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Advances in Differential Equations
Advances in Differential Equations
Takasao, Keisuke
In this paper, we consider the Allen-Cahn equation with constraint. In 1994, Chen and
Elliott [9] studied the asymptotic behavior of the solution of the Allen-Cahn equation
with constraint. They proved that the zero level set of the solution converges to the
classical solution of the mean curvature flow under the suitable conditions on initial
data. In 1993, Ilmanen [20] proved the existence of the mean curvature flow via the
Allen-Cahn equation without constraint in the sense of Brakke. We proved the same
conclusion for the Allen-Cahn equation with constraint.
Shimizu, Senjo; Yagi, Shintaro
A basic model for incompressible two-phase flows with phase transitions where the
interface is nearly flat in the case of non-equal densities is considered. Local
well-posedness of the model in $L_p$ in a time setting $L_q$ in a space setting was proved
in [30] under a smallness assumption for the initial data. In this paper, we remove the
smallness assumption for the initial data.
Mitra, Debanjana; Ramaswamy, Mythily; Raymond, Jean-Pierre
In this paper, we study the local stabilization of one dimensional compressible
Navier-Stokes equations around a constant steady solution $(\rho_s, u_s)$, where
$\rho_s>0, u_s\neq 0$. In the case of periodic boundary conditions, we determine a
distributed control acting only in the velocity equation, able to stabilize the system,
locally around $(\rho_s, u_s)$, with an arbitrary exponential decay rate. In the case
of Dirichlet boundary conditions, we determine boundary controls for the velocity and for
the density at the inflow boundary, able to stabilize the system, locally around $(\rho_s,
u_s)$, with an arbitrary exponential decay rate.
Murata, Minoru; Tsuchida, Tetsuo
We consider positive solutions of elliptic partial differential equations on non-compact
domains of Riemannian manifolds. We establish general theorems which determine Martin
compactifications and Martin kernels for a wide class of elliptic equations in skew
product form, by thoroughly exploiting parabolic Martin kernels for associated parabolic
equations developed in [35] and [25]. As their applications, we explicitly determine the
structure of all positive solutions to a Schrödinger equation and the Martin boundary
of the product of Riemannian manifolds. For their sharpness, we show that the Martin
compactification of ${\mathbb R}^2$ for some Schrödinger equation is so much
distorted near infinity that no product structures remain.
Suzuki, Takuya
This paper shows the analyticity of semigroups generated by higher order divergence type
elliptic operators in $L^{\infty}$ spaces in $C^{1}$ domains which may be unbounded. For
this purpose, we establish resolvent estimates in $L^{\infty}$ spaces by a contradiction
argument based on a blow-up method. Our results yield the $L^{\infty}$ analyticity of
solutions of parabolic equations for $C^{1}$ domains.
Schnaubelt, Roland
We develop a wellposedness and regularity theory for a large class of quasilinear
parabolic problems with fully nonlinear dynamical boundary conditions. Moreover, we
construct and investigate stable and unstable local invariant manifolds near a given
equilibrium. In a companion paper, we treat center, center-stable and center-unstable
manifolds for such problems and investigate their stability properties. This theory
applies e.g. to reaction-diffusion systems with dynamical boundary conditions and to the
two-phase Stefan problem with surface tension.
Ruzhansky, Michael; Suragan, Durvudkhan
We prove local refined versions of Hardy's and Rellich's inequalities as well as of
uncertainty principles for sums of squares of vector fields on bounded sets of smooth
manifolds under certain assumptions on the vector fields. We also give some explicit
examples, in particular, for sums of squares of vector fields on Euclidean spaces and for
sub-Laplacians on stratified Lie groups.
Benoit, Mesognon-Gireau
We prove, in this paper, a long time existence result for a modified Boussinesq-Peregrine
equation in dimension $1$, describing the motion of Water Waves in shallow water, in the
case of a non flat bottom. More precisely, the dimensionless equations depend strongly on
three parameters $\epsilon,\mu,\beta$ measuring the amplitude of the waves, the
shallowness and the amplitude of the bathymetric variations, respectively. For the
Boussinesq-Peregrine model, one has small amplitude variations ($\epsilon = O(\mu)$). We
first give a local existence result for the original Boussinesq Peregrine equation as
derived by Boussinesq ([9], [8]) and Peregrine ([22]) in all dimensions. We then introduce
a new model which has formally...
Quittner, Pavol
If $p>1+2/n$, then the equation $u_t-\Delta u = u^p, \quad x\in{\mathbb R}^n,\ t>0,$
possesses both positive global solutions and positive solutions which blow up in finite
time. We study the large time behavior of radial positive solutions lying on the
borderline between global existence and blow-up.
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.