weak solutions for an inviscid two phase flow model in physical vacuum abstract we investigate a free boundary problem for the inviscid two phase flow model with moving physical vacuum boundary which is a degenerate and characteristic hyperbolic system of conservation laws based on a suitable transformation a special degenerate parabolic approximation and delicate uniform energy estimates we prove the existence and uniqueness of the local weak solutions this can be regarded as a generalization of the results in coutand and shkoller 2011 5 from single phase gas model to two phase gas liquid model

singular behavior of vacuum states for compressible fluids in this survey paper we will present the recent work on the study of the compressible fluids with vacuum states by illustrating its interesting and singular behavior through some systems of fluid dynamics that is euler equations euler poisson equations and navier stokes equations the main concern is the well posedness of the problem when vacuum presents and the singular behavior of the solution near the interface separating the vacuum and the gas furthermore the relation of the solutions for the gas dynamics with vacuum to those of the boltzmann equation will also be discussed in fact the results obtained so far for vacuum states are far from being complete and satisfactory therefore this paper can only be served as an introduction to this interesting field which has many open and challenging mathematical problems moreover the problems considered here are limited to the author s interest and knowledge in this area

on the vacuum state for the equations of isentropic gas dynamics condition c which is dictated by physics is a source of essential difficulties in the mathematical treatment of this problem there have been extensive studies on the smooth solutions for equations e 2 61 these studies are concerned with states p u in the region strictly away from the vacuum that is p l u p0 for some fixed positive constant p a necessary condition for the existence of a global bounded continuous solution is

interaction of rarefaction waves and vacuum in a convex duct the motion of the supersonic compressible flow governed by the euler system in a two dimensional convex duct is studied the rarefaction waves in the compressible flow propagate and reflect on the walls of the convex duct so that interaction occurs and a vacuum may appear the existence of the global piecewise smooth solution to the steady euler system in the interaction region is established meanwhile the appearance of a vacuum is carefully considered it is found that a vacuum is always adjacent to one of the walls and the appearance of a vacuum depends on the limit of the slope of the wall at the infinity

well posedness of 1 d compressible euler poisson equations with physical vacuum abstract this paper is concerned with the 1 d compressible euler poisson equations with moving physical vacuum boundary condition it is usually used to describe the motion of a self gravitating inviscid gaseous star the local well posedness of classical solutions is established in the case of the adiabatic index 1 3

well posedness in smooth function spaces for moving boundary 1 d compressible euler equations in physical vacuum we prove well posedness for the 3 d compressible euler equations with moving physical vacuum boundary with an equation of state given by the so called gamma gas law for gamma 1 the physical vacuum singularity requires the sound speed c to go to zero as the square root of the distance to the moving boundary and thus creates a degenerate and characteristic hyperbolic free boundary system wherein the density vanishes on the free boundary the uniform kreiss lopatinskii condition is violated and manifest derivative loss ensues nevertheless we are able to establish the existence of unique solutions to this system on a short time interval which are smooth in sobolev spaces all the way to the moving boundary and our estimates have no derivative loss with respect to initial data our proof is founded on an approximation of the euler equations by a degenerate parabolic regularization obtained from a specific choice of a degenerate artificial viscosity term chosen to preserve as much of the geometric structure of the euler equations as possible we first construct solutions to this degenerate parabolic regularization using a new higher order hardy type inequality we then establish estimates for solutions to this degenerate parabolic system which are independent of the artificial viscosity parameter solutions to the compressible euler equations are found in the limit as the artificial viscosity tends to zero our regular solutions can be viewed as degenerate viscosity solutions out methodology can be applied to many other systems of degenerate and characteristic hyperbolic systems of conservation laws

a priori estimates for the free boundary 3d compressible euler equations in physical vacuum we prove a priori estimates for the three dimensional compressible euler equations with moving physical vacuum boundary with an equation of state given by p c for 1 the vacuum condition necessitates the vanishing of the pressure and hence density on the dynamic boundary which creates a degenerate and characteristic hyperbolic free boundary system to which standard methods of symmetrizable hyperbolic equations cannot be applied

a priori estimates for 3d incompressible current vortex sheets we consider the free boundary problem for current vortex sheets in ideal incompressible magneto hydrodynamics it is known that current vortex sheets may be at most weakly neutrally stable due to the existence of surface waves solutions to the linearized equations the existence of such waves may yield a loss of derivatives in the energy estimate of the solution with respect to the source terms however under a suitable stability condition satisfied at each point of the initial discontinuity and a flatness condition on the initial front we prove an a priori estimate in sobolev spaces for smooth solutions with no loss of derivatives the result of this paper gives some hope for proving the local existence of smooth current vortex sheets without resorting to a nash moser iteration such result would be a rigorous confirmation of the stabilizing effect of the magnetic field on kelvin helmholtz instabilities which is well known in astrophysics

global classical solutions to the one dimensional free boundary problem for compressible non isothermal liquid crystal flow with large data abstract in this paper we study the free boundary problem of compressible non isothermal liquid crystal flow in dimension one for the vacuum interface condition that is the density is assumed to vanish and connect to vacuum continuously we prove global existence and uniqueness of classical solutions to the liquid crystal system with large initial data

brief paper state estimation in the presence of bounded disturbances this contribution proposes a robust recursive algorithm for the state estimation of linear models with unknown but bounded disturbances corrupting both the state and measurement vectors a novel approach based on state bounding techniques is presented the proposed algorithm can be decomposed into two steps time updating and observation updating that uses a switching estimation kalman like gain matrix particular emphasis will be given to the design of a weighting factor that ensures the stability of the estimation error

