uniqueness and asymptotic stability of riemann solutions for the compressible euler equations we prove the uniqueness of riemann solutions in the class of entropy solutions in l bvloc for the 3 3 system of compressible euler equations under usual assumptions on the equation of state for the pressure which imply strict hyperbolicity of the system and genuine nonlinearity of the first and third characteristic families in particular if the riemann solutions consist of at most rarefaction waves and contact discontinuities we show the global l2 stability of the riemann solutions even in the class of entropy solutions in l with arbitrarily large oscillation for the 3 3 system we apply our framework established earlier to show that the uniqueness of riemann solutions implies their inviscid asymptotic stability under l1 perturbation of the riemann initial data as long as the corresponding solutions are in l and have local bounded total variation satisfying a natural condition on its growth with time no specific reference to any particular method for constructing the entropy solutions is made our uniqueness result for riemann solutions can easily be extended to entropy solutions u x t piecewise lipschitz in x for any t 0

wave interactions in nonlinear elastic strings we study a system modeling the dynamics of a nonlinear elastic string this is a 6 6 system of hyperbolic conservation laws which is degenerate in that two wave families have multiplicity two we construct the wave curves for this problem and solve the riemann problem we then give a detailed analysis of elementary wave interactions leading to a glimm theorem and describe features of the system when the total variation is large there are complicated wave patterns including infinitely many interactions in finite time and both three and four resonances may be present

the large time stability of sound waves we demonstrate the existence of solutions to the full 3 3 system of compressible euler equations in one space dimension up to an arbitrary timet 0 in the case when the initial data has arbitrarily large total variation and sufficiently small supnorm the result applies to periodic solutions of the euler equations a nonlinear model for sound wave propagation in gas dynamics our analysis establishes a growth rate for the total variation that depends on a new length scaled that we identify in the problem this length scale plays no role in 2 2 systems or any system possessing a full set of riemann coordinates nor in the small total variation problem forn n systems the cases originally addressed by glimm in 1965 recent work by a number of authors has demonstrated that when the total variation is sufficiently large solutions of 3 3 systems of conservation laws can in general blow up in finite time independent of the supnorm due to amplifying instabilities created by the non trivial lie algebra of the vector fields that define the elementary waves for the large total variation problem there is an interaction between large scale effects that amplify and small scale effects that are stable and we show that the length scale on which this interaction occurs isd in the limitd we recover glimm s theorem and we observe that there exist linearly degenerate systems within the class considered for which the growth rate we obtain is sharp

sup norm stability for glimm s scheme we consider the cauchy problem for a general n n system of conservation laws existence of solutions was proved by glimm using his celebrated random choice scheme in this paper we obtain a third order interaction estimate analagous to that obtained by glimm for 2 2 systems by using this estimate and identifying a global cancellation effect we obtain l stability for solutions generated by glimm s scheme as an immediate consequence we have l1 stability and l decay obtained by temple for 2 2 systems 1993 john wiley sons inc

blowup with small bv data in hyperbolic conservation laws we construct weak solutions of 3 3 conservation laws which blow up in finite time the system is strictly hyperbolic at every state in the solution and the data can be chosen to have arbitrarily small total variation this is thus an example where glimm s existence theorem fails to apply and it implies the necessity of uniform hyperbolicity in glimm s theorem because our system is very simple we can carry out explicit calculations and understand the global geometry of wave curves

multidimensional rovella like attractors in a joint work with a castro v pinheiro both from federal univ of bahia and m j pacifico federal university of rio de janeiro we construct a multidimensional flow exhibiting a rovella like attractor a compact transitive invariant set with an equilibrium accumulated by regular orbits and a partially hyperbolic splitting of the tangent bundle with a multidimensional non uniformly expanding direction moreover this attractor has a physical measure with full support which is a u gibbs state as in the 3 dimensional rovella attractor this example is not robust this introduces a class of multidimensional dynamics where benedicks carleson arguments can be applied to get persistent non uniform expansion

