Showing papers in "Indiana University Mathematics Journal in 2000"

Asymptotic L1-decay of solutions of the porous medium equation to self-similarity

Abstract: We consider the flow of gas in an N-dimensional porous medium with initial density v0NxO 0. The density vNx;tO then satisfies the nonlinear degenerate parabolic equa- tion vt E —v m where m> 1 is a physical constant. Assum- ing that R N1 Cj x j 2 Ov0NxOdx < 1, we prove that vNx;tO be- haves asymptotically, as t !1 , like the Barenblatt-Pattle solu- tion VNjxj;tO. We prove that the L 1 -distance decays at a rate t 1=NNNC2Om NO .M oreover, if N E1, we obtain an explicit time decay for the L 1 -distance at a suboptimal rate. The method we use is based on recent results we obtained for the Fokker-Planck equation (2), (3).

Some Sharp Isoperimetric Theorems for Riemannian Manifolds

Abstract: We prove that a region of small prescribed volume in a smooth, compact Riemannian manifold has at least as much perimeter as a round ball in the model space form, using dif- ferential inequalities and the Gauss-Bonnet-Chern theorem with boundary term. First we show that a minimizer is a nearly round sphere. We also provide some new isoperimetric inequalities in surfaces.

Q spaces of several real variables

Abstract: For alpha is an element of (-infinity, infinity), let Q(alpha)(R-n) be the space of all measurable functions with sup[l(I)](2 alpha -n) integral (I)integral (I)/f(x) - f(y)/(2)//x-y/(n+2 alpha) dxd ...

3D-2D Asymptotic Analysis for Inhomogeneous Thin Films

Abstract: A dimension reduction analysis is undertaken using ?-convergence techniques within a relaxation theory for 3D nonlinear elastic thin domains of the form where ! is a bounded domain of R 2 and f\" is an \"-dependent proole. An abstract representation of the eeective 2D energy is obtained, and speciic characterizations are found for nonhomogeneous plate models, periodic pro-les, and within the context of optimal design for thin lms.

Finite-time blow-up of solutions of some long-wave unstable thin film equations

Abstract: We consider the family of long–wave unstable lubrication equations ht = −(hhxxx)x − (hhx)x with m ≥ 3. Given a fixed m ≥ 3, we prove the existence of a weak solution that becomes singular in finite time. Specifically, given compactly supported nonnegative initial data with negative energy, there is a time T ∗ < ∞, determined by m and the H norm of the initial data, and a compactly supported nonnegative weak solution such that lim supt→T∗ ||h(·, t)||L∞ = lim supt→T∗ ||h(·, t)||H1 = ∞. We discuss the relevance of these singular solutions to an earlier conjecture [Comm Pure Appl Math 51:625–661, 1998] on when finite–time singularities are possible for long–wave unstable lubrication equations.

Diffusive BGK approximations for nonlinear multidimensional parabolic equations

Abstract: We introduce a class of discrete velocity BGK type approximations to multidimensional scalar nonlinearly diiusive conservation laws. We prove the well-posedness of these models, a priori bounds and kinetic entropy inequalities that allow to pass into the limit towards the unique entropy solution recently obtained by Carrillo. Examples of such BGK models are provided.

Two-weight, weak-type norm inequalities for fractional integrals, Calderon-Zygmund operators and commutators

Abstract: We give Ap-type conditions which are sucient for the two-weight, weak-type (p,p) inequalities for fractional integral operators, Calderon-Zygmund operators and commutators. For fractional integral operators, this solves a problem posed by Sawyer and Wheeden (28). At the heart of all of our proofs is an inequality relating the Hardy-Littlewood maximal function and the sharp maximal function which is strongly reminiscent of the good- inequality of Feerman and Stein (13).

Hypersurfaces of prescribed curvature in Lorentzian manifolds

Abstract: The existence of closed hypersurfaces of prescribed curvature in globally hyperbolic Lorentzian manifolds is proved, provided there are barriers. . 0 INTRODUCTION Consider the problem of finding a closed hypersurface of prescribed curvature F in a complete (n+1)-dimensional manifold N. To be more precise, let Ω be a connected open subset of N, f ∈ C2,α(Ω̄), F a smooth, symmetric function defined in an open cone Γ ⊂ Rn, then we look for a hypersurface M ⊂ Ω such that F|M = f(x) for all x ∈ M, (0.1) where F|M means that F is evaluated at the vector (κi(x)) the components of which are the principal curvatures of M. The prescribed function f should satisfy natural structural conditions, e.g., if Γ is the positive cone and the hypersurface M is supposed to be convex, then f should be positive, but no further, merely technical, conditions should be imposed. If N is a Riemannian manifold, then the problem has been solved in the case when F = H, the mean curvature, where in addition n had to be small, and N conformally flat, cf. [7], and for curvature functions F of class (K), no restrictions on n, cf. [4, 6]. We also refer to [5], where more special situations are considered, and the bibliography therein. 1125 Indiana University Mathematics Journal c ©, Vol. 49, No. 3 (2000)

