vanishing of vacuum states and blow up phenomena of the compressible navier stokes equations the navier stokes systems for compressible fluids with density dependent viscosities are considered in the present paper these equations in particular include the ones which are rigorously derived recently as the saint venant system for the motion of shallow water from the navier stokes system for incompressible flows with a moving free surface 14 these compressible systems are degenerate when vacuum state appears we study initial boundary value problems for such systems for both bounded spatial domains or periodic domains the dynamics of weak solutions and vacuum states are investigated rigorously

brief input to state stability of pd controlled robotic systems a sufficient condition for input to state stability is presented for systems admitting a lyapunov function with negative semi definite derivative an application to pd controlled robotic manipulators is also described

compressible navier stokes equations with degenerate viscosity coefficient and vacuum in this paper we consider the compressible navier stokes equations for isentropic flow of finite total mass when the initial density is either of compact or infinite support the viscosity coefficient is assumed to be a power function of the density so that the cauchy problem is well posed new global existence results are established when the density function connects to the vacuum states continuously for this some new a priori estimates are obtained to take care of the degeneracy of the viscosity coefficient at vacuum we will also give a non global existence theorem of regular solutions when the initial data are of compact support in eulerian coordinates which implies singularity forms at the interface separating the gas and vacuum

global existence of solutions for compressible navier stokes equations with vacuum abstract in this paper we will investigate the global existence of solutions for the one dimensional compressible navier stokes equations when the density is in contact with vacuum continuously more precisely the viscosity coefficient is assumed to be a power function of density i e a where a and are positive constants new global existence result is established for 0 1 when the initial density appears vacuum in the interior of the gas which is the novelty of the presentation

strong convergence to global solutions for multidimensional flows of compressible viscous fluids with polytropic equations of state and discontinuous initial data we extend to general polytropic pressures p k 1 the existence theory of 8 for isothermal 1 flows of navier stokes fluids in two and three space dimensions with fairly general initial data specifically we require that the initial density be close to a constant in l2 and l and that the initial velocity be small in l2 and bounded in l2n in two dimensions the l2 norms must be weighted slightly solutions are obtained as limits of approximate solutions corresponding to mollified initial data the key point is that the approximate densities are shown to converge strongly so that nonlinear pressures can be accommodated even in the absence of any uniform regularity information for the approximate densities

asymptotic tracking of a reference state for systems with a feedforward structure we consider the problem of asymptotic tracking for a system which can be written in a feedforward form the data of a bounded reference state trajectory is assumed our solution relies on a lyapunov construction the time varying state feedbacks obtained are bounded and ensure the global uniform asymptotic stability when the signals are periodic and the local exponential stability of the reference state trajectory of the closed loop system

existence and uniqueness of global weak solution to a two phase flow model with vacuum in this paper we study a two phase liquid gas model with constant viscosity coefficient when both the initial liquid and gas masses connect to vacuum continuously just as in evje and karlsen commun pure appl anal 8 1867 1894 2009 and evje et al nonlinear anal 70 3864 3886 2009 the gas is assumed to be polytropic whereas the liquid is treated as an incompressible fluid we use a new technique to get the upper and lower bounds of gas and liquid masses n and m then we get the global existence of weak solution by the line method also we obtain the uniqueness of the weak solution

discontinuous solutions of the navier stokes equations for multidimensional flows of heat conducting fluids we prove the global existence of weak solutions of the navier stokes equations for compressible heat conducting fluids in two and three space dimensions when the initial density is close to a constant in l 2 l the initial temperature is close to a constant in l 2 and the initial velocity is small in h s l 4 where s 0 when n 2 and when n 3 the l p norms must be weighted slightly when n 2 in particular the initial data may be discontinuous across a hypersurface of n a great deal of qualitative information about the solution is obtained for example we show that the velocity vorticity and temperature are relatively smooth in positive time as is the effective viscous flux f which is the divergence of the velocity minus a certain multiple of the pressure we find that f plays a central role in the entire analysis particularly in closing the required energy estimates and in understanding rates of regularization near the initial layer moreover f is precisely the quantity through which the hyperbolicity of the corresponding equations for inviscid fluids shows itself an effect which is crucial for obtaining time independent pointwise bounds for the density

existence of global weak solutions for a 2d viscous shallow water equations and convergence to the quasi geostrophic model we consider a two dimensional viscous shallow water model with friction term existence of global weak solutions is obtained and convergence to the strong solution of the viscous quasi geostrophic equation with free surface term is proven in the well prepared case the ill prepared data case is also discussed