nonlinear stability of expanding star solutions of the radially symmetric mass critical euler poisson system we prove nonlinear stability of compactly supported expanding star solutions of the mass critical gravitational euler poisson system these special solutions were discovered by goldreich and weber in 1980 the expansion rate of such solutions can be either self similar or non self similar linear and we treat both types an important outcome of our stability results is the existence of a new class of global in time radially symmetric solutions which are not homologous and therefore not encompassed by the existing works using lagrangian coordinates we reformulate the associated free boundary problem as a degenerate quasilinear wave equation on a compact spatial domain the problem is mass critical with respect to an invariant rescaling and the analysis is carried out in similarity variables 2017 wiley periodicals inc

optimal estimation theory for dynamic systems with set membership uncertainty an overview in many problems such as linear and nonlinear regressions parameter and state estimation of dynamic systems state space and time series prediction interpolation smoothing and functions approximation one has to evaluate some unknown variable using available data the data are always associated with some uncertainty and it is necessary to evaluate how this uncertainty affects the estimated variables typically the problem is approached assuming a probabilistic description of uncertainty and applying statistical estimation theory an interesting alternative approach referred to as set membership or unknown but bounded ubb error description has been investigated since the late 1960s in this approach uncertainty is described by an additive noise which is known only to have given integral typically l2 of l1 or componentwise l bounds in this paper we review the main results of this theory with special attention to the most recent advances obtained in the case of componentwise bounds

well posedness for the motion of an incompressible liquid with free surface boundary we study the motion of an incompressible perfect liquid body in vacuum this can be thought of as a model for the motion of the ocean or a star the free surface moves with the velocity of the liquid and the pressure vanishes on the free surface this leads to a free boundary problem for euler s equations where the regularity of the boundary enters to highest order we prove local existence in sobolev spaces assuming a physical condition related to the fact that the pressure of a fluid has to be positive

existence and stability of compressible current vortex sheets in three dimensional magnetohydrodynamics compressible vortex sheets are fundamental waves along with shocks and rarefaction waves in entropy solutions to multidimensional hyperbolic systems of conservation laws understanding the behavior of compressible vortex sheets is an important step towards our full understanding of fluid motions and the behavior of entropy solutions for the euler equations in two dimensional gas dynamics the classical linearized stability analysis on compressible vortex sheets predicts stability when the mach number m sqrt 2 and instability when m sqrt 2 for the euler equations in three dimensions every compressible vortex sheet is violently unstable and this instability is the analogue of the kelvin helmholtz instability for incompressible fluids the purpose of this paper is to understand whether compressible vortex sheets in three dimensions which are unstable in the regime of pure gas dynamics become stable under the magnetic effect in three dimensional magnetohydrodynamics mhd one of the main features is that the stability problem is equivalent to a free boundary problem whose free boundary is a characteristic surface which is more delicate than noncharacteristic free boundary problems another feature is that the linearized problem for current vortex sheets in mhd does not meet the uniform kreiss lopatinskii condition these features cause additional analytical difficulties and especially prevent a direct use of the standard picard iteration to the nonlinear problem in this paper we develop a nonlinear approach to deal with these difficulties in three dimensional mhd we first carefully formulate the linearized problem for the current vortex sheets to show rigorously that the magnetic effect makes the problem weakly stable and establish energy estimates especially high order energy estimates in terms of the nonhomogeneous terms and variable coefficients then we exploit these results to develop a suitable iteration scheme of the nash moser hormander type to deal with the loss of the order of derivative in the nonlinear level and establish its convergence which leads to the existence and stability of compressible current vortex sheets locally in time in three dimensional mhd

on global multidimensional supersonic flows with vacuum states at infinity in this paper we are concerned with the global existence and stability of a smooth supersonic flow with vacuum state at infinity in a three dimensional infinitely long divergent nozzle the flow is described by a three dimensional steady potential equation which is multi dimensional quasilinear hyperbolic but degenerate at infinity with respect to the supersonic direction and whose linearized part admits the form partial t 2 frac 1 1 t 2 gamma 1 partial 1 2 partial 2 2 frac 2 gamma 1 1 t partial t for 1 gamma 2 from the physical point of view due to the expansive geometric property of the divergent nozzle and the mass conservation of gases the moving gases in the nozzle will gradually become rarefactive and tend to vacuum states at infinity which implies that such a smooth supersonic flow should be globally stable for small perturbations since there are no strong resulting compressions in the motion of the flow we will confirm such a global stability phenomenon by rigorous mathematical proofs and further show that there do not exist vacuum domains in any finite parts of the nozzle

separation of trajectories and its relation to entropy for intermittent systems with a zero lyapunov exponent i t where is determined by the nonlinearity of the map in the vicinity of marginally unstable fixed points the mean of is determined by the infinite invariant density using semianalytical arguments we calculate the infinite invariant density for the pomeaumanneville map and with it we obtain excellent agreement between numerical simulation and theory we show that is equal to krengel s entropy and to the complexity calculated by the lempel ziv compression algorithm this generalized pesin s identity shows that and krengel s entropy are the natural generalizations of usual lyapunov exponent and entropy for these systems

on the value of information in system identification bounded noise case assuming instantaneous bounds on the noise system parameter identification is formulated as membership set estimation problem sequential algorithms are constructed to estimate the membership sets of the parameters which are consistent with the measurements and the noise constraints the important new feature of the proposed algorithms is their ability to ignore redundant data the efficient data extraction property of the new algorithms is achieved with small computational effort and with improved performance when compared to the least square algorithm the convergence properties and the notion of identifiability in the set theoretic context are also studied

existence of compressible current vortex sheets variable coefficients linear analysis we study the initial boundary value problem resulting from the linearization of the equations of ideal compressible magnetohydrodynamics and the rankine hugoniot relations about an unsteady piecewise smooth solution this solution is supposed to be a classical solution of the system of magnetohydrodynamics on either side of a surface of tangential discontinuity current vortex sheet under some assumptions on the unperturbed flow we prove an energy a priori estimate for the linearized problem since the tangential discontinuity is characteristic the functional setting is provided by the anisotropic weighted sobolev space w21 despite the fact that the constant coefficients linearized problem does not meet the uniform kreiss lopatinskii condition the estimate we obtain is without loss of smoothness even for the variable coefficients problem and nonplanar current vortex sheets the result of this paper is a necessary step in proving the local in time existence of current vortex sheet solutions of the nonlinear equations of magnetohydrodynamics

