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Communications on Pure & Applied Analysis

2003 , Volume 2 , Issue 1

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A sharp Sobolev inequality on Riemannian manifolds
YanYan Li and Tonia Ricciardi
2003, 2(1): 1-31 doi: 10.3934/cpaa.2003.2.1 +[Abstract](244) +[PDF](335.1KB)
Let $(M,g)$ be a smooth compact Riemannian manifold without boundary of dimension $n\ge 6$. We prove that

$||u||_{L^{2^*}(M,g)}^2 \le K^2\int_M\{|\nabla_{g} u|^2+c(n)R_{g} u^2\}dv_g +A||u||_{L^{2n/(n+2)}(M,g)}^2,$

for all $u\in H^1(M)$, where $2^*=2n/(n-2)$, $c(n)=(n-2)/[4(n-1)]$, $R_g$ is the scalar curvature, $K^{-1}=$ inf $\|\nabla u\|_{L^2(\mathbb R^n)}\|u\|_{L^{2n/(n-2)}(\mathbb R^n)}^{-1}$ and $A>0$ is a constant depending on $(M,g)$ only. The inequality is sharp

Polynomial upper bounds for the instability of the nonlinear Schrödinger equation below the energy norm
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao
2003, 2(1): 33-50 doi: 10.3934/cpaa.2003.2.33 +[Abstract](247) +[PDF](260.8KB)
We continue the study (initiated in [18]) of the orbital stability of the ground state cylinder for focussing non-linear Schrödinger equations in the $H^s(\R^n)$ norm for $1-\varepsilon < s < 1$, for small $\varepsilon$. In the $L^2$-subcritical case we obtain a polynomial bound for the time required to move away from the ground state cylinder. If one is only in the $H^1$-subcritical case then we cannot show this, but for defocussing equations we obtain global well-posedness and polynomial growth of $H^s$ norms for $s$ sufficiently close to 1.
A note on singular limits to hyperbolic systems of conservation laws
Stefano Bianchini
2003, 2(1): 51-64 doi: 10.3934/cpaa.2003.2.51 +[Abstract](281) +[PDF](203.4KB)
In this note we consider two different singular limits to hyperbolic system of conservation laws, namely the standard backward schemes for non linear semigroups and the semidiscrete scheme. Under the assumption that the rarefaction curve of the corresponding hyperbolic system are straight lines, we prove the stability of the solution and the convergence to the perturbed system to the unique solution of the limit system for initial data with small total variation. The method used here to estimate the source terms is based on the calculus of residues.
Positive solutions for a class of reaction-diffusion systems
A. Dall'Acqua
2003, 2(1): 65-76 doi: 10.3934/cpaa.2003.2.65 +[Abstract](357) +[PDF](199.2KB)
In this paper we will derive existence of positive solutions for a system of two coupled superlinear elliptic equations with Dirichlet boundary condition. We will use a topological method; in fact the class of systems that we will study does not allow a variational approach. After establishing an a priori estimate for the solutions, we will obtain existence by a continuation method.
Breakdown of $C^1$ solution to the Cauchy problem for quasilinear hyperbolic systems with characteristics with constant multiplicity
Libin Wang
2003, 2(1): 77-89 doi: 10.3934/cpaa.2003.2.77 +[Abstract](231) +[PDF](208.1KB)
In this paper we consider the Cauchy problem for quasilinear hyperbolic systems with characteristics with constant multiplicity. Suppose that characteristics with constant multiplicity ($>$1) are linearly degenerate only at $u=0$, if there is a genuinely nonlinear simple characteristic which does not have certain "monotonicity" and the initial data possess some decaying properties, we obtain the blow-up result for the $C^1$ solution to the Cauchy problem.
Analysis and numerical approximation of a class of two-way diffusions
G. Wei and P. Clifford
2003, 2(1): 91-99 doi: 10.3934/cpaa.2003.2.91 +[Abstract](274) +[PDF](255.9KB)
We consider a class of two-way diffusions with reflecting boundary conditions. We show that the problem can be reduced to the investigation of the solution of an Abel integral equation and the solution of two classical one-way diffusion problems. We approximate the solution of the integral equation by the product of a piecewise constant function and the known solution of the problem with infinite boundaries. A numerical solution of high accuracy is then obtained by solving a stable linear system.
On some problems related to de Giorgi’s conjecture
Changfeng Gui
2003, 2(1): 101-106 doi: 10.3934/cpaa.2003.2.101 +[Abstract](269) +[PDF](167.3KB)
We study the uniqueness of minimizers for the Allen-Cahn energy and the nonexistence of monotone stationary solutions for the Allen-Cahn equation with double well potentials of different depths.
$L^1$ continuous dependence for the Euler equations of compressible fluids dynamics
Paola Goatin and Philippe G. LeFloch
2003, 2(1): 107-137 doi: 10.3934/cpaa.2003.2.107 +[Abstract](311) +[PDF](266.2KB)
We prove the $L^1$ continuous dependence of entropy solutions for the $2 \times 2$ (isentropic) and the $3\times 3$ (non-isentropic) systems of inviscid fluid dynamics in one-space dimension. We follow the approach developed by the second author for solutions with small total variation to general systems of conservation laws in [11, 14]. For the systems of fluid dynamics under consideration here, our estimates are more precise and we cover entropy solutions with large total variation.

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