on spherically symmetric solutions of the compressible isentropic navier stokes equations we prove the global existence of weak solutions to the cauchy problem for the compressible isentropic navier stokes equations in n n 2 3 when the cauchy data are spherically symmetric the proof is based on the exploitation of the one dimensional feature of symmetric solutions and use of a new multidimensional property induced by the viscous flux the present paper extends lions existence theorem 15 to the case 1 n for spherically symmetric initial data where is the specific heat ratio in the pressure n 3 2 for n 2 and n 9 5 for n 3 dedicated to professor rolf leis on the occasion of his 70th birthday

blow up of smooth highly decreasing at infinity solutions to the compressible navier stokes equations abstract we prove that the smooth solutions to the cauchy problem for the navier stokes equations with conserved total mass finite total energy and finite momentum of inertia lose the initial smoothness within a finite time in the case of space of dimension 3 or greater even if the initial data are not compactly supported the cases of isentropic and incompressible fluids are also considered

a nested matrosov theorem and persistency of excitation for uniform convergence in stable nonautonomous systems a new infinitesimal sufficient condition is given for uniform global asymptotic stability ugas for time varying nonlinear systems it is used to show that a certain relaxed persistency of excitation condition called uniform spl delta persistency of excitation u spl delta pe is sufficient for uniform global asymptotic stability in certain situations u spl delta pe of the right hand side of a time varying differential equation is also shown to be necessary under a uniform lipschitz condition the infinitesimal sufficient condition for ugas involves the inner products of the flow field with the gradients of a finite number of possibly sign indefinite locally lipschitz lyapunov like functions these inner products are supposed to be bounded by functions that have a certain nested or triangular negative semidefinite structure this idea is reminiscent of a previous idea of matrosov who supplemented a lyapunov function having a negative semidefinite derivative with an additional function having a derivative that is definitely nonzero where the derivative of the lyapunov function is zero for this reason we call the main result a nested matrosov theorem the utility of our results on stability analysis is illustrated through the well known case study of the nonholonomic integrator

further results on strict lyapunov functions for rapidly time varying nonlinear systems we explicitly construct global strict lyapunov functions for rapidly time varying nonlinear control systems the lyapunov functions we construct are expressed in terms of oftentimes more readily available lyapunov functions for the limiting dynamics which we assume are uniformly globally asymptotically stable this leads to new sufficient conditions for uniform global exponential uniform global asymptotic and input to state stability of fast time varying dynamics we also construct strict lyapunov functions for our systems using a strictification approach we illustrate our results using several examples

global classical solutions to the 2d compressible mhd equations with large data and vacuum abstract in this paper we study the global well posedness of classical solutions to the 2d compressible mhd equations with large initial data and vacuum with the assumption const and 1 va gant kazhikhov model for the viscosity coefficients the global existence and uniqueness of classical solutions to the initial value problem is established on the torus t 2 and the whole space r 2 with vacuum or non vacuum far fields these results generalize the previous ones for the va gant kazhikhov model of compressible navier stokes equations

on blowup of classical solutions to the compressible navier stokes equations in this paper we study the finite time blow up of smooth solutions to the compressible navier stokes system when the initial data contain vacuums we prove that any classical solutions of viscous compressible fluids without heat conduction will blow up in finite time as long as the initial data has an isolated mass group see definition 2 2 the results hold regardless of either the size of the initial data or the far fields being vacuum or not this improves the blowup results of xin comm pure appl math 51 229 240 1998 by removing the crucial assumptions that the initial density has compact support and the smooth solution has finite total energy furthermore the analysis here also yields that any classical solutions of viscous compressible fluids without heat conduction in bounded domains or periodic domains will blow up in finite time if the initial data have an isolated mass group satisfying some suitable conditions

various lipschitz like properties for functions and sets i directional derivative and tangential characterizations in this work we introduce for extended real valued functions defined on a banach space x the concept of k directionally lipschitzian behavior where k is a bounded subset of x for different types of sets k e g zero singleton or compact the k directionally lipschitzian behavior recovers well known concepts in variational analysis locally lipschitzian directionally lipschitzian or compactly epi lipschitzian properties respectively characterizations of this notion are provided in terms of the lower dini subderivatives we also adapt the concept for sets and establish characterizations of the mentioned behavior in terms of the bouligand tangent cones the special case of convex functions and sets is also studied