global classical solutions to the 1 d vacuum free boundary problem for full compressible navier stokes equations with large data the vacuum free boundary problem of one dimensional non isentropic compressible navier stokes equations with large initial data is investigated in this paper the fluid is initially assumed to occupy a finite interval and connect to the vacuum continuously at the free boundary which is often considered in the gas vacuum interface problem using the method of lagrangian particle path we derive some point wise estimates and weighted spatial and time energy estimates for the classical solutions then the global existence and uniqueness of classical solutions are shown and the expanding speed for the free boundary is proved to be finite the main difficulty of this problem is the degeneracy of the system near the free boundary previous results are only for the solutions with low regularity cf g q chen and m kratka commun partial differ equations 27 907 943 2002

an application of the nash moser theorem to the vacuum boundary problem of gaseous stars abstract we have been studying spherically symmetric motions of gaseous stars with physical vacuum boundary governed either by the euler poisson equations in the non relativistic theory or by the einstein euler equations in the relativistic theory the problems are to construct solutions whose first approximations are small time periodic solutions to the linearized problem at an equilibrium and to construct solutions to the cauchy problem near an equilibrium these problems can be solved when 1 1 is an integer where is the adiabatic exponent of the gas near the vacuum by the formulation by r hamilton of the nash moser theorem we discuss on an application of the formulation by j t schwartz of the nash moser theorem to the case in which 1 1 is not an integer but sufficiently large

identification and application of bounded parameter models abstract the place of parameter bounding algorithms in identification methodology is discussed such algorithms provide a radical alternative to the computation of parameter point estimates and covariances instead of p d f or mean and covariance for the noise and prior parameter estimates they require bounds in an essentially deterministic model formulation the paper is a preliminary review of some standard tasks namely experiment design testing for outliers toleranced prediction and worst case control design in the context of parameter bounding identification computational requirements of these tasks are examined and the limitations of existing parameter bounding algorithms are noted

number of first passage times as a measurement of information for weakly chaotic systems we consider a general class of maps of the interval having lyapunov subexponential instability xt x0 exp t x0 t where t grows sublinearly as t we outline here a scheme j stat phys 154 988 2014 whereby the choice of a characteristic function automatically defines the map equation and corresponding growth rate t this matching approach is based on the infinite measure property of such systems we show that the average information that is necessary to record without ambiguity a trajectory of the system tends to t suitably extending the kolmogorov sinai entropy and pesin s identity for such systems information behaves like a random variable for random initial conditions its statistics obeying a universal mittag leffler law we show that for individual trajectories information can be accurately inferred by the number of first passage times through a given turbulent phase space cell this enables us to calculate far more efficiently lyapunov exponents for such systems lastly we also show that the usual renewal description of jumps to the turbulent cell usually employed in the literature does not provide the real number of entrances there our results are supported by exhaustive numerical simulations

large time behavior of solutions for compressible euler equations with damping in r 3 we consider the long time behavior and optimal decay rates of global strong solutions for the isentropic compressible euler equations with damping in r3 in the present paper when the regular initial data belong to some sobolev space hl r3 b 1 s r3 with l 4 and s 0 1 we show that the density of the system converges to its equilibrium state at the rates 1 t 34 s2 in the l2 norm or 1 t 32 s2 in the l norm respectively the momentum of the system decays at the rates 1 t 54 s2 in the l2 norm or 1 t 2 s2 in the l norm respectively which are shown to be optimal for the compressible euler equations with damping

optimal algorithms theory for robust estimation and prediction this paper deals with the theory of optimal algorithms for problems which cannot be solved exactly the theory developed allows for the derivation of new and interesting results in parameter estimation and in time series prediction in situations where no reliable statistical hypothesis can be made on the functions and modeling errors involved but only a bound on them is known in particular the derivation of computationally simple optimal algorithms for these two problems is investigated the practical effectiveness of the algorithms obtained is illustrated by several numerical examples

the free boundary euler equations with large surface tension abstract we study the free boundary euler equations with surface tension in three spatial dimensions showing that the equations are well posed if the coefficient of surface tension is positive then we prove that under natural assumptions the solutions of the free boundary motion converge to solutions of the euler equations in a domain with fixed boundary when the coefficient of surface tension tends to infinity

well posedness in smooth function spaces for the moving boundary three dimensional compressible euler equations in physical vacuum we prove well posedness for the three dimensional compressible euler equations with moving physical vacuum boundary with an equation of state given by p c for 1 the physical vacuum singularity requires the sound speed c to go to zero as the square root of the distance to the moving boundary and thus creates a degenerate and characteristic hyperbolic free boundary system wherein the density vanishes on the free boundary the uniform kreiss lopatinskii condition is violated and manifest derivative loss ensues nevertheless we are able to establish the existence of unique solutions to this system on a short time interval which are smooth in sobolev spaces all the way to the moving boundary and our estimates have no derivative loss with respect to initial data our proof is founded on an approximation of the euler equations by a degenerate parabolic regularization obtained from a specific choice of a degenerate artificial viscosity term chosen to preserve as much of the geometric structure of the euler equations as possible we first construct solutions to this degenerate parabolic regularization using a higher order version of hardy s inequality we then establish estimates for solutions to this degenerate parabolic system which are independent of the artificial viscosity parameter solutions to the compressible euler equations are found in the limit as the artificial viscosity tends to zero our regular solutions can be viewed as degenerate viscosity solutions our methodology can be applied to many other systems of degenerate and characteristic hyperbolic systems of conservation laws