sup norm estimates in glimm s method abstract we give a proof that total variation u0 1 can be replaced by sup u0 1 in glimm s method whenever a coordinate system of riemann invariants is present the argument is somewhat simpler but in the same spirit as that given by glimm in his celebrated paper of 1965

infinitesimal lyapunov functions for singular flows we present an extension of the notion of infinitesimal lyapunov function to singular flows and from this technique we deduce a characterization of partial sectional hyperbolic sets in absence of singularities we can also characterize uniform hyperbolicity these conditions can be expressed using the space derivative dx of the vector field x together with a field of infinitesimal lyapunov functions only and are reduced to checking that a certain symmetric operator is positive definite at the tangent space of every point of the trapping region

convergence of the viscosity method for isentropic gas dynamics a convergence theorem for the method of artificial viscosity applied to the isentropic equations of gas dynamics is established convergence of a subsequence in the strong topology is proved without uniform estimates on the derivatives using the theory of compensated compactness and an analysis of progressing entropy waves

nonlinear resonance for quasilinear hyperbolic equation the purpose of this paper is to study the wave behavior of hyperbolic conservation laws with a moving source resonance occurs when the speed of the source is too close to one of the characteristic speeds of the system for the nonlinear system characteristic speeds depend on the basic dependence variables and resonance gives rise to nonlinear interactions which lead to rich wave phenomena motivated by physical examples a scalar model is proposed and analyzed to describe the qualitative behavior of waves for a general system in resonance with the source analytical understanding is used to design a numerical scheme based on the random choice method an important physical example is transonic gas flow through a nozzle this analysis provides a transparent and revealing qualitative understanding of wave behavior of gas flow including such phenomena as nonlinear stability instability and changing types of waves

shock free solutions of the compressible euler equations we study the structure of shock free solutions of the compressible euler equations with large data we describe conditions under which the rarefactive compressive character of solutions changes and conditions under which the vacuum is formed asymptotically we present several new examples of shock free solutions which demonstrate a large variety of behaviors

billiards with pesin region of measure one we give a description of a large class of plane billiards with pesin region of measure one open conditions including properly those founded by wojtkowski w1 forc4 focusing boundaries are obtained lyapunov s forms introduced by lewowicz are used

global entropy solutions to weakly nonlinear gas dynamics abstract entropy weak solutions with bounded periodic initial data are considered for the system of weakly nonlinear gas dynamics through a modified glimm scheme an approximate solution sequence is constructed and then a priori estimates are provided with the methods of approximate characteristics and approximate conservation laws which gives not only the existence and uniqueness but also the uniform total variation bounds for the entropy solutions

crack identification in a stepped beam carrying a rigid disk abstract in this study a simple algorithm based on a mathematical model is proposed to identify crack location and depth in a stepped cantilever euler bernoulli beam carrying a rigid disk at its tip the mathematical model that describes the lateral vibration of the beam is derived using the assumed mode method that coalesces with the lagrange s equation a massless torsional spring whose stiffness depend on the severity of the crack is used as a crack model using this crack modeling method combined with the assumed mode method the crack effect is introduced to the system flexibility as global additional structural flexibility for the assumed mode method the mode shapes for two uniform beams connected by a massless torsional spring simulating the cracked beam are adopted as trial functions the proposed identification algorithm utilizes the first three natural frequencies shift of the beam caused by a crack to estimate its location and depth in addition the proposed mathematical model is used to illustrate the effect of the crack depth and its location on the dynamic characteristics of the system using the commercial finite element fe software ansys 8 0 three dimensional finite element analysis fea is carried out to show the accuracy of the derived mathematical model and to demonstrate the reliability of the proposed crack identification algorithm the analysis showed consistency with the assumed mode results it showed that the error in concurrent prediction of crack depth and its location using the proposed algorithm is about 10