remarks on blow up of smooth solutions to the compressible fluid with constant and degenerate viscosities abstract in this paper we will show the blow up of smooth solutions to the cauchy problem for the full compressible navier stokes equations and isentropic compressible navier stokes equations with constant and degenerate viscosities in arbitrary dimensions under some restrictions on the initial data in particular the results hold true for the full compressible euler equations and isentropic compressible euler equations and the blow up time can be computed in a more precise way it is not required that the initial data has compact support or contains vacuum in any finite regions moreover we will give a simplified and unified proof on the blow up results to the classical solutions of the full compressible navier stokes equations without heat conduction by xin 41 and with heat conduction by cho jin 5

global classical solution of the cauchy problem to 1d compressible navier stokes equations with large initial data in this paper we prove that the 1d cauchy problem of the compressible navier stokes equations admits a unique global classical solution u if the viscosity 1 with 0 the initial data can be arbitrarily large and may contain vacuum some new weighted estimates of the density and velocity are obtained when deriving higher order estimates of the solution

a free boundary problem for compressible hydrodynamic flow of liquid crystals in one dimension abstract in this paper we consider the free boundary problem for a simplified version of ericksen leslie equations modeling the compressible hydrodynamic flow of nematic liquid crystals in dimension one we obtain both existence and uniqueness of global classical solutions provided that the initial density is away from vacuum

blow up of viscous heat conducting compressible flows abstract we show the blow up of strong solution of viscous heat conducting flow when the initial density is compactly supported this is an extension of z xin s result z xin blow up of smooth solutions to the compressible navier stokes equations with compact density comm pure appl math 51 1998 229 240 to the case of positive heat conduction coefficient but we do not need any information for the time decay of total pressure nor the lower bound of the entropy we control the lower bound of second moment by total energy and obtain the exact relationship between the size of support of initial density and the existence time we also provide a sufficient condition for the blow up in case that the initial density is positive but has a decay at infinity

existence results for viscous polytropic fluids with vacuum abstract we consider the full navier stokes equations for viscous polytropic fluids with nonnegative thermal conductivity we prove the existence of unique local strong solutions for all initial data satisfying some compatibility condition the initial density need not be positive and may vanish in an open set moreover our results hold for both bounded and unbounded domains

blow up criterion for the compressible magnetohydrodynamic equations with vacuum abstract in this paper the 3 d compressible magnetohydrodynamic mhd equations with initial vacuum or infinite electric conductivity is considered we prove that the l norms of the deformation tensor d u and the absolute temperature control the possible blow up see 18 23 of strong solutions especially for the non resistive mhd system when the magnetic diffusion vanishes this conclusion means that if a solution of the compressible mhd equations is initially regular and loses its regularity at some later time then the formation of singularity must be caused by losing the bound of d u or as the critical time approaches the viscosity coefficients are only restricted by the physical conditions our criterion see 1 17 is similar to 17 for 3 d incompressible euler equations and to 12 for 3 d compressible isentropic navier stokes equations

global well posedness of classical solutions with large oscillations and vacuum to the three dimensional isentropic compressible navier stokes equations we establish the global existence and uniqueness of classical solutions to the cauchy problem for the isentropic compressible navier stokes equations in three spatial dimensions with smooth initial data that are of small energy but possibly large oscillations with constant state as far field which could be either vacuum or nonvacuum the initial density is allowed to vanish and the spatial measure of the set of vacuum can be arbitrarily large in particular the initial density can even have compact support these results generalize previous results on classical solutions for initial densities being strictly away from vacuum and are the first for global classical solutions that may have large oscillations and can contain vacuum states 2012 wiley periodicals inc

blowup criterion for viscous baratropic flows with vacuum states we prove that the maximum norm of the deformation tensor of velocity gradients controls the possible breakdown of smooth strong solutions for the 3 dimensional 3d barotropic compressible navier stokes equations more precisely if a solution of the 3d barotropic compressible navier stokes equations is initially regular and loses its regularity at some later time then the loss of regularity implies the growth without bound of the deformation tensor as the critical time approaches our result is the same as ponce s criterion for 3 dimensional incompressible euler equations ponce in commun math phys 98 349 353 1985 in addition initial vacuum states are allowed in our cases

