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SIAM J. on Mathematical Analysis

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2000

Volume 31, Issue 6, pp. 1175-1404


Interface Behavior of Compressible Navier--Stokes Equations with Vacuum

Tao Luo, Zhouping Xin, and Tong Yang

SIAM J. Math. Anal. 31, pp. 1175-1191 (17 pages) | Cited 26 times

Online Publication Date: August 01, 2006

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In this paper, we study a one-dimensional motion of viscous gas near vacuum with (or without) gravity. We are interested in the case that the gas is in contact with the vacuum at a finite interval. This is a free boundary problem for the one-dimensional isentropic Navier--Stokes equations, and the free boundaries are the interfaces separating the gas from vacuum, across which the density changes continuously. The regularity and behavior of the solutions near the interfaces and expanding rate of the interfaces are studied. Smoothness of the solutions is discussed. The uniqueness of the weak solutions to the free boundary problem is also proved.

Group Analysis of Differential Equations and Generalized Functions

Michael Kunzinger and Michael Oberguggenberger

SIAM J. Math. Anal. 31, pp. 1192-1213 (22 pages) | Cited 7 times

Online Publication Date: August 01, 2006

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We present an extension of the methods of classical Lie group analysis of differential equations to equations involving generalized functions (in particular: distributions). A suitable framework for such a generalization is provided by Colombeau's theory of algebras of generalized functions. We show that under some mild conditions on the differential equations, symmetries of classical solutions remain symmetries for generalized solutions. Moreover, we introduce a generalization of the infinitesimal methods of group analysis that allows us to compute symmetries of linear and nonlinear differential equations containing generalized function terms. Thereby, the group generators and group actions may be given by generalized functions themselves.

Discrete and Continuous Dirichlet-to-Neumann Maps in the Layered Case

David V. Ingerman

SIAM J. Math. Anal. 31, pp. 1214-1234 (21 pages) | Cited 5 times

Online Publication Date: August 01, 2006

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Every sufficiently regular nonnegative function $\gamma$ (conductivity) on the closed unit disk $\overline{\mathbb{D}}$ induces the Dirichlet-to-Neumann map $\Lambda_{\gamma}$ on functions on $\partial\mathbb{D}$. The main inverse problems are to give a characterization of the maps $\Lambda_{\gamma}$ and to find out when $\Lambda_{\gamma}$ uniquely determines $\gamma$. In this paper we consider the case of conductivities that are constant on circles centered at the origin, and a discrete analogue of this so called i layered case. We characterize a closure of the set of the layered Dirichlet-to-Neumann maps in terms of their kernels and spectra. We give sharp conditions for the uniqueness in the discrete inverse problem, and conditions on $\gamma$ for the uniqueness in the continuous problem that we conjecture to be sharp. The characterization in terms of the spectra shows that continuous Dirichlet-to-Neumann maps can be viewed as limits of the discrete Dirichlet-to-Neumann maps. The characterization in terms of the kernels supports the conjecture in [D. Ingerman and J. Morrow, SIAM J. Math. Anal., 29 (1998), pp. 106--115] that the alternating property essentially characterizes continuous Dirichlet-to-Neumann maps. The characterizations above give a new interpretation of connections between positive measures, positive definite functions, and analytic functions that map the right half-plane to itself in the Bochner and Herglotz theorems.

KAM Theory and a Partial Justification of Greene's Criterion for Nontwist Maps

Amadeu Delshams and Rafael de la Llave

SIAM J. Math. Anal. 31, pp. 1235-1269 (35 pages) | Cited 25 times

Online Publication Date: August 01, 2006

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We consider perturbations of integrable, area preserving nontwist maps of the annulus (those are maps in which the twist condition changes sign). These maps appear in a variety of applications, notably transport in atmospheric Rossby waves.
We show in suitable two-parameter families the persistence of critical circles (invariant circles whose rotation number is the maximum of all the rotation numbers of points in the map) with Diophantine rotation number. The parameter values with critical circles of frequency $\omega_0$ lie on a one-dimensional analytic curve.
Furthermore, we show a partial justification of Greene's criterion: If analytic critical curves with Diophantine rotation number $\omega_0$ exist, the residue of periodic orbits (that is, one fourth of the trace of the derivative of the return map minus 2) with rotation number converging to $\omega_0$ converges to zero exponentially fast. We also show that if analytic curves exist, there should be periodic orbits approximating them and indicate how to compute them.
These results justify, in particular, conjectures put forward on the basis of numerical evidence in [D. del Castillo-Negrete, J.M. Greene, and P.J. Morrison, Phys. D., 91 (1996), pp. 1--23]. The proof of both results relies on the successive application of an iterative lemma which is valid also for 2d-dimensional exact symplectic diffeomorphisms. The proof of this iterative lemma is based on the deformation method of singularity theory.

Global Behavior of the Cauchy Problem for Some Critical Nonlinear Parabolic Equations

J. A. Aguilar Crespo and I. Peral Alonso

SIAM J. Math. Anal. 31, pp. 1270-1294 (25 pages) | Cited 5 times

Online Publication Date: August 01, 2006

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The paper deals with the following nonlinear parabolic problem: u_t- \lp u &=& \lambda \dfrac {u^{p-1}}{|x|^p} \quad x \in \ren, \, t > 0, \\ u(x,0) &=& u_0 (x) \ge 0, where 1 < p < N and $ \lambda > 0 $. The existence or nonexistence (blow-up) of global solution is analyzed. Also the finite time extinction for solutions in the case 1 < p<2 in bounded and unbounded domains is studied. This behavior depends on the relationships between $ \lambda $, N, p and the integrability of u0.