well posedness for compressible euler equations with physical vacuum singularity an important problem in the theory of compressible gas flows is to understand the singular behavior of vacuum states the main difficulty lies in the fact that the system becomes degenerate at the vacuum boundary where the characteristic speeds u c coincide and have unbounded spatial derivative since c behaves like x 1 2 close to the boundary in this paper we overcome this difficulty by presenting a new formulation and new energy spaces we establish the local in time well posedness of one dimensional compressible euler equations for isentropic flows with the physical vacuum singularity in some spaces adapted to the singularity 2009 wiley periodicals inc

exact invariant measures how the strength of measure settles the intensity of chaos the aim of this paper is to show how to extract dynamical behavior and ergodic properties from deterministic chaos with the assistance of exact invariant measures on the one hand we provide an approach to deal with the inverse problem of finding nonlinear interval maps from a given invariant measure then we show how to identify ergodic properties by means of transitions along the phase space via exact measures on the other hand we discuss quantitatively how infinite measures imply maps having subexponential lyapunov instability weakly chaotic as opposed to finite measure ergodic maps which are fully chaotic in addition we provide general solutions of maps for which infinite invariant measures are exactly known throughout the interval a demand from this field finally we give a simple proof that infinite measure implies universal mittag leffler statistics of observables rather than narrow distributions typically observed in finite measure ergodic maps

existence and stability of multidimensional shock fronts in the vanishing viscosity limit in this paper we present a new approach to the study of linear and nonlinear stability of inviscid multidimensional shock waves under small viscosity perturbation yielding optimal estimates and eventually an extension to the viscous case of the celebrated theorem of majda on existence and stability of multidimensional shock waves more precisely given a curved lax shock solution u0 to a hyperbolic system of conservation laws we construct nearby viscous shock solutions ue to a parabolic viscous perturbation of the hyperbolic system which converge to u0 as viscosity e 0 and satisfy an appropriate conormal version of majda s stability estimate

lyapunov statistics and mixing rates for intermittent systems we consider here a recent conjecture stating that correlation functions and tail probabilities of finite time lyapunov exponents would have the same power law decay in weakly chaotic systems we demonstrate that this conjecture fails for a generic class of maps of the pomeau manneville type we show further that typically the decay properties of such tail probabilities do not provide significant information on key aspects of weakly chaotic dynamics such as ergodicity and instability regimes our approaches are firmly based on rigorous results particularly the aaronson darling kac theorem and are also confirmed by exhaustive numerical simulations

on nonlinear asymptotic stability of the lane emden solutions for the viscous gaseous star problem abstract this paper proves the nonlinear asymptotic stability of the lane emden solutions for spherically symmetric motions of viscous gaseous stars if the adiabatic constant lies in the stability range 4 3 2 it is shown that for small perturbations of a lane emden solution with same mass there exists a unique global in time strong solution to the vacuum free boundary problem of the compressible navier stokes poisson system with spherical symmetry for viscous stars and the solution captures the precise physical behavior that the sound speed is c 1 2 holder continuous across the vacuum boundary provided that lies in 4 3 2 the key is to establish the global in time regularity uniformly up to the vacuum boundary which ensures the large time asymptotic uniform convergence of the evolving vacuum boundary density and velocity to those of the lane emden solution with detailed convergence rates and detailed large time behaviors of solutions near the vacuum boundary in particular it is shown that every spherical surface moving with the fluid converges to the sphere enclosing the same mass inside the domain of the lane emden solution with a uniform convergence rate and the large time asymptotic states for the vacuum free boundary problem 1 1 2a 1 1 2b 1 1 2c 1 1 2d 1 1 2e 1 1 2f are determined by the initial mass distribution and the total mass to overcome the difficulty caused by the degeneracy and singular behavior near the vacuum free boundary and coordinates singularity at the symmetry center the main ingredients of the analysis consist of combinations of some new weighted nonlinear functionals involving both lower order and higher order derivatives and space time weighted energy estimates the constructions of these weighted nonlinear functionals and space time weights depend crucially on the structures of the lane emden solution the balance of pressure and gravitation and the dissipation finally the uniform boundedness of the acceleration of the vacuum boundary is also proved

global existence of smooth solutions and convergence to barenblatt solutions for the physical vacuum free boundary problem of compressible euler equations with damping for the physical vacuum free boundary problem with the sound speed being c1 2 holder continuous near vacuum boundaries of the one dimensional compressible euler equations with damping the global existence of the smooth solution is proved which is shown to converge to the barenblatt self similar solution for the porous media equation with the same total mass when the initial datum is a small perturbation of the barenblatt solution the pointwise convergence with a rate of density the convergence rate of velocity in the supremum norm and the precise expanding rate of the physical vacuum boundaries are also given the proof is based on a construction of higher order weighted functionals with both space and time weights capturing the behavior of solutions both near vacuum states and in large time an introduction of a new ansatz higher order nonlinear energy estimates and elliptic estimates 2016 wiley periodicals inc

well posedness for the motion of a compressible liquid with free surface boundary we study the motion of a compressible perfect liquid body in vacuum this can be thought of as a model for the motion of the ocean or a star the free surface moves with the velocity of the liquid and the pressure vanishes on the free surface this leads to a free boundary problem for euler s equations where the regularity of the boundary enters to highest order we prove local existence in sobolev spaces assuming a physical condition related to the fact that the pressure of a fluid has to be positive

global existence and large time behavior for the system of compressible adiabatic flow through porous media in r 3 abstract the system of compressible adiabatic flow through porous media is considered in r 3 in the present paper the global existence and uniqueness of classical solutions are obtained when the initial data is near its equilibrium we also show that the pressure of the system converges to its equilibrium state at the same l 2 rate 1 t 3 4 as the navier stokes equations without heat conductivity but the velocity of the system decays at the l 2 rate 1 t 5 4 which is faster than the l 2 rate 1 t 3 4 for the navier stokes equations without heat conductivity duan and ma 2008 3