indefinite lyapunov functions for input to state stability of impulsive systems abstract this paper presents a new lyapunov method for the input to state stability iss and integral iss iiss of impulsive systems the approach is proposed on the basis of an indefinite lyapunov function rather than a negative definite one when the continuous dynamics have a lyapunov function with an indefinite time derivative and the discrete dynamics are destabilizing the impulsive system is iss if an average dwell time adt condition is satisfying numerical examples show their effectiveness and advantages

global well posedness of the cauchy problem of two dimensional compressible navier stokes equations in weighted spaces abstract in this paper we study the global well posedness of classical solution to 2d cauchy problem of the compressible navier stokes equations with large initial data and vacuum it is proved that if the shear viscosity is a positive constant and the bulk viscosity is the power function of the density that is with 3 then the 2d cauchy problem of the compressible navier stokes equations on the whole space r 2 admits a unique global classical solution u which may contain vacuums in an open set of r 2 note that the initial data can be arbitrarily large to contain vacuum states various weighted estimates of the density and velocity are obtained in this paper and these self contained estimates reflect the fact that the weighted density and weighted velocity propagate along with the flow

mean value inequalities in hilbert space we establish a new mean value theorem applicable to lower semicontinuous functions on hilbert space a novel feature of the result is its multidirectionality it compares the value of a function at a point to its values on a set we then discuss some refinements and consequences of the theorem including applications to calculus flow invariance and generalized solutions to partial differential equations rtsumt on etablit un nouveau theoreme de la valeur moyenne qui s applique aux fonctions semicontinues inferieurement sur un espace de hilbert on deduit plusieurs consequences du resultat portant par exemple sur les fonctions monotones et sur les solutions generalisees des equations aux derivees partielles

blow up of smooth solutions to the compressible magnetohydrodynamic flows abstract in this paper we prove the blow up phenomena of smooth solutions to the cauchy problem for the full compressible magnetohydrodynamic equations and isentropic compressible magnetohydrodynamic equations with constant and degenerate viscosities under some restrictions on the initial data in particular our results do not require that the initial data have compact support or contain vacuum in any finite region

robust stabilization of the angular velocity of a rigid body the problem of robust control for the angular velocity of a rigid body subject to external disturbances is addressed it is shown that if the disturbances are matched there exists a control law attenuating the effect of the disturbances whereas in the case of non matched disturbances no such a feedback law generically exists hence a new concept of disturbance attenuation is introduced and it is proved that the aforementioned problem is solvable in this weaker sense explicit expressions of the control laws solving the proposed robust stabilization problems are given

approximate mean value theorem for upper subderivatives the purpose of this paper is to give a mean value theorem for a lower semicontinuous function f on a banach space using the upper subderivatives defined by rockafellar l as will be shown by example section 4 it may happen that the subdifferential af x is empty for all x in the closed line segment a 61 where a and b belong to x therefore we cannot formulate the mean value theorem in such a form as that obtained in 2 and 3 for other generalized derivatives we prove among other factors that there exists a sequence xk converging to some point x e a 61 x b such that the sets df xj are nonempty and

a smooth variational principle with applications to subdifferentiability and to differentiability of convex functions we show that typically lower semicontinuous functions on a banach space densely inherit lower subderivatives of the same degree of smoothness as the norm in particular every continuous convex function on a space with a g teaux weak hadamard frechet smooth renorm is densely g teaux weak hadamard frechet differentiable our technique relies on a more powerful analogue of ekeland s variational principle in which the function is perturbed by a quadratic like function this smooth variational principle has very broad applicability in problems of nonsmooth analysis

on the existence of local classical solutions to the navier stokes equations with degenerate viscosities in this paper we consider the three dimensional cauchy problem of compressible navier stokes equations with degenerate viscosities depending on density with the initial data containing vacuum at far fields we prove the local existence of regular solutions with some compatibility conditions the regular solution is classical due to the smoothing effect of the velocity u in this paper we consider the three dimensional cauchy problem of compressible navier stokes equations with degenerate viscosities depending on density with the initial data containing vacuum at far fields we prove the local existence of regular solutions with some compatibility conditions the regular solution is classical due to the smoothing effect of the velocity u

a lyapunov approach to cascade interconnection of integral input to state stable systems this technical note deals with the stability of cascade interconnection of integral input to state iiss time varying systems a new technique to construct smooth lyapunov functions of cascaded systems is proposed from the construction sufficient conditions for internal stability and stability with respect to external signals are derived one of the derived conditions is a trade off between slower convergence of the driving system and steeper input growth of the driven system the trade off is no more necessary if the speed of convergence of the driven system is not radially vanishing the results are related to trajectory based approaches and small gain techniques for feedback interconnection the difference between the feedback case and the cascade case is viewed from the requirement on convergence speed of autonomous parts

