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### Open Access Journals

NHM

Flow of two phases in a heterogeneous porous medium is modeled by a scalar conservation law with a discontinuous coefficient. As solutions of conservation laws with discontinuous coefficients depend explicitly on the underlying small scale effects, we consider a model where the relevant small scale effect is dynamic capillary pressure. We prove that the limit of vanishing dynamic capillary pressure exists and is a weak solution of the corresponding scalar conservation law with discontinuous coefficient. A robust numerical scheme for approximating the resulting limit solutions is introduced. Numerical experiments show that the scheme is able to approximate interesting solution features such as propagating non-classical shock waves as well as discontinuous standing waves efficiently.

DCDS

We prove the existence of a positive solution in $W_{loc}^{2,q}$ for a semilinear elliptic integro-differential problem in $\mathbb{R}^N.$
The integral operator of the equation depends on a nonlinear function that is singular in the origin. Moreover, we prove
that the averages of the solution and its gradient on the balls $\{x\in\mathbb{R}^N; |x| \le R\}, R>0,$
vanish as $R\to \infty.$

DCDS

We consider the high order Camassa-Holm equation, which is a non linear dispersive equation of the fifth order. We prove that as the diffusion and dispersion parameters tends to zero, the solutions converge to the entropy ones of a scalar conservation law. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the $L^p$ setting.

DCDS-S

We consider the Ibragimov-Shabat equation, which contains nonlinear dispersive effects. We prove
that as the diffusion parameter tends to zero, the solutions of the dispersive equation converge to discontinuous weak solutions of a scalar conservation law. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the $L^p$ setting.

CPAA

Let $\Omega$ be a smooth bounded domain in $R^N$ and let
\begin{eqnarray}
Lu=\sum_{j,k=1}^N \partial_{x_j}\left(a_{jk}(x)\partial_{x_k} u\right),
\end{eqnarray}
in $\Omega$ and
\begin{eqnarray}
Lu+\beta(x)\sum\limits_{j,k=1}^N a_{jk}(x)\partial_{x_j} u n_k+\gamma (x)u-q\beta(x)\sum_{j,k=1}^{N-1}\partial_{\tau_k}\left(b_{jk}(x)\partial_{\tau_j}u\right)=0,
\end{eqnarray}
on $\partial\Omega$
define a generalized Laplacian on $\Omega$ with a Wentzell boundary condition involving a generalized Laplace-Beltrami operator on the boundary.
Under some smoothness and positivity conditions on the coefficients, this defines a nonpositive selfadjoint operator, $-S^2$, on a suitable Hilbert space.
If we have a sequence of such operators $S_0,S_1,S_2,...$ with corresponding coefficients
\begin{eqnarray}
\Phi_n=(a_{jk}^{(n)},b_{jk}^{(n)}, \beta_n,\gamma_n,q_n)
\end{eqnarray}
satisfying $\Phi_n\to\Phi_0$ uniformly as $n\to\infty$, then $u_n(t)\to u_0(t)$ where $u_n$ satisfies
\begin{eqnarray}
i\frac{du_n}{dt}=S_n^m u_n,
\end{eqnarray}
or
\begin{eqnarray}
\frac{d^2u_n}{dt^2}+S_n^{2m} u_n=0,
\end{eqnarray}
or
\begin{eqnarray}
\frac{d^2u_n}{dt^2}+F(S_n)\frac{du_n}{dt}+S_n^{2m} u_n=0,
\end{eqnarray}
for $m=1,2,$ initial conditions independent of $n$, and for certain nonnegative functions $F$.
This includes Schrödinger equations, damped and undamped wave equations, and telegraph equations.

NHM

We consider the Kawahara-Korteweg-de Vries equation, which contains nonlinear dispersive effects. We prove
that as the dispersion parameter tends to zero, the solutions of the dispersive equation converge to discontinuous weak
solutions of the Burgers equation.
The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the $L^p$ setting.

DCDS

We consider a shallow water equation of Camassa-Holm type, which contains nonlinear dispersive effects. We prove
that as the diffusion parameter tends to zero, the solution of the dispersive equation converges to the unique entropy
solution of a scalar conservation law.
The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the $L^p$ setting.

DCDS

In this paper we prove the existence and regularity of positive solutions of the homogeneous Dirichlet problem
\begin{equation*}
-Δ u=g(x,u) in \Omega, u=0 on ∂ \Omega,
\end{equation*}
where $g(x,u)$ can be singular as $u\rightarrow0^+$ and $0\le g(x,u)\le\frac{\varphi_0(x)}{u^p}$ or $0\le$ $ g(x,u)$ $\le$ $\varphi_0(x)(1+\frac{1}{u^p})$, with $\varphi_0 \in L^m(\Omega), 1 ≤ m.$
There are no assumptions on the monotonicity of $g(x,\cdot)$ and the existence of super- or sub-solutions.

DCDS

We study a non-relativistic charged quantum particle moving
in a bounded open set $\Omega\subset\R^3$ with smooth boundary under the action of
a zero-range potential. In the electrostatic case the standing wave solutions take the form
$\psi(t,x)=u(x)e^{-i\omega t}$ where $u$ formally satisfies
$-\Delta u+\alpha\varphi u-\beta\delta_{x_0} u=\omega u$ and the electric potential $\varphi$ is given by
$-\Delta\varphi = u^2$. We introduce the definition of ground state. We show the existence of such solutions for each $\beta>0$ and the compactness as $\beta\to 0$.

DCDS

We show existence of a unique, regular global solution of the parabolic-elliptic system
$u_t +f(t,x,u)_x+g(t,x,u)+P_x=(a(t,x) u_x)_x$ and
$-P_{x x}+P=h(t,x,u,u_x)+k(t,x,u)$ with initial data
$u|_{t=0} = u_0$. Here inf$_(t,x) a(t,x)>0$. Furthermore, we show that the solution is
stable with respect to variation in the initial data $u_0$ and the functions $f$, $g$ etc.
Explicit stability estimates are provided.
The regularized generalized Camassa--Holm equation is a special case of the model we discuss.

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