expanding large global solutions of the equations of compressible fluid mechanics without any symmetry assumptions on the initial data we construct global in time unique solutions to the vacuum free boundary three dimensional isentropic compressible euler equations when the adiabatic exponent gamma lies in the interval 1 frac 5 3 our initial data lie sufficiently close to the expanding compactly supported affine motions recently constructed by sideris and they satisfy the physical vacuum boundary condition

global stability and decay for the classical stefan problem the classical one phase stefan problem describes the temperature distribution in a homogeneous medium undergoing a phase transition such as ice melting to water this is accom plished by solving the heat equation on a time dependent domain whose boundary is transported by the normal derivative of the temperature along the evolving and a priori unknown free boundary we establish a global in time stability result for nearly spherical geometries and small tempera tures using a novel hybrid methodology which combines energy estimates decay estimates and hopf type inequalities 1 1 the problem formulation we consider the problem of global existence and asymptotic stability of classical solutions to the classical stefan problem describing the evolving free boundary between the liquid and solid phases the temperature of the liquid p t x and the a priori unknown moving phase boundary t must satisfy the following system of equations

parameter bounds for armax models from records with bounded errors in variables the errors in variables problem is that of parameter estimation where all observed variables are corrupted by noise so far the computation of parameter bounds for dynamical models has been performed using equation error and output error approaches in this paper parameter bounds for autoregres sive moving average exogenous armax models are derived on the assumption that both input and output are affected by bounded noise parameter bounding by both the bounded equation error and the bounded errors in variables approach is outlined together with the approximation of the feasible parameter region by an orthotope outer bounding set and is tested on simulated data

local existence for the free boundary problem for nonrelativistic and relativistic compressible euler equations with a vacuum boundary condition we study the free boundary problem for the equations of compressible euler equations with a vacuum boundary condition our main goal is to recover in eulerian coordinates the earlier well posedness result obtained by lindblad 11 for the isentropic euler equations and extend it to the case of full gas dynamics for technical simplicity we consider the case of an unbounded domain whose boundary has the form of a graph and make short comments about the case of a bounded domain we prove the local in time existence in sobolev spaces by the technique applied earlier to weakly stable shock waves and characteristic discontinuities 5 21 it contains in particular the reduction to a fixed domain using the good unknown of alinhac 1 and a suitable nash moser type iteration scheme a certain modification of such an approach is caused by the fact that the symbol associated to the free surface is not elliptic this approach is still directly applicable to the relativistic version of our problem in the setting of special relativity and we briefly discuss its extension to general relativity 2009 wiley periodicals inc

optimal inner bounds of feasible parameter set in linear estimation with bounded noise some problems arising in parameter estimation theory with unknown but bounded measurement errors are discussed in this theory a key role is played by the feasible parameter set i e the set of all parameter values consistent with the system model and the error bounds if a linear relationship between parameters and measurements is assumed this set is a polytope an exact representation of this polytope may be too complex for practical use and approximate descriptions in terms of simple shaped sets contained in the feasible parameter set inner bounds have shown to be useful in several applications balls in l sub infinity norms boxes l sub 2 norms ellipsoids and l sub 1 norms diamonds are used as bounding sets results are given on the computation of maximal balls when their shape is either known or partially free a numerical example is given in which the results of a computation of inner boxes of fixed shape and fixed orientation are given and compared

compressible euler equations with vacuum abstract in this paper we first show that the regular solutions of compressible euler equations in r 3 with damping will not be global if the initial density function has compact support this implies that c 2 cannot be smooth across the boundary separating the gas and the vacuum after a finite time where c is the speed of sound then we study the local existence of solutions for isentropic gas flow in r when c 0 1 is smooth across using the energy method and the characteristics method

thermodynamic phase transitions for pomeau manneville maps we study phase transitions in the thermodynamic description of pomeau manneville intermittent maps from the point of view of infinite ergodic theory which deals with diverging measure dynamical systems for such systems we use a distributional limit theorem to provide both a powerful tool for calculating thermodynamic potentials as also an understanding of the dynamic characteristics at each instability phase in particular topological pressure and renyi entropy are calculated exactly for such systems finally we show the connection of the distributional limit theorem with non gaussian fluctuations of the algorithmic complexity proposed by gaspard and wang proc natl acad sci usa 85 4591 1988

fractional fokker planck equation solution and application recently metzler et al phys rev lett 82 3563 1999 introduced a fractional fokker planck equation ffpe describing a subdiffusive behavior of a particle under the combined influence of external nonlinear force field and a boltzmann thermal heat bath in this paper we present the solution of the ffpe in terms of an integral transformation the transformation maps the solution of ordinary fokker planck equation onto the solution of the ffpe and is based on levy s generalized central limit theorem the meaning of the transformation is explained based on the known asymptotic solution of the continuous time random walk ctrw we investigate in detail i a force free particle ii a particle in a uniform field and iii a particle in a harmonic field we also find an exact solution of the ctrw and compare the ctrw result with the corresponding solution of the ffpe the relation between the fractional first passage time problem in an external nonlinear field and the corresponding integer first passage time is given an example of the one dimensional fractional first passage time in an external linear field is investigated in detail the ffpe is shown to be compatible with the scher montroll approach for dispersive transport and thus is applicable in a large variety of disordered systems the simple ffpe approach can be used as a practical tool for a phenomenological description of certain types of complicated transport phenomena

fractional diffusion and levy stable processes anomalous diffusion in which the mean square distance between diffusing quantities increases faster than linearly in time has been observed in all manner of physical and biological systems from macroscopic surface growth to dna sequences herein we relate the cause of this nondiffusive behavior to the statistical properties of an underlying process using an exact statistical model this model is a simple two state process with long time correlations and is shown to produce a random walk described by an exact fractional diffusion equation fractional diffusion equations describe anomalous transport and are shown to have exact solutions in terms of fox functions including l evy ensuremath alpha stable processes in the superdiffusive domain 1 21