global strong solutions to radial symmetric compressible navier stokes equations with free boundary in this paper we consider the two dimensional barotropic compressible navier stokes equations with stress free boundary condition imposed on the free surface as the viscosity coefficients satisfies 2 1 we establish the existence of global strong solution for arbitrarily large spherical symmetric initial data even if the density vanishes across the free boundary in particular we show that the density is strictly positive and bounded from the above and below in any finite time if the initial density is strictly positive and the free boundary propagates along the particle path and expand outwards at an algebraic rate

entropy bounded solutions to the one dimensional compressible navier stokes equations with zero heat conduction and far field vacuum abstract the entropy is one of the fundamental states of a fluid and in the viscous case the equation that it satisfies is highly singular in the region close to the vacuum in spite of its importance in the gas dynamics the mathematical analyses on the behavior of the entropy near the vacuum region were rarely carried out in particular in the presence of vacuum either at the far field or at some isolated interior points it was unknown whether the entropy remains its boundedness the results obtained in this paper indicate that the ideal gases retain their uniform boundedness of the entropy locally or globally in time if the vacuum occurs at the far field only and the density decays slowly enough at the far field precisely we consider the cauchy problem to the one dimensional full compressible navier stokes equations without heat conduction and establish the local and global existence and uniqueness of entropy bounded solutions in the presence of vacuum at the far field only it is also shown that different from the case that with compactly supported initial density the compressible navier stokes equations with slowly decaying initial density can propagate the regularities in inhomogeneous sobolev spaces

lagrange structure and dynamics for solutions to the spherically symmetric compressible navier stokes equations the compressible navier stokes system cns with density dependent viscosity coefficients is considered in multi dimension the prototype of the system is the viscous saint venat model for the motion of shallow water a spherically symmetric weak solution to the free boundary value problem for cns with stress free boundary condition and arbitrarily large data is shown to exist globally in time with the free boundary separating fluids and vacuum and propagating at finite speed as particle path which is continuous away from the symmetry center detailed regularity and lagrangian structure of this solution have been obtained in particular it is shown that the particle path is uniquely defined starting from any non vacuum region away from the symmetry center along which vacuum states shall not form in any finite time and the initial regularities of the solution is preserved starting from any non vacuum point at a later on time a particle path is also uniquely defined backward in time which either reaches at some initial non vacuum point or stops at a small middle time and connects continuously with vacuum in addition the free boundary is shown to expand outward at an algebraic rate in time and the fluid density decays and tends to zero almost everywhere away from the symmetry center as the time grows up this finally leads to the formation of vacuum state almost everywhere as the time goes to infinity

on the dynamics of navier stokes equations for a shallow water model in this paper we study a free boundary problem of one dimensional compressible navier stokes equations with a density dependent viscosity which include in particular a shallow water model under some suitable assumptions on the initial data we obtain the global existence uniqueness and the large time behavior of weak solutions in particular it is shown that a stationary wave pattern connecting a gas to the vacuum continuously is asymptotically stable for small initial general perturbations

the initial value problem for the compressible navier stokes equations without heat conductivity abstract in this paper we are concerned with the global existence and convergence rates of strong solutions for the compressible navier stokes equations without heat conductivity in r 3 the global existence and uniqueness of strong solutions are established by the delicate energy method under the condition that the initial data are close to the constant equilibrium state in h 2 framework furthermore if additionally the initial data belong to l 1 the optimal convergence rates of the solutions in l 2 norm and convergence rates of their spatial derivatives in l 2 norm are obtained

global weak solutions for a viscous liquid gas model with transition to single phase gas flow and vacuum abstract this work deals with a viscous two phase liquid gas model relevant to the flow in wells and pipelines the liquid is treated as an incompressible fluid whereas the gas is assumed to be polytropic the model is rewritten in terms of lagrangian coordinates and is studied in a free boundary setting where the liquid and gas masses are of compact support initially and continuous at the boundary consequently the initial masses involve a transition to single phase gas flow and vacuum at the boundary an appropriate balance between pressure and viscous forces is identified which allows obtaining pointwise upper and lower estimates of masses these estimates rely on the assumption of a certain relation between the rate of degeneracy of the viscosity coefficient and the rate that determines how fast the initial masses are vanishing at the boundary by combining these estimates with basic energy type of estimates higher order regularity estimates are obtained the existence of global weak solutions is then proved by showing compactness for a class of semi discrete approximations