decay with a rate for noncompactly supported solutions of conservation laws abstract this document shows that solutions of the cauchy problem for systems of two conservation laws decay in the supnorm at a rate that depends only on the l sub 1 norm of the initial data this implies that the dissipation due to the entropy dominates the nonlinearities in the problem at a rate depending only on the l sub 1 norm of the initial data the main estimate requires an analysis of approximate characteristics for its proof a general framework is developed for the study of approximate characteristics and the main estimate is obtained for an arbitrary number of equations keywords riemann problem random choice method decay stability continuous dependence conservation laws nonlinear partial differential equations

lyapunov functions and cone families we describe systematically the relation between lyapunov functions and nonvanishing lyapunov exponents both for maps and flows this includes a brief survey of the existing results in the area in particular we consider separately the cases of nonpositive and arbitrary lyapunov functions thus yielding optimal criteria for negativity and positivity of the lyapunov exponents of linear cocycles over measure preserving transformations moreover we describe converse results of these criteria with the explicit construction of eventually strict lyapunov functions for any map or flow with nonzero lyapunov exponents we also construct examples showing that in general the existence of an eventually strict invariant cone family does not imply the existence of an eventually strict lyapunov function

dominated splitting for exterior powers and singular hyperbolicity abstract we show that an invariant splitting for the tangent map to a smooth flow over a compact invariant subset is dominated if and only if the exterior power of the tangent map admits an invariant dominated splitting for a c 1 vector field x on a 3 manifold we obtain singular hyperbolicity using only the tangent map dx of x and a family of indefinite and non degenerate quadratic forms without using the associated flow x t and its derivative d x t as a consequence we show the existence of adapted metrics for singular hyperbolic sets for three dimensional c 1 vector fields

vibration of a beam with a breathing crack a continuous cracked beam vibration theory is used for the prediction of changes in transverse vibration of a simply supported beam with a breathing crack the equation of motion and the boundary conditions of the cracked beam considered as a one dimensional continuum were used the eigenfrequency changes due to a breathing edge crack are shown to depend on the bi linear character of the system the associated linear problems are solved over their respective domains of definition and the solutions are matched through the initial conditions the changes in vibration frequencies for a fatigue breathing crack are smaller than the ones caused by open cracks the method has been tested for the evaluation of the lowest natural frequency of lateral vibration for beams with a single edge breathing crack experimental results from aluminium beams with fatigue cracks are used for comparison with the analytical results

resonant one dimensional nonlinear geometric optics abstract in this paper we study the existence and resonant interaction of oscillatory wave trains in one space dimension giving a rigorous proof of the validity of the corresponding expansions of weakly nonlinear optics we consider both semilinear and quasilinear systems the latter before shock formation some important features of the study are the following 1 we prove the existence of families of exact solutions which have asymptotic expansions governed by weakly nonlinear optics equations with variable coefficients nonconstant background fields and nonlinear phases are permitted our weak transversality hypotheses allow us to justify expansions where even a formal theory did not exist before 2 we make a detailed study of resonances the geometry associated with such resonances is related to the theory of planar webs 3 we study the smoothness of the profiles their regularity is ruled by a sum law analogous to that describing the propagation of singularities in one dimension 4 the expansions are justified up to the breakdown of the profiles which coincides with a suitably defined breakdown for exact solutions

robust transitive singular sets for 3 flows are partially hyperbolic attractors or repellers inspired by lorenz remarkable chaotic flow we describe in this paper the structure of all c 1 robust transitive sets with singularities for flows on closed 3 manifolds they are partially hyperbolic with volume expanding central direction and are either attractors or repellers in particular any c 1 robust attractor with singularities for flows on closed 3 manifolds always has an invariant foliation whose leaves are forward contracted by the flow and has positive lyapunov exponent at every orbit showing that any c 1 robust attractor resembles a geometric lorenz attractor