on the finite time splash and splat singularities for the 3 d free surface euler equations we prove that the 3 d free surface incompressible euler equations with regular initial geometries and velocity fields have solutions which can form a finite time splash or splat singularity first introduced in castro et al splash singularity for water waves http arxiv org abs 1106 2120v2 2011 wherein the evolving 2 d hypersurface the moving boundary of the fluid domain self intersects at a point or on surface such singularities can occur when the crest of a breaking wave falls unto its trough or in the study of drop impact upon liquid surfaces our approach is founded upon the lagrangian description of the free boundary problem combined with a novel approximation scheme of a finite collection of local coordinate charts as such we are able to analyze a rather general set of geometries for the evolving 2 d free surface of the fluid we do not assume the fluid is irrotational and as such our method can be used for a number of other fluid interface problems including compressible flows plasmas as well as the inclusion of surface tension effects

on positively invariant sets for discrete time linear systems with disturbance an application of maximal disturbance sets this note concerns the investigation of maximal sets with respect to which a set is positively invariant and given a set a some properties of spl delta p i sets are analyzed the approach uses one step of the backward recursion introduced in d alessandro and de santis 1991 where feasibility and optimality conditions for linear dynamical input and or state constrained systems were analyzed in section ii the author gives the main definitions and some basic results in the subsequent section iii some properties of spl delta p i sets are investigated the particular case of polyhedral sets is analyzed in section iv in section v the author shows an application to linear systems with parametric uncertainties in the dynamic matrix a the author develops a technique to determine a set in the parameters space such that for any value of the parameter in the set a given set in the state space is positively invariant in the last section a numerical example is developed

from weakly chaotic dynamics to deterministic subdiffusion via copula modeling copula modeling consists in finding a probabilistic distribution called copula whereby its coupling with the marginal distributions of a set of random variables produces their joint distribution the present work aims to use this technique to connect the statistical distributions of weakly chaotic dynamics and deterministic subdiffusion more precisely we decompose the jumps distribution of geisel thomae map into a bivariate one and determine the marginal and copula distributions respectively by infinite ergodic theory and statistical inference techniques we verify therefore that the characteristic tail distribution of subdiffusion is an extreme value copula coupling mittag leffler distributions we also present a method to calculate the exact copula and joint distributions in the case where weakly chaotic dynamics and deterministic subdiffusion statistical distributions are already known numerical simulations and consistency with the dynamical aspects of the map support our results

optimality of central and projection algorithms for bounded uncertainty this paper investigates optimality properties of central and projection algorithms for linear problems in the field of system identification in a context in which uncertainty is described in a deterministic rather than statistical way particular attention is devoted to least squares algorithms when the measurement noise is assumed to be unknown but bounded in a hilbert norm a major contribution of this paper consists in proving that least squares algorithms enjoy strong optimality properties on the contrary it is pointed out that these properties do not hold for other frequently used projection algorithms such as least absolute values or minimax algorithms corresponding to a description of the measurement error in h or l norm respectively

estimation theory for nonlinear models and set membership uncertainty this chapter studies the problem of estimating a given function of a vector of unknowns called the problem element by using measurements depending non linearly on the problem element and affected by unknown but bounded noise assuming that both the solution sought and the measurements depend polynomially on the unknown problem element a method is given to compute the axis aligned box of minimal volume containing the feasible solution set i e the set of all unknowns consistent with the actual measurements and the given bound on the noise the center of this box is a point estimate of the solution which enjoys useful optimality properties the sides of the box represent the intervals of possible variation of the estimates important problems like parameter estimation of exponential models time series prediction with arma models and parameter estimates of discrete time state space models can be formalized and solved by using the developed theory

estimation of model quality this paper provides an introduction to recent work on the problem of quantifying errors in the estimation of models for dynamic systems this is a very large field we therefore concentrate on approaches that have been motivated by the need for reliable models for control system design this will involve a discussion of efforts that go under the titles of estimation in th worst case estimation estimation in l1 and stochastic embedding of undermodelling a central theme of this survey is to examine these new methods with reference to the classic bias variance tradeoff in model structure selection

properties of least squares estimates in set membership identification the properties of the least square algorithm in the identification of linear time invariant discrete time systems in the presence of unknown but bounded errors are investigated it is known that the least squares algorithm enjoys strong optimality properties in the case of power bounded l 2 errors while it may be far from optimal in the case of pointwise l bounded errors exact expressions of the local and global worst case errors of the least squares estimates are derived using these results general conditions assuring convergence to zero of the global worst case error are given it is also shown that the least squares algorithm may be optimal in l 1 and l identification of fir systems with pointwise bounded errors

smooth solutions of the one dimensional compressible euler equation with gravity abstract we study one dimensional motions of polytropic gas governed by the compressible euler equations the problem on the half space under a constant gravity gives an equilibrium which has free boundary touching the vacuum and the linearized approximation at this equilibrium gives time periodic solutions but it is difficult to justify the existence of long time true solutions for which this time periodic solution is the first approximation the situation is in contrast to the problem of free motions without gravity the reason is that the usual iteration method for quasilinear hyperbolic problem cannot be used because of the loss of regularities which causes from the touch with the vacuum due to this reason we try to find a family of solutions expanded by a small parameter and apply the nash moser theorem to justify this expansion note that the application of nash moser theorem is necessary for the sake of conquest of the trouble with loss of regularities and the justification of the applicability requires a very delicate analysis of the problem

