NHM
Convergence of vanishing capillarity approximations for scalar conservation laws with discontinuous fluxes
Giuseppe Maria Coclite Lorenzo di Ruvo Jan Ernest Siddhartha Mishra
Networks & Heterogeneous Media 2013, 8(4): 969-984 doi: 10.3934/nhm.2013.8.969
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.
keywords: discontinuous fluxes Conservation laws capillarity approximation.
DCDS
Positive solutions of an integro-differential equation in all space with singular nonlinear term
Giuseppe Maria Coclite Mario Michele Coclite
Discrete & Continuous Dynamical Systems - A 2008, 22(4): 885-907 doi: 10.3934/dcds.2008.22.885
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 $R\to \infty.$
keywords: asymptotic behavior Integro-differential equations in all space singular nonlinearity existence of positive solutions
DCDS
A note on the convergence of the solution of the high order Camassa-Holm equation to the entropy ones of a scalar conservation law
Giuseppe Maria Coclite Lorenzo Di Ruvo
Discrete & Continuous Dynamical Systems - A 2017, 37(3): 1247-1282 doi: 10.3934/dcds.2017052

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.

keywords: Singular limit compensated compactness Camassa-Holm equation entropy condition
DCDS-S
A singular limit problem for the Ibragimov-Shabat equation
Giuseppe Maria Coclite Lorenzo di Ruvo
Discrete & Continuous Dynamical Systems - S 2016, 9(3): 661-673 doi: 10.3934/dcdss.2016020
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.
keywords: entropy condition. compensated compactness Singular limit Ibragimov-Shabat equation
CPAA
Continuous dependence in hyperbolic problems with Wentzell boundary conditions
Giuseppe Maria Coclite Angelo Favini Gisèle Ruiz Goldstein Jerome A. Goldstein Silvia Romanelli
Communications on Pure & Applied Analysis 2014, 13(1): 419-433 doi: 10.3934/cpaa.2014.13.419
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.
keywords: Wentzell boundary conditions higher order boundary operators. continuous dependence Wave equation semigroup approximation
NHM
A singular limit problem for conservation laws related to the Kawahara-Korteweg-de Vries equation
Giuseppe Maria Coclite Lorenzo di Ruvo
Networks & Heterogeneous Media 2016, 11(2): 281-300 doi: 10.3934/nhm.2016.11.281
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.
keywords: Singular limit Kawahara-Korteweg-de Vries equation entropy condition. compensated compactness
DCDS
A note on the convergence of the solutions of the Camassa-Holm equation to the entropy ones of a scalar conservation law
Giuseppe Maria Coclite Lorenzo di Ruvo
Discrete & Continuous Dynamical Systems - A 2016, 36(6): 2981-2990 doi: 10.3934/dcds.2016.36.2981
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.
keywords: compensated compactness Camassa-Holm equation entropy condition. Singular limit
DCDS
On a Dirichlet problem in bounded domains with singular nonlinearity
Giuseppe Maria Coclite Mario Michele Coclite
Discrete & Continuous Dynamical Systems - A 2013, 33(11&12): 4923-4944 doi: 10.3934/dcds.2013.33.4923
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.
keywords: Dirichlet boundary conditions. existence of positive solutions Singular nonlinear problems
DCDS
Ground states of the Schrödinger-Maxwell system with dirac mass: Existence and asymptotics
Giuseppe Maria Coclite Helge Holden
Discrete & Continuous Dynamical Systems - A 2010, 27(1): 117-132 doi: 10.3934/dcds.2010.27.117
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 $\beta\to 0$.
keywords: point interaction. Schrödinger-Maxwell system
DCDS
Wellposedness for a parabolic-elliptic system
Giuseppe Maria Coclite Helge Holden Kenneth H. Karlsen
Discrete & Continuous Dynamical Systems - A 2005, 13(3): 659-682 doi: 10.3934/dcds.2005.13.659
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.
keywords: wellposedness Camassa-Holm equation. Parabolic-elliptic system

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