weak stability in the global norm for systems of hyperbolic conservation laws we prove that solutions for systems of two conservation laws which are generated by glimm s method are weakly stable in the global l norm the method relies on a previous decay result of the author together with a new estimate for the l1 lipschitz constant that relates solutions at different times the estimate shows that this constant can be bounded by the supnorm of the solution and is proved for any number of equations the techniques do not rely on the existence of a family of entropies and moreover the results would generalize immediately to more than two equations if one were to establish the stability of solutions in the supnorm for more than two equations

long time existence of entropy solutions to the one dimensional non isentropic euler equations with periodic initial data the non isentropic euler system with periodic initial data in mathbb r 1 is studied by analyzing wave interactions in a framework of specially chosen riemann invariants generalizing glimm s functionals and applying the method of approximate conservation laws and approximate characteristics an mathcal o varepsilon 2 lower bound is established for the life span of the entropy solutions with initial data that possess varepsilon variation in each period

a functional integral approach to shock wave solutions of euler equations with spherical symmetry forn n systems of conservation laws in one dimension without source terms the existence of global weak solutions was proved by glimm 1 glimm constructed approximate solutions using a difference scheme by solving a class of riemann problems

modal analysis of a cracked beam abstract this paper addresses the problem of vibrations of a cracked beam in general the motion of such a beam can be very complex this phenomenon can be attributed to the presence of the non linearity due to the opening and closing of cracks the focus of this paper is the modal analysis of a cantilever beam with a transverse edge crack the non linearity mentioned above has been modelled as a piecewise linear system in an attempt to define effective natural frequencies for this piecewise linear system the idea of a bilinear frequency is utilized the bilinear frequency is obtained by computing the associated frequencies of each of the linear pieces of the piecewise linear system the finite element method is used to obtain the natural frequencies in each linear region in order better to understand the essential non linear dynamics of the cracked beam a piecewise linear two degree of freedom model is studied perturbation methods are used to obtain the non linear normal modes of vibration and the associated period of the motion results of this piecewise linear model problem are shown to justify the definition of the bilinear frequency as the effective natural frequency it is therefore expected that calculating piecewise mode shapes and bilinear frequencies is useful for understanding the dynamics of the infinite degree of freedom cracked beam

srb measures for partially hyperbolic systems whose central direction is mostly expanding we construct sinai ruelle bowen srb measures supported on partially hyperbolic sets of diffeomorphisms the tangent bundle splits into two invariant subbundles one of which is uniformly contracting under the assumption that the complementary subbundle is non uniformly expanding if the rate of expansion lyapunov exponents is bounded away from zero then there are only finitely many srb measures our techniques extend to other situations including certain maps with singularities or critical points as well as diffeomorphisms having only a dominated splitting and no uniformly hyperbolic subbundle

existence and stability of supersonic euler flows past lipschitz wedges it is well known that when the vertex angle of a straight wedge is less than the critical angle there exists a shock front emanating from the wedge vertex so that the constant states on both sides of the shock front are supersonic since the shock front at the vertex is usually strong especially when the vertex angle of the wedge is large then a global flow is physically required to be governed by the isentropic or adiabatic euler equations in this paper we systematically study two dimensional steady supersonic euler i e nonpotential flows past lipschitz wedges and establish the existence and stability of supersonic euler flows when the total variation of the tangent angle functions along the wedge boundaries is suitably small we develop a modified glimm difference scheme and identify a glimm type functional by naturally incorporating the lipschitz wedge boundary and the strong shock front and by tracing the interaction not only between the boundary and weak waves but also between the strong shock front and weak waves to obtain the required bv estimates these estimates are then employed to establish the convergence of both approximate solutions to a global entropy solution and corresponding approximate strong shock fronts emanating from the vertex to the strong shock front of the entropy solution the regularity of strong shock fronts emanating from the wedge vertex and the asymptotic stability of entropy solutions in the flow direction are also established