the interaction between quasilinear elastodynamics and the navier stokes equations the interaction between a viscous fluid and an elastic solid is modeled by a system of parabolic and hyperbolic equations coupled to one another along the moving material interface through the continuity of the velocity and traction vectors we prove the existence and uniqueness locally in time of strong solutions in sobolev spaces for quasilinear elastodynamics coupled to the incompressible navier stokes equations unlike our approach in 5 for the case of linear elastodynamics we cannot employ a fixed point argument on the nonlinear system itself and are instead forced to regularize it by a particular parabolic artificial viscosity term we proceed to show that with this specific regularization we obtain a time interval of existence which is independent of the artificial viscosity together with a priori estimates we identify the global solution in both phases as well as the interface motion as a weak limit in strong norms of our sequence of regularized problems

well posedness of the water waves equations 1 1 presentation of the problem the water waves problem for an ideal liquid consists of describing the motion of the free surface and the evolution of the velocity field of a layer of perfect incompressible irrotational fluid under the influence of gravity in this paper we restrict our attention to the case when the surface is a graph parameterized by a function x where t denotes the time variable and x xi xd rd the horizontal spatial variables the method developed here works equally well for any integer d 1 but the only physically relevant cases are of course d 1 and d 2 the layer of fluid is also delimited from below by a not necessarily flat bottom parameterized by a time independent function b x we denote by q t the fluid domain at time t the incompressibility of the fluids is expressed by 1 1 div v 0 in qu t 0

well posedness of the free surface incompressible euler equations with or without surface tension we provide a new method for treating free boundary problems in perfect fluids and prove local in time well posedness in sobolev spaces for the free surface incompressible 3d euler equations with or without surface tension for arbitrary initial data and without any irrotationality assumption on the fluid this is a free boundary problem for the motion of an incompressible perfect liquid in vacuum wherein the motion of the fluid interacts with the motion of the free surface at highest order

well posedness in sobolev spaces of the full water wave problem in 2 d we consider the motion of the interface of 2 d irrotational incompressible inviscid water wave with air above water and surface tension zero we show that the interface is always not accelerating into the water region normal to itself as rapidly as the normal acceleration of gravity as long as the interface is nonself intersect we therefore obtain the existence and uniqueness of solutions of the full water wave problem locally in time for any initial interface which is nonself intersect including the case that the interface is of multiple heights

on the expanding configurations of viscous radiation gaseous stars thermodynamic model abstract in this work we study the stability of the expanding configurations of radiation gaseous stars such expanding configurations exist for a thermodynamic model given as a class of self similar solutions to the associated dynamic system with viscosity coefficients satisfying 2 3 0 for the monatomic gas that is the bulk viscosity is vanishing with respect to small perturbations this work shows that the linearly expanding homogeneous solutions are stable for a large expanding rate this is an extensive study of the result in 14 by had ic and jang

a new formulation of the 3d compressible euler equations with dynamic entropy remarkable null structures and regularity properties we derive a new formulation of the 3d compressible euler equations exhibiting remarkable null structures and regularity properties our results hold for an arbitrary equation of state which yields the pressure in terms of the density and the entropy in non vacuum regions where the speed of sound is positive our work here is an extension of our prior joint work with j luk in which we derived a similar new formulation in the special case of a barotropic fluid that is when the equation of state depends only on the density the new formulation comprises covariant wave equations for the cartesian components of the velocity and the logarithmic density coupled to a transport equation for the specific vorticity defined to be vorticity divided by density transport equations for the entropy and its gradient and some additional transport divergence curl type equations involving special combinations of the derivatives of the solution variables the good geometric structures in the equations allow one to use the full power of the vectorfield method in treating the wave part of the system in a forthcoming application we will use the new formulation to give a sharp constructive proof of finite time shock formation tied to the intersection of acoustic wave characteristics for solutions with nontrivial vorticity and entropy at the singularity in the present article we derive the new formulation and provide an overview of the central role that it plays in the proof of shock formation although the equations are significantly more complicated than they are in the barotropic case they enjoy many of same remarkable features including i all derivative quadratic inhomogeneous terms are null forms relative to the acoustical metric which is the lorentzian metric driving the propagation of sound waves and ii the transport divergence curl type equations allow one to show that the entropy is one degree more differentiable than the velocity and that the vorticity is exactly as differentiable as the velocity assuming that the initial data enjoy the same gain in regularity this represents a gain of one derivative compared to standard estimates this gain of a derivative which seems to be new for the entropy is essential for closing the energy estimates in our forthcoming proof of shock formation since the second derivatives of the entropy and the first derivatives of the vorticity appear as inhomogeneous terms in the wave equations

feasible parameter set for linear models with bounded errors in all variables abstract when all observed variables of a model are affected by noise parameter estimation is known as the errors in variables problem while parameter bounding methods and algorithms have been extensively developed in the case of exactly known regressor variables little attention has been paid to the bounded errors in variables problem in this paper a formal proof of a previous result on the description of the feasible parameter region for linear models in the presence of bounded errors in all variables is given topological features of the feasible parameter region such as convexity and connectedness are also discussed

estimation theory and uncertainty intervals evaluation in presence of unknown but bounded errors linear families of models and estimators the problem of parameter estimation and of the evaluation of related uncertainty intervals is considered in the case that a probabilistic description of noise and errors is not available of suitable but only a bound on them is known in the present paper only linear parametrizations and estimators are considered very simple and computationally feasible algorithms are derived for evaluating two different types of uncertainty intervals estimates uncertainty intervals parameter uncertainty intervals the relationships between the eui s and the pui s are established and the solution to the problem of the minimum uncertainty intervals estimator is given the latter can be obtained by means of a simple linear programming algorithm

on the free boundary problem of three dimensional incompressible euler equations in this paper we consider in three dimensions the motion of a general inviscid incompressible fluid with a free interface that separates the fluid region from the vacuum we assume that the fluid region is below the vacuum and that there is no surface tension on the free surface then we prove the local well posedness of the free boundary problem in sobolev space provided that there is no self intersection point on the initial surface and under the stability assumption that with being restricted to the initial surface 2007 wiley periodicals inc

