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2000

Volume 31, Issue 5, pp. 941-1174


Nonclassical Shocks and Kinetic Relations: Strictly Hyperbolic Systems

Brian T. Hayes and Philippe G. LeFloch

SIAM J. Math. Anal. 31, pp. 941-991 (51 pages) | Cited 11 times

Online Publication Date: August 01, 2006

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We consider strictly hyperbolic systems of conservation laws whose characteristic fields are not genuinely nonlinear, and we introduce a framework for the nonclassical shocks generated by diffusive or diffusive-dispersive approximations. A nonclassical shock does not fulfill the Liu entropy criterion and turns out to be undercompressive.
We study the Riemann problem in the class of solutions satisfying a single entropy inequality, the only such constraint available for general diffusive-dispersive approximations. Each non-genuinely nonlinear characteristic field admits a two-dimensional wave set, instead of the classical one-dimensional wave curve. In specific applications, these wave sets are narrow and resemble the classical curves. We find that even in strictly hyperbolic systems, nonclassical shocks with arbitrarily small amplitudes occur. The Riemann problem can be solved uniquely using nonclassical shocks, provided an additional constraint is imposed: we stipulate that the entropy dissipation across any nonclassical shock be a given constitutive function. We call this admissibility criterion a kinetic relation, by analogy with similar laws introduced in material science for propagating phase boundaries. In particular, the kinetic relation may be expressed as a function of the propagation speed. It is derived from traveling waves and, typically, depends on the ratioof the diffusion and dispersion parameters.

Uniform Asymptotic Formula for Orthogonal Polynomials with Exponential Weight

W. Y. Qiu and R. Wong

SIAM J. Math. Anal. 31, pp. 992-1029 (38 pages) | Cited 4 times

Online Publication Date: August 01, 2006

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Let {pn(x)}_{n\ge 0}$ be the set of orthonormal polynomials with respect to the exponential weight w(x)=e-v(x), where v(x)=x2m + ... is a monic polynomial of degree 2m with $m \ge 2$ and is even. An asymptotic approximation is obtained for pn(x), as $n \to \infty$, which holds uniformly for $0 \le x \le O(n^{1/2m})$. As a corollary, a three-term asymptotic expansion is also derived for the zeros of these polynomials.

Nonlinear Pyramid Transforms Based on Median-Interpolation

David L. Donoho and Thomas P. Y. Yu

SIAM J. Math. Anal. 31, pp. 1030-1061 (32 pages) | Cited 11 times

Online Publication Date: August 01, 2006

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We introduce a nonlinear refinement subdivision scheme based on median-inter-polation. The scheme constructs a polynomial interpolating adjacent block medians of an underlying object. The interpolating polynomial is then used to impute block medians at the next finer triadic scale. Perhaps surprisingly, expressions for the refinement operator can be obtained in closed-form for the scheme interpolating by polynomials of degree D=2. Despite the nonlinearity of this scheme, convergence and regularity can be established using techniques reminiscent of those developed in analysis of linear refinement schemes.
The refinement scheme can be deployed in multiresolution fashion to construct a nonlinear pyramid and an associated forward and inverse transform. In this paper we discuss the basic properties of these transforms and their possible use in removing badly non-Gaussian noise. Analytic and computational results are presented to show that in the presence of highly non-Gaussian noise, the coefficients of the nonlinear transform have much better properties than traditional wavelet coefficients.

Orthonormal Ridgelets and Linear Singularities

David L. Donoho

SIAM J. Math. Anal. 31, pp. 1062-1099 (38 pages) | Cited 26 times

Online Publication Date: August 01, 2006

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We construct a new orthonormal basis for $L^2({\Bbb R}^2)$, whose elements are angularly integrated ridge functions---{\it orthonormal ridgelets}. The basis elements are smooth and of rapid decay in the spatial domain, and in the frequency domain are localized near angular wedges which, at radius $r = 2^j$, have radial extent $\Delta r \approx 2^j$ and angular extent $\Delta \theta \approx 2\pi/2^{j}$.
Orthonormal ridgelet expansions expose an interesting phenomenon in nonlinear approximation: they give very efficient approximations to objects such as $1_{\{ x_1\cos\theta+ x_2\sin\theta > a\}} \ e^{-x^2_1-x^2_2}$ which are smooth away from a discontinuity along a line. The orthonormal ridgelet coefficients of such objects are sparse: they belong to every $\ell^p$, p > 0. This implies that simple thresholding in the ridgelet orthobasis is, in a certain sense, a near-ideal nonlinear approximation scheme for such objects.
Orthonormal ridgelets may be viewed as L2 substitutes for approximation by sums of ridge functions, and so can perform many of the same tasks as the ridgelet systems constructed by Candès [Ph.D. Thesis, Department of Statistics, Stanford University, Stanford, CA, 1998; Appl. Comput. Harmon. Anal., 6 (1999), pp. 197--218]. Orthonormal ridgelets make available the machinery of orthogonal decompositions, which is not available for ridge functions as they are not in $L^2({\Bbb R}^2)$.
The ridgelet orthobasis is constructed as the isometric image of a special wavelet basis for Radon space; as a consequence, ridgelet analysis is equivalent to a special wavelet analysis in the Radon domain. This means that questions of ridgelet analysis of linear singularities can be answered by wavelet analysis of point singularities. At the heart of our nonlinear approximation result is the study of a certain tempered distribution on ${\Bbb R}^2$ defined formally by S(u,v) = |v|^{-1/2} \sigma(u/|v|)$ with $\sigma$ a certain smooth bounded function; this is singular at (u,v) = (0,0) and $C^\infty$ elsewhere. The key point is that the analysis of this point singularity by tensor Meyer wavelets yields sparse coefficients at high frequencies; this is reflected in the sparsity of the ridgelet coefficients and the good nonlinear approximation properties of the ridgelet basis.