A spectral mapping theorem and invariant manifolds for nonlinear Schrödinger equations

Abstract: A spectral mapping theorem is proved that resolves a key problem in applying invariant manifold the- orems to nonlinear Schr- odinger type equations. The theo- rem is applied to the operator that arises as the linearization of the equation around a standing wave solution. We cast the problem in the context of space-dependent nonlineari- ties that arise in optical waveguide problems. The result is, however, more generally applicable including to equations in higher dimensions and even systems. The consequence is that stable, unstable, and center manifolds exist in the neighborhood of a (stable or unstable) standing wave, such as a waveguide mode, under simple and commonly verifiable spectral conditions.

Existence of classical solutions to a free boundary problem for the p-Laplace operator: (II) the interior convex case

Abstract: In this paper, we prove the existence of convex classical solutions for a Bernoulli-type free boundary problem, in the interior of a convex domain. The governing operator considered is the p-Laplac ...

Hamilton-Jacobi-Bellman equations with fast gradient-dependence

Abstract: We investigate existence, uniqueness, and regular- ity properties for a class of H-J-B equations arising in non-linear control problems with unbounded controls. These equations involve Hamiltonians which are superlinear in the adjoint vari- able, and they have been already studied in the case when the growth in the adjoint variable is, in a sense, uniform with re- spect to the state variable. For instance, this is the case of the linear-quadratic problem. On the contrary, our results concern Hamiltonians that are superlinear in the adjoint variable, pos- sibly not uniformly with respect to the state variable. Actually, this is the general situation one has to deal with when consid- ering optimal control problems with a nonlinear dynamics (e.g. by slightly perturbing the linear quadratic problem). We also investigate situations where the fast growth of the Hamilton- ian in the adjoint variable degenerates into a very discontinuity. Such Hamiltonians arise quite naturally in those optimal con- trol problems where, roughly speaking, the dynamics and the cost display the same growth in the control variable.

Geometric optics with caustic crossing for some nonlinear Schrodinger equations

Abstract: We underscore nonlinear phenomena when oscillations give rise to a caustic. We consider a family of nonlinear Schrodinger equations, for which several notions of critical index appear, as for the description of the propagation outside the caustic on the one hand, and the caustic crossing on the other hand. The propagation is similar to that known by WKB method, and the caustic crossing is either the same as in the linear case, or described in terms of scattering operators. A uniform description is obtained by a generalization of the use of Lagrangian integrals. .

Second-order Self-similar Identities and Multifractal Decompositions

Abstract: Motivated by the study of convolutions of the Cantor measure, we set up a framework for computing the mul- tifractal L q -spectrum NqO, q> 0, for certain overlapping self- similar measures which satisfy a family of second-order identities introduced by Strichartz et al. We apply our results to the family of iterated function systems Sjx E N1=mOxCUNm 1Oma=j, j E 0, 1, ::: ,m, wherem is an odd integer, and obtain closed formulas defining NqO, q> 0, for the associated self-similar measures. As a result, we can show thatNqO is diVerentiable on N0;1O and justify the multifractal formalism in the regionq>0. Furthermore, expressions for the HausdorV and entropy dimen- sions of these measures can also be derived. By lettingmE 3, we obtain all these results for the 3-fold convolution of the standard Cantor measure.

On the dimension of the attractor for the non-homogeneous Navier-Stokes equations in non-smooth domains

Abstract: This paper concerns the two-dimensional NavierStokes equations in a Lipschitz domain Ω with nonhomogeneous boundary condition u = φ on ∂Ω. Assuming φ ∈ L∞(∂Ω), we establish the existence of the universal attractor, and show that its dimension is bounded by c1G + c2Re, where G is the Grashof number and Re the Reynolds number.