blow up criterion for the 3d compressible non isentropic navier stokes equations without thermal conductivity abstract in this paper we prove a blow up criterion in terms of the density and the pressure p for the strong solutions with vacuum to cauchy problem of the 3d compressible non isentropic navier stokes equations without thermal conductivity more precisely we show that the strong solution exists globally if the norm p l 0 t r 3 is bounded from above

global smooth solutions of the compressible navier stokes equations with density dependent viscosity abstract in this paper we study a free boundary problem for the viscous compressible heat conducting one dimensional real fluids more precisely the viscosity is assumed to be a power function of density i e where denotes the density of fluids and is a positive constant in addition the equations of state include and are more general than perfect flows which only depend linearly on temperature the global existence uniqueness of smooth solutions is established with 0 1 2 for general large initial data which improves the previous results moreover it is also shown that the solutions will not develop vacuum mass concentration or heat concentration in a finite time provided the initial data are bounded and smooth and do not contain vacuum

global classical large solutions to 1d compressible navier stokes equations with density dependent viscosity and vacuum abstract in this paper we investigate an initial boundary value problem for 1d compressible isentropic navier stokes equations with large initial data density dependent viscosity external force and vacuum making full use of the local estimates of the solutions in cho and kim 2006 3 and the one dimensional properties of the equations and the sobolev inequalities we get a unique global classical solution u where c 1 0 t h 1 0 1 and u h 1 0 t h 2 0 1 for any t 0 as it is pointed out in xin 1998 31 that the smooth solution u c 1 0 t h 3 r 1 t is large enough of the cauchy problem must blow up in finite time when the initial density is of nontrivial compact support it seems that the regularities of the solutions we obtained can be improved which motivates us to obtain some new estimates with the help of a new test function 2 u t t such as lemmas 3 2 3 6 this leads to further regularities of u where c 1 0 t h 3 0 1 u h 1 0 t h 3 0 1 it is still open whether the regularity of u could be improved to c 1 0 t h 3 0 1 with the appearance of vacuum since it is not obvious that the solutions in c 1 0 t h 3 0 1 to the initial boundary value problem must blow up in finite time

time varying exponential stabilization of a rigid spacecraft with two control torques rigid body models with two controls cannot be locally asymptotically stabilized by continuous feedbacks which are functions of the state only this impossibility no longer holds when the feedback is also a function of time and time varying asymptotically stabilizing feedbacks have already been proposed however due to the smoothness of the feedbacks the convergence rate is only polynomial in this paper exponential convergence is obtained by considering time varying feedbacks which are only continuous

stability of periodically time varying systems this paper proposes a novel approach to stability analysis of discrete time nonlinear periodically time varying systems the contributions are as follows firstly a relaxation of standard lyapunov conditions is derived this leads to a less conservative lyapunov function that is required to decrease at each period rather than at each time instant secondly for linear periodic systems with constraints it is shown that compared to standard lyapunov theory the novel concept of periodic lyapunov functions allows for the calculation of a larger estimate of the region of attraction an example illustrates the effectiveness of the developed theory

further remarks on strict input to state stable lyapunov functions for time varying systems we study the stability properties of a class of time varying non linear systems we assume that non strict input to state stable iss lyapunov functions for our systems are given and posit a mild persistency of excitation condition on our given lyapunov functions which guarantee the existence of strict iss lyapunov functions for our systems next we provide simple direct constructions of explicit strict iss lyapunov functions for our systems by applying an integral smoothing method we illustrate our constructions using a tracking problem for a rotating rigid body

the failure on continuous dependence on initial data for the navier stokes equations of compressible flow it is shown that physical solutions of the navier stokes equations for one dimensional compressible flow need not depend continuously on their initial data at least when vacuum states are allowed specifically two fluid regions initially separated by a third region of very low density 6 are considered it is shown that as delta to 0 the unique solutions corresponding to delta 0 do not in fact converge to a physical solution but rather to a nonphysical weak solution in which the two fluids cannot collide independent of their initial velocities and whose separate momenta need not be conserved a particular consequence is that solutions of the cavity problem delta 0 are not unique