Resolution in Dynamic Emission Tomography

Jean Maeght and Dominikus Noll

SIAM J. Math. Anal. 31, pp. 1100-1120 (21 pages) | Cited 4 times

Online Publication Date: August 01, 2006

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Based on a two-dimensional (2-D) Fourier analysis of the attenuated Radon transform and a 2-D version of the Shannon sampling theorem, we investigate the problem of resolution in dynamic emission tomography. As a result we provide guidelines on how to acquire and on how to filter the projection data.

Second Order Singular Perturbation Models for Phase Transitions

Irene Fonseca and Carlo Mantegazza

SIAM J. Math. Anal. 31, pp. 1121-1143 (23 pages) | Cited 10 times

Online Publication Date: August 01, 2006

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Singular perturbation models involving a penalization of the first order derivatives have provided a new insight into the role played by surface energies in the study of phase transitions problems. It is known that if $W:{\mathbb R}^d \to [0,+\infty)$ grows at least linearly at infinity and it has exactly two potential wells of level zero at $a, b \in {\mathbb R}^d$, then the $\Gamma(L^1)$-limit of the family of functionals $$ {\mathcal F}_\varepsilon(u):= \begin{cases} \int_{\Omega} \left(\frac{W(u)}{\varepsilon}+\varepsilon \vert \nabla u\vert^2\right) \, dx & \hbox{ if $u\in W^{1,2}(\Omega;{\mathbb R}^d)$, }\\ \\ +\infty & \hbox{ if $u\in L^1(\Omega;{\mathbb R}^d)\setminus W^{1,2}(\Omega;{\mathbb R}^d)$, } \end{cases} $$ where $\Omega$ is a bounded, open set in ${\mathbb R}^N$, is given by $$ {\mathcal F}(u):= \begin{cases} {\rm {\bf m }}\; {\rm Per}_{\Omega} (\{u = a\}) & \hbox{ if $u\in BV(\Omega;\{a,b\})$, }\\ +\infty & \hbox{ otherwise, } $$ for a suitable constant {\bf m} depending on the energy density W. In this paper, and motivated by the study of phase transitions for nonlinear elastic materials, the $\Gamma(L^1)$-limit is obtained in the case where in ${\mathcal F}_\varepsilon(u)$ the penalization term $\varepsilon \vert \nabla u\vert^2$ is replaced by $\varepsilon^3 \vert \nabla^2 u\vert^2$, for $u \in W^{2,2}(\Omega;{\mathbb R}^d)$. The resulting functional is of the same form as $ {\mathcal F}(u)$ above.

Vanishing Shear Viscosity in the Equations of Compressible Fluids for the Flows with the Cylinder Symmetry

Hermano Frid and Vladimir Shelukhin

SIAM J. Math. Anal. 31, pp. 1144-1156 (13 pages) | Cited 5 times

Online Publication Date: August 01, 2006

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We analyze the question of the limit process when the shear viscosity goes to zero for global solutions to the Navier--Stokes equations for compressible heat conductive fluids for the flows which are invariant over cylindrical sheets.

Asymptotics of the Fast-Diffusion Equation with Critical Exponent

Victor A. Galaktionov, Lambertus A. Peletier, and Juan L. Vazquez

SIAM J. Math. Anal. 31, pp. 1157-1174 (18 pages) | Cited 6 times

Online Publication Date: August 01, 2006

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We study the large-time behavior of the solutions of the initial-value problem for the nonlinear diffusion equation $$ u_t=\nabla\cdot (u^{-\s}\nabla u) \quad \mbox{in ${\bf R}^n \times {\bf R}_+$} \leqno{{\rm (ND)}} $$ in dimensions $ n \geq 3$ with nonnegative initial data $u(x,0)\in L^1({\bf R}^n)$ when the exponent takes on the \textit{critical value} $\sigma =2/n$. This represents a borderline case in the study of the problem and offers marked qualitative and technical differences with the neighboring cases $\sigma \approx 2/n$, $\sigma \neq 2/n$. In particular, it marks the transition between two completely different asymptotic behavior types. It is known that solutions exist globally in time and conserve the L1-norm for this problem. We prove that they decay exponentially in time with a complicated law: $$ \log \|u(\cdot,t)\|_\infty \sim - \kappa M^{-2/(n-2)} t^{n/(n-2)} \quad \mbox{as}\quad t \to \infty, $$ where $M=\int u(x,0)dx$ is the conserved mass and the constant $\kappa > 0$ depends only on the dimension n. This strongly differs from the comparatively simple self-similar asymptotics of the case $\sigma <2/n$.
The description is split into an inner and an \textit{outer} region, conveniently matched at a transition layer. The analysis of the outer region can be done independently and the behavior is governed by a first-order conservation law which acts as the reduced asymptotic equation. The uniqueness theory for first-order conservation laws is one of the great contributions of S. N. Kruzhkov to mathematics. The behavior in the inner parabolic region is then studied by means of a semiconvexity argument which makes it possible to translate into this region the precise behavior from the outer region.
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