Supersonic flow past a symmetrically curved cone

Abstract: We study the conic shock waves for 3-dimensional steady irrotational isentropic supersonic flow against a sharp symmetrically curved conic projectile. An approximate solution was constructed and the existence of a conic shock wave solution is established by linear iteration.

The global attractor and finite determining modes for the Navier-Stokes equations of compressible flow with singular initial data

Abstract: We prove the existence of a compact global attractor for the Navier-Stokes equations of compressible flow in one space dimension, and we estimate the number of its determining modes in terms of the force and a scale invariant of the system. Of particular interest is the construction of the underlying phase space, particularly for the fluid density: we consider densities ρ for which the Lebesgue decomposition of the distribution derivative ρx has singular part with countable support and AC part in L . Remarkably, the solution operator respects this decomposition sufficiently to permit an analysis of the time–evolution of the separate components. We also show that, in the attractor, densities are in H and velocities are in H, so that, in a very precise sense, singularities in the density and in the velocity gradient disappear in the time–asymptotic limit. These results are based on a new existence and uniqueness theory for these equations in which we obtain time–independent estimates for solutions with large, discontinuous initial data, and large external forces. A crucial point in this existence theory is a new, uncontingent estimate for the viscous flux, reminiscent of the Oleinik entropy condition for scalar conservation laws. ∗ This research was supported in part by the NSF under grant No. DMS-9703703. §

Growth of Degree for Iterates of Rational Maps in Several Variables

Abstract: We investigate maps on P 2 and C 3 for which the algebraic degrees of the iterates, dm, are not constant nor grow like d m . These maps are called B-maps. Answering a question posed by Ghys, we will show here that there are only countably many sequences {dm} of natural numbers for which there exists a rational map f on P 2 with deg f m = dm ,f or allm ∈ N. We show here also that there are indeed infinitely many possible sequences {dm} with d1 > 2.

L^1 stability of patterns of non-interacting large shock waves

Abstract: We consider a strictly hyperbolic n × n system of conservation laws in one space dimension ut + f(u)x = 0, together with Cauchy initial data u(0, x) = ū(x), that is a small BV ∩ L perturbation of fixed Riemann data (u− 0 , u 0 ). We a priori assume that the latter problem is solved by M large shocks (2 ≤ M ≤ n) of different characteristic families, each of them Majda stable and Lax compressive. We prove that under a suitable Finiteness Condition the problem has a unique solution defined globally in space and time, while a stronger Stability Condition guarantees the existence of a Lipschitz semigroup of solutions.

A Priori Estimates for Solutions of Fully Nonlinear Equations with Convex Level Set

Abstract: We derive an ap rioriC 2,α estimate for solutions of the fully non-linear elliptic equation F( D 2 u) = 0, provided the level set Σ ={ M | F( M) = 0} satisfies: (a) Σ ∩{ M | Tr M = t} is strictly convex for all constants t; (b) the angle between the identity matrix I and the normal Fij to Σ is strictly positive on the non-convex part of Σ. Moreover, we do not need any convexity assumption on F in the course of the proof for the two dimensional case, as the classical result indicates.