a vacuum problem for the one dimensional compressible navier stokes equations with density dependent viscosity this paper is concerned with the free boundary problem for the one dimensional compressible navier stokes equations with density dependent viscosity a local in time existence result is established when the initial density is of compact support and connects to the vacuum continuously c 2002 elsevier science usa

global existence and asymptotic behavior for the 3d compressible navier stokes equations without heat conductivity in a bounded domain abstract in this paper we investigate the global existence and uniqueness of strong solutions to the initial boundary value problem for the 3d compressible navier stokes equations without heat conductivity in a bounded domain with slip boundary the global existence and uniqueness of strong solutions are obtained when the initial data is near its equilibrium in h 2 furthermore the exponential convergence rates of the pressure and velocity are also proved by delicate energy methods

finite time blow up of compressible navier stokes equations on half space or outside a fixed ball abstract in this paper we consider the initial boundary value problem to the compressible navier stokes equations for ideal gases without heat conduction in the half space or outside a fixed ball in r n with n 1 we prove that any classical solutions u in the class c 1 0 t h m m n 2 2 with bounded from below initial entropy and compactly supported initial density which allows to touch the physical boundary must blow up in finite time as long as the initial mass is positive this paper extends the classical result by xin 1998 19 in which the cauchy problem is considered to the case that with physical boundary

blowup of smooth solutions to the compressible navier stokes equation with compact density we present a sufficient condition on the blowup of smooth solutions to the compressible navier stokes equations in arbitrary space dimensions with initial density of compact support as an immediate application it is shown that any smooth solutions to the compressible navier stokes equations for polytropic fluids in the absence of heat conduction will blow up in finite time as long as the initial densities have compact support and an upper bound which depends only on the initial data on the blowup time follows from our elementary analysis immediately another implication is that there is no global small decay in time or even bounded in the case that all the viscosity coefficients are positive smooth solutions to the compressible navier stokes equations for polytropic fluids no matter how small the initial data are as long as the initial density is of compact support this is in contrast to the classical theory of global existence of small solutions to the same system with initial data being a small perturbation of a constant state that is not a vacuum the blowup of smooth solutions to the compressible euler system with initial density and velocity of compact support is a simple consequence of our argument 1998 john wiley sons inc

input to state stability of nonlinear systems based on an indefinite lyapunov function abstract this study investigated the input to state stability iss and integral iss iiss of nonlinear systems and discovered new sufficient conditions for them these conditions are more relaxed than others in that they employ an indefinite lyapunov function rather than a negative definite one thus a nonlinear time varying system satisfying them has uniform asymptotic stability two numerical examples show their effectiveness and advantages

strong lyapunov functions for systems satisfying the conditions of la salle we present a construction of a strong lyapunov function whose derivative is negative definite along the solutions of the system using another weak lyapunov function whose derivative along the solutions of the system is negative semidefinite the construction can be carried out if a lie algebraic condition that involves the weak lyapunov function and the system vector field is satisfied our main result extends to general nonlinear systems the strong lyapunov function construction presented in a previous paper that was valid only for homogeneous systems

initial boundary value problems for the equations of motion of compressible viscous and heat conductive fluids the equations of motion of compressible viscous and heat conductive fluids are investigated for initial boundary value problems on the half space and on the exterior domain of any bounded region the global solution in time is proved to exist uniquely and approach the stationary state ast provided the prescribed initial data and the external force are sufficiently small

global strong solutions to the two dimensional full compressible navier stokes equations with vacuum abstract in this paper we consider the global existence of strong solutions to the two dimensional viscous compressible and heat conducting navier stokes equations on the square domain 0 1 2 based on the blow up criterion and uniform estimates we prove that the strong solution exists globally in time if the initial mass is small for the constant coefficients of viscosity and heat conductivity

formation of singularities in solutions to the compressible radiation hydrodynamics equations with vacuum abstract we study the cauchy problem for multi dimensional compressible radiation hydrodynamics equations with vacuum first we present some sufficient conditions on the blow up of smooth solutions in multi dimensional space then we obtain the invariance of the support of density for the smooth solutions with compactly supported initial mass density by the property of the system under the vacuum state based on the above mentioned results we prove that we cannot get a global classical solution no matter how small the initial data are as long as the initial mass density is of compact support finally we will see that some of the results that we obtained are still valid for the isentropic flows with degenerate viscosity coefficients as well as for one dimensional case