Identification of Two-Phase Free Boundary Arising in Plasma Physics

June-Yub Lee and Jin Keun Seo

SIAM J. Math. Anal. 31, pp. 1295-1306 (12 pages)

Online Publication Date: August 01, 2006

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We try to estimate the shape and the location of two-phase free boundary which has been studied in [A. Friedman and Y. Liu, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 22 (1995), pp. 375--448] to model a stationary magnetohydrodynamics system. A sufficient condition is obtained to check whether a test disk is included in the plasma region D surrounded by a two-phase free boundary. In the test disk technique, only two simply verifiable conditions are used and indispensableness of the conditions is demonstrated using an example. The technique is applicable to select some of test disks placed in the domain $\Omega$, which gives a rough guess on the shape of plasma region. Next we draw some geometrical properties of plasma region D when the domain $\Omega$ possesses a kind of convexity. It is proved that if $\Omega$ itself contains the mirror image of the right portion $\{x \in \Omega : x \cdot \xi > t \}$ of the domain with respect to a line $\{x: x \cdot \xi = t \}$ for all t > t0, then so does the plasma region.

Global Continuation via Higher-Gradient Regularization and Singular Limits in Forced One-Dimensional Phase Transitions

Timothy J. Healey and Hansjörg Kielhöfer

SIAM J. Math. Anal. 31, pp. 1307-1331 (25 pages) | Cited 5 times

Online Publication Date: August 01, 2006

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We consider a standard "higher-gradient" model for forced phase transitions in one-dimensional, shape-memory solids. We prescribe a parameter-dependent body forcing. The component of the potential energy corresponding to conventional elasticity is characterized by a nonconvex stored energy function of the strain. Our main goal is to show that global solution branches of the regularized problem converge to a global branch of weak solutions in the limit of vanishing "capillarity" (the coefficient of the higher-gradient term). The existence of global branches for the regularized, semilinear problem is routine, based upon the Leray--Schauder degree. In the physically meaningful case when the body force is everywhere nonnegative, we obtain uniform a priori bounds via a subtle maximum principle. This together with topological connectivity arguments yields the existence of global branches of weak solutions to the zero-capillarity problem. Moreover, by examining the singular limits of various supplementary conservation laws (satisfied by all solutions of the regularized problem), we show that the above-mentioned weak solutions also minimize the potential energy of the zero-capillarity problem.

A Complete Solution Characterizing Smooth Refinable Functions

Vladimir Protasov

SIAM J. Math. Anal. 31, pp. 1332-1350 (19 pages) | Cited 2 times

Online Publication Date: August 01, 2006

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In this paper we study univariate two-scale refinement equations $\varphi(x)=\sum_{k \in \zed} c_k\varphi\break(2x-k)$, where the coefficients $c_k \in \cx$ satisfy an exponential decay assumption. We show that any refinement equation that has a smooth solution can be reduced to the well-studied case of complete sum rules: $\sum_k(-1)^kk^nc_k = 0,\ \ n=0, . . . L, where L depends on regularity of the solution. This result makes it possible to extend previously known results on refinable functions and subdivision schemes from the case of complete sum rules to the general case. As a corollary we obtain sharp necessary conditions for the existence of smooth refinable functions and the convergence of corresponding cascade algorithms. Other applications concern polynomial spaces spanned by integer translates of a refinable function and one special property of linear operators associated to refinement equations.

The Mechanism of the Polarizational Mode Instability in Birefringent Fiber Optics

Yi A. Li and Keith Promislow

SIAM J. Math. Anal. 31, pp. 1351-1373 (23 pages) | Cited 6 times

Online Publication Date: August 01, 2006

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We show that the soliton solutions of the integrable Manakov equation exhibit an instability under arbitrarily small Hamiltonian perturbations. The instability arises from eigenvalues embedded in the essential spectrum of the associated linearized operators; these eigenvalues are dislodged by smooth perturbations. Specifically we consider perturbations which arise in fiber optics as a result of birefringence, including the so-called four-wave mixing term. Employing the Evans function and a Dirichlet expansion on the stable manifold of the linearized system, we obtain rigorous perturbation results and compute the stability diagram of the fast wave for all physical values of the birefringent parameters, using a novel numerical scheme derived from the Dirichlet expansion.

Convergence of Meissner Minimizers of the Ginzburg--Landau Energy of Superconductivity as $\kappa\to +\infty$

A. Bonnet, S. J. Chapman, and R. Monneau

SIAM J. Math. Anal. 31, pp. 1374-1395 (22 pages) | Cited 5 times

Online Publication Date: August 01, 2006

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The Meissner solution of a smooth cylindrical superconducting domain subject to a uniform applied axial magnetic field is examined. Under an additional convexity condition the uniqueness of the Meissner solution is proved. It is then shown that it is a local minimizer of the Ginzburg--Landau energy ${\cal E}_\kappa$. For applied fields less than a critical value, the existence of the Meissner solution is proved for large enough Ginzburg--Landau parameter $\kappa$. Moreover it is proved that the Meissner solution converges to a local minimizer of a certain energy ${\cal E}_\infty$ in the limit as $\kappa \rightarrow \infty$. Finally, it is proved that for $\kappa$ large enough the Meissner solution is not a global minimizer of ${\cal E}_\kappa$.

Generic Hopf Bifurcation From Lines of Equilibria Without Parameters: II. Systems of Viscous Hyperbolic Balance Laws

Bernold Fiedler and Stefan Liebscher

SIAM J. Math. Anal. 31, pp. 1396-1404 (9 pages) | Cited 1 time

Online Publication Date: August 01, 2006

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We investigate viscous shock profiles of the Riemann problem for systems of hyperbolic balance laws. Even strictly hyperbolic flux terms together with a nonoscillating kinetic part can lead to oscillating viscous shock profiles. They appear near a Hopf-like bifurcation point of the traveling wave equation.
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