well posedness in sobolev spaces of the full water wave problem in 3 d we consider the motion of the interface separating an inviscid incompressible irrotational fluid from a region of zero density in three dimensional space we assume that the fluid region is below the vacuum the fluid is under the influence of gravity and the surface tension is zero assume that the density of mass of the fluid is one the gravitational field is 0 0 1 the free interface is t at time t 0 and the fluid occupies the region t the motion of the fluid is described by vt v v 0 0 1 p on t t 0 euler s equation 1 1 div v 0 on t t 0 incompressible 1 2 curl v 0 on t t 0 irrotational 1 3 where v v1 v2 v3 is the fluid velocity p is the fluid pressure since we neglect the surface tension the pressure is zero on the interface so on the interface p 0 on t 1 4 1 v is tangent to the free surface t t 1 5 we want to find solutions of system 1 1 1 5 taking prescribed initial data such that for every fixed t 0 t approaches the xy plane at infinity and v x y z t 0 vt x y z t 0 as x y z since the fluid is assumed irrotational incompressible we can reduce the study of the entire motion to the motion of the free surface the above model is a 3 d water wave model it is generally known that when surface tension is neglected the motion of the interface between an inviscid fluid and vacuum under the influence of gravity can be subject to taylor instability 8 22 in a previous work 24 we studied the 2 dimensional water wave model we showed that for a 2 d water wave the sign condition relating to taylor instability always holds for nonself intersecting interface that is the motion of the interface is not subject to taylor instability we showed further that the 2 d full nonlinear water wave problem is uniquely solvable in sobolev spaces locally in time for any initially nonself intersecting interface earlier works on the well posedness in sobolev spaces of the 2 d water wave problem include nalimov 18 yosihara 25 and walter craig 6 where the main results concern the well posedness of the

fundamental solution of the tempered fractional diffusion equation in this paper we consider the space time fractional diffusion equation dt u x t k dx u x t 0 x r t 0 with the tempered riemann liouville derivative of order 0 1 in space and the caputo derivative of order 0 1 in time the fundamental solution which turns out to be a spatial probability density function is given in computable series form as well as in integral representation the spatial moments of the probability density function are determined explicitly for an arbitrary order n 0 moreover green s function of the untempered neutral fractional diffusion equation is analyzed in view of absolute and relative extreme points at the end of this article we point out a remarkably and important integral representation for accurate evaluation of the m wright mainardi function m x of order 0 1 and arguments x r0

lagrangian formulation and a priori estimates for relativistic fluid flows with vacuum abstract we study the evolution of a compressible fluid surrounded by vacuum and introduce a new symmetrization in lagrangian coordinates that allows us to encompass both relativistic and non relativistic fluid flows the problem under consideration is a free boundary problem of central interest in compressible fluid dynamics and from the mathematical standpoint the main challenge to be overcome lies in the loss of regularity in the fluid variables near the free boundary based on our lagrangian formulation we establish the necessary a priori estimates in weighted sobolev spaces which are adapted to this loss of regularity

local existence with physical vacuum boundary condition to euler equations with damping abstract in this paper we consider the local existence of solutions to euler equations with linear damping under the assumption of physical vacuum boundary condition by using the transformation introduced in lin and yang methods appl anal 7 3 2000 495 to capture the singularity of the boundary we prove a local existence theorem on a perturbation of a planar wave solution by using littlewood paley theory and justifies the transformation introduced in liu and yang 2000 in a rigorous setting

geometry and a priori estimates for free boundary problems of the euler s equation in this paper we derive estimates to the free boundary problem for the euler equation with surface tension and without surface tension provided the rayleigh taylor sign condition holds we prove that as the surface tension tends to 0 when the rayleigh taylor condition is satisfied solutions converge to the euler flow with zero surface tension 2007 wiley periodicals inc

alternative numerical computation of one sided levy and mittag leffler distributions we consider here the recently proposed closed form formula in terms of the meijer g functions for the probability density functions g ensuremath alpha x of one sided l evy stable distributions with rational index ensuremath alpha l k with 0l ensuremath alpha l1 since one sided l evy and mittag leffler distributions are known to be related this formula could also be useful for calculating the probability density functions ensuremath rho ensuremath alpha x of the latter we show however that the formula is computationally inviable for fractions with large denominators being unpractical even for some modest values of l and k we present a fast and accurate numerical scheme based on an early integral representation due to mikusinski for the evaluation of g ensuremath alpha x and ensuremath rho ensuremath alpha x their cumulative distribution function and their derivatives for any real index ensuremath alpha ensuremath in 0 1 as an application we explore some properties of these probability density functions in particular we determine the location and value of their maxima as functions of the index ensuremath alpha we show that ensuremath alpha ensuremath approx 0 567 and 0 605 correspond respectively to the one sided l evy and mittag leffler distributions with shortest maxima we close by discussing how our results can elucidate some recently described dynamical behavior of intermittent systems

well posedness for the motion of physical vacuum of the three dimensional compressible euler equations with or without self gravitation this paper concerns the well posedness theory of the motion of a physical vacuum for the compressible euler equations with or without self gravitation first a general uniqueness theorem of classical solutions is proved for the three dimensional general motion second for the spherically symmetric motions without imposing the compatibility condition of the first derivative being zero at the center of symmetry a new local in time existence theory is established in a functional space involving less derivatives than those constructed for three dimensional motions in coutand et al commun math phys 296 559 587 2010 coutand and shkoller arch ration mech anal 206 515 616 2012 jang and masmoudi well posedness of compressible euler equations in a physical vacuum 2008 by constructing suitable weights and cutoff functions featuring the behavior of solutions near both the center of the symmetry and the moving vacuum boundary