartheta},\\
-\Delta v=|x|^{\alpha} |u|^{p-1}u,
\end{cases}\quad \mbox{in}\;\ \mathbb{R}^N,
\end{equation*}where $N \ge 5$, $\alpha >-4$, $
0 \le \beta < N-4$, $p>1$ and $\vartheta=1$.
We first apply Pohozaev identity to construct a
monotonicity formula
and reveal their certain equivalence relation.
By the use of
{\it Pohozaev identity}, {\it monotonicity formula}
of solutions together with a {\it blowing down}
sequence,
we prove Liouville-type theorems for stable solutions
(whether positive or sign-changing) of the weighted elliptic system
in the higher dimension.
Boulakia, M.; Guerrero, S.
In this paper, we consider an elastic structure immersed in a compressible
viscous fluid. The motion of the fluid
is described by the compressible
Navier-Stokes equations whereas the motion
of the structure is given by the
nonlinear Saint-Venant Kirchhoff model.
For this model, we prove the existence and
uniqueness of regular solutions defined
locally in time. To do so, we first
rewrite the nonlinearity in the elasticity
equation in an adequate way. Then, we
introduce a linearized problem and prove
that this problem admits a unique regular
solution. To obtain time regularity on the
solution, we use energy estimates on the
unknowns and their successive derivatives
in time and to
obtain spatial regularity, we use elliptic
estimates. At...
Ascanelli, Alessia; Boiti, Chiara; Zanghirati, Luisa
We consider $p$-evolution equations, for $p\geq2$, with complex valued coefficients. We prove that a necessary condition for $H^\infty$ well-posedness
of the associated Cauchy problem is that the imaginary part of the coefficient of the subprincipal part (in the sense of Petrowski) satisfies a decay estimate as $|x|\to+\infty$.
Kondratyev, Stanislav; Monsaingeon, Léonard; Vorotnikov, Dmitry
We introduce a new optimal transport distance between nonnegative finite Radon measures with possibly different masses. The
construction is based on non-conservative continuity equations and a corresponding modified Benamou-Brenier formula. We establish various topological and
geometrical properties of the resulting metric space, derive some formal Riemannian structure, and develop differential calculus following F. Otto's approach. Finally, we apply
these ideas to identify a model of animal dispersal proposed by MacCall and Cosner as a gradient flow in our formalism and obtain new long-time convergence results.
Yamazaki, Kazuo
We study the three-dimensional stochastic nonhomogeneous magnetohydrodynamics system with random external forces that involve feedback, i.e.,
multiplicative noise, and are non-Lipschitz. We prove the existence of a global martingale solution via a semi-Galerkin approximation scheme with stochastic
calculus and applications of Prokhorov's and Skorokhod's theorems. Furthermore, using de Rham's theorem for processes, we prove the existence of the pressure term.
Ao, Weiwei; Wei, Juncheng; Yao, Wei
We study uniqueness of sign-changing radial solutions for the following semi-linear elliptic equation
\begin{align*}
\Delta u-u+|u|^{p-1}u=0\quad{\rm{in}}\
\mathbb{R}^N,\quad u\in H^1(\mathbb{R}^N),
\end{align*}
where $1 < p < \frac{N+2}{N-2}$, $N\geq3$.
It is well-known that this equation has a unique positive radial solution. The existence of sign-changing radial solutions with exactly $k$
nodes is also known. However, the uniqueness of such solutions is open. In this paper, we show that such sign-changing radial
solution is unique when $p$ is close to $\frac{N+2}{N-2}$. Moreover, those solutions are non-degenerate, i.e., the kernel of
the linearized operator is exactly $N$-dimensional.
De Anna, Francesco
In this paper, we obtain a result about the global existence of weak solutions to the d-dimensional Boussinesq-Navier-Stokes system,
with viscosity dependent on temperature. The initial temperature is only supposed to be bounded, while the initial velocity
belongs to some critical Besov Space, invariant to the scaling of this system. We suppose the viscosity close enough to a positive constant, and the
$L^\infty$-norm of their difference plus the Besov norm of the horizontal component of the initial velocity is supposed to be exponentially small with respect
to the vertical component of the initial velocity. In the preliminaries, and in the appendix, we consider some...
Zhou, Deqin; Zhang, Bing-Yu
In this paper, we consider
the initial boundary value problem (IBVP) of the Hamiltonian
fifth-order KdV equation posed on a finite interval $(0,L)$,
$$
\begin{cases}
\partial_t u-\partial_x^{5}u= c_1
u\partial_x u+ c_2 u^2 \partial _x u +
2b\partial_x u\partial_x^{2}u+bu\partial_x^{3}
u, \quad x\in (0,L), \ t>0 \\ u(0,x)=\phi (x) ,
\ x\in (0,L)\\
u(t,0)=\partial_x u(t,0)=u(t,L)=\partial_x
u(t,L)=\partial_x^{2}u(t,L)=0, \quad t>0,
\end{cases}
$$
and show that, given $0\leq s\leq 5$
and $T>0$, for any $\phi \in H^s (0,L) $
satisfying the natural compatibility
conditions, the IBVP admits a unique solution
$$
u\in L^{\infty}_{loc} (\mathbb R^+; H^s(0,L))\cap
L^2 _{loc}(\mathbb R^+; H^{s+2} (0,L)).
$$
Moreover, the corresponding solution map
is shown to be locally Lipschtiz continuous
from $L^2 (0,L)$ to
$L^{\infty}(0,T; L^2 (0,L))\cap L^2 (0,T; H^2 (0,L))$ and
from $H^5 (0,L)$ to
$L^{\infty}(0,T; H^5...
Bellec, Stevan; Colin, Mathieu
In this paper, we present a long time existence theory
for a new enhanced Boussinesq-Type system with
constant bathymetry written in a conservative form.
We also prove the existence of solitary wave for
a large class of asymptotic models, including
Beji-Nadaoka, Madsen-Sorensen and Nwogu equations.
Furthermore, we give a procedure to calculate
numerically these particular solutions and we
present some effective computations.
Kozlov, Vladimir; Nazarov, Sergei A.
The spectral problem of anisotropic elasticity
with traction-free boundary conditions is considered in a bounded
domain with a spatial cusp having its vertex at the origin and given
by triples $(x_1,x_2,x_3)$ such that $x_3^{-2}(x_1,x_2) \in \omega$,
where $\omega$ is a two-dimensional Lipschitz domain with a compact
closure. We show that there exists a threshold $\lambda_\dagger>0$
expressed explicitly in terms of the elasticity constants and the
area of $\omega$ such that the continuous spectrum coincides with the
half-line $[\lambda_\dagger,\infty)$, whereas the interval
$[0,\lambda_\dagger)$ contains only the discrete spectrum.
The asymptotic formula for solutions to this spectral
problem near cusp's vertex is also derived.
A principle feature of this asymptotic formula is the
dependence of...