BV solutions for a class of viscous hyperbolic systems

Abstract: The paper is concerned with the Cauchy problem for a nonlinear, strictly hyperbolic system with small viscosity: ut +A(u)ux = e uxx, u(0, x) = ū(x). (∗) We assume that the integral curves of the eigenvectors ri of the matrix A are straight lines. On the other hand, we do not require the system (∗) to be in conservation form, nor do we make any assumption on genuine linearity or linear degeneracy of the characteristic fields. In this setting we prove that, for some small constant η0 > 0 the following holds. For every initial data ū ∈ L with Tot.Var.{ū} 0. The total variation of u(t, ·) satisfies a uniform bound, independent of t, e. Moreover, as e → 0+, the solutions u(t, ·) converge to a unique limit u(t, ·). The map (t, ū) 7→ Stū . = u(t, ·) is a Lipschitz continuous semigroup on a closed domain D ⊂ L of functions with small total variation. This semigroup is generated by a particular Riemann Solver, which we explicitly determine. The above results can also be applied to strictly hyperbolic systems on a Riemann manifold. Although these equations cannot be written in conservation form, we show that the Riemann structure uniquely determines a Lipschitz semigroup of “entropic” solutions, within a class of (possibly discontinuous) functions with small total variation. The semigroup trajectories can be obtained as the unique limits of solutions to a particular parabolic system, as the viscosity coefficient approaches zero. The proofs rely on some new a priori estimates on the total variation of solutions for a parabolic system whose components drift with strictly different speeds. 0 1 Introduction Consider a strictly hyperbolic n× n system of conservation laws in one space dimension: ut + f(u)x = 0. (1.1) For initial data with small total variation, the global existence of weak solutions was proved in [8]. Moreover, the uniqueness and stability of entropy admissible BV solutions was recently established in a series of papers [3,4,5,6]. A long standing open question is whether these discontinuous solutions can be obtained as vanishing viscosity limits. More precisely, given a smooth initial data ū : IR 7→ IR with small total variation, consider the parabolic Cauchy problem u(0, x) = ū(x). (1.2) ut +A(u)ux = e uxx. (1.3) Here A(u) . = Df(u) is the Jacobian matrix of f and e > 0. It is then natural to expect that, as e→ 0, the solution u of (1.2)-(1.3) converges to the unique entropy weak solution u of (1.1)-(1.2). Unfortunately, no general theorem in this direction is yet known. Some of the main results available in the literature are listed below. 1) In the case of a scalar conservation law, the entropic solutions of (1.1) determine a semigroup which is contractive w.r.t. the L-distance. In this case, a general convergence theorem for vanishing viscosity approximations was proved in the classical work of Kruzhkov [12]. 2) For various 2 × 2 systems, if a uniform L∞–bound on all functions u is available, one can consider a weak limit u ⇀ u. By a compensated compactness argument introduced by DiPerna [7], it then follows that u is actually a weak solution of the nonlinear system (1.1). For a comprehensive discussion of the compensated compactness method and its applications to conservation laws, see [18]. 3) For n × n Temple class systems, a proof of the convergence of the viscous solutions u to a solution of (1.1) can be found in [17,18]. 4) Assume that all characteristic fields of the system (1.1) are linearly degenerate. Then every solution with small total variation which is initially smooth remains smooth for all positive times [2]. Clearly such solution can be obtained as limit of vanishing viscosity approximations. By a density argument it follows that every weak solution of (1.1) with sufficiently small total variation is a limit of viscous approximations. 5) For a general n× n strictly hyperbolic system, let u be a piecewise smooth entropic solution of (1.1) with jumps along a finite number of smooth curves in the t-x plane. Thanks to this 1 additional regularity assumptions on u, it was proved in [10] that there exists a family of viscous solutions u converging to u in Lloc as e→ 0. From our point of view, the major difficulty toward a general proof of the convergence u → u lies in deriving an a priori estimate on the total variation of the solution of (1.2)-(1.3), uniformly valid as e → 0. To fix the ideas, assume ū ∈ C∞ c with Tot.Var.(ū) sufficiently small. Performing the rescalings t 7→ t/e, x 7→ x/e, the Cauchy problem becomes ut +A(u)ux = uxx, (1.4) u(0, x) = ū(ex). (1.5) Observe that, as e → 0, the initial data u(0, ·) has constant total variation, all of its derivatives approach zero, but its L-norm approaches infinity. We thus need estimates on the total variation of a solution u(t, ·) of (1.4) which are independent of the L-norm of the initial data. To illustrate the heart of the matter, let us denote by λ1(u) < · · · < λn(u) the eigenvalues of the n × n Jacobian matrix A(u) . = Df(u), and call l, . . . , l, r1, . . . , rn, its left and right eigenvectors, normalized so that ∣∣ri(u)∣∣ ≡ 1, l(u) · rj(u) = { 1 if i = j, 0 if i 6= j. (1.6) The directional derivative of a function φ = φ(u) in the direction of the eigenvector ri is written ri • φ(u) . = lim h→0 φ ( u+ hri(u) ) − φ(u) h . Moreover, by ux . = l(u) · ux we denote the i-th component of the gradient ux w.r.t. the basis of right eigenvectors {r1, . . . , rn}. Recalling (1.6), this implies ux = ∑

Lower bound of the number of the Rayleigh resonances for arbitrary body

Abstract: We study the system of linear elasticity in an exterior domain in R with Neumann boundary conditions. We prove an optimal lower bound for the asymptotic distribution of the resonances near the real axis due to the existence of Rayleigh waves. To this end we construct real quasimodes as the eigenvalues of an elliptic first order ΨDO on the boundary. In case of a convex boundary, we prove that the resonance states are asymptotically close to the quasimode states and we use this to find explicitly the leading term of the resonance states (the Rayleigh waves) on the boundary.