global existence in critical spaces for compressible navier stokes equations we investigate global strong solutions for isentropic compressible fluids with initial data close to a stable equilibrium we obtain the existence and uniqueness of a solution in a functional setting invariant by the scaling of the associated equations more precisely the initial velocity has the same critical regularity index as for the incompressible homogeneous navier stokes equations and one more derivative is needed for the density we point out a smoothing effect on the velocity and a l 1 decay on the difference between the density and the constant reference state the proof lies on uniform estimates for a mixed hyperbolic parabolic linear system with a convection term

remarks on the breakdown of smooth solutions for the 3 d euler equations the authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3 d euler equations in other words if a solution of the euler equations is initially smooth and loses its regularity at some later time then the maximum vorticity necessarily grows without bound as the critical time approaches equivalently if the vorticity remains bounded a smooth solution persists

expressions for subdifferential and optimization problems defined by nonsmooth homogeneous functions abstract in this paper we prove a theoretical expression for subdifferentials of lower semicontinuous and homogeneous functions the theoretical expression is a generalization of the euler formula for differentiable homogeneous functions as applications of the generalized euler formula we consider constrained optimization problems defined by nonsmooth positively homogeneous functions in smooth banach spaces some results concerning karush kuhn tucker points and necessary optimality conditions for the optimization problems are obtained

global well posedness for compressible navier stokes equations with highly oscillating initial velocity in this paper we prove global well posedness for compressible navier stokes equations in the critical functional framework with the initial data close to a stable equilibrium this result allows us to construct global solutions for the highly oscillating initial velocity the proof relies on a new estimate for the hyperbolic parabolic system with convection terms 2010 wiley periodicals inc

global asymptotic controllability implies input to state stabilization the main problem addressed in this paper is the design of feedbacks for globally asymptotically controllable gac control affine systems that render the closed loop systems input to state stable iss with respect to actuator errors extensions for fully nonlinear gac systems with actuator errors are also discussed our controllers have the property that they tolerate small observation noise as well

brief strict lyapunov functions for time varying systems uniformly asymptotically stable periodic time varying systems for which is known a lyapunov function with a derivative along the trajectories non positive and negative definite in the state variable on non empty open intervals of the time are considered for these systems strict lyapunov functions are constructed

global behavior of spherically symmetric navier stokes poisson system with degenerate viscosity coefficients in this paper we study a free boundary problem for compressible spherically symmetric navier stokes poisson equations with degenerate viscosity coefficients and without a solid core under certain assumptions that are imposed on the initial data we obtain the global existence and uniqueness of the weak solution and give some uniform bounds with respect to time of the solution moreover we obtain some stabilization rate estimates of the solution the results show that such a system is stable under small perturbations and could be applied to the astrophysics

non existence of classical solutions with finite energy to the cauchy problem of the compressible navier stokes equations the well posedness of classical solutions with finite energy to the compressible navier stokes equations cns subject to arbitrarily large and smooth initial data is a challenging problem in the case when the fluid density is away from vacuum strictly positive this problem was first solved for the cns in either one dimension for general smooth initial data or multi dimension for smooth initial data near some equilibrium state that is small perturbation antontsev et al in boundary value problems in mechanics of nonhomogeneous fluids north holland publishing co amsterdam 1990 kazhikhov in sibirsk mat zh 23 60 64 1982 kazhikhov et al in prikl mat meh 41 282 291 1977 matsumura and nishida in proc jpn acad ser a math sci 55 337 342 1979 j math kyoto univ 20 67 104 1980 commun math phys 89 445 464 1983 in the case that the flow density may contain a vacuum the density can be zero at some space time point it seems to be a rather subtle problem to deal with the well posedness problem for cns the local well posedness of classical solutions containing a vacuum was shown in homogeneous sobolev space without the information of velocity in l2 norm for general regular initial data with some compatibility conditions being satisfied initially cho et al in j math pures appl 9 83 243 275 2004 cho and kim in j differ equ 228 377 411 2006 manuscr math 120 91 129 2006 choe and kim in j differ equ 190 504 523 2003 and the global existence of a classical solution in the same space is established under the additional assumption of small total initial energy but possible large oscillations huang et al in commun pure appl math 65 549 585 2012 however it was shown that any classical solutions to the compressible navier stokes equations in finite energy inhomogeneous sobolev space cannot exist globally in time since it may blow up in finite time provided that the density is compactly supported xin in commun pure appl math 51 229 240 1998 in this paper we investigate the well posedess of classical solutions to the cauchy problem of navier stokes equations and prove that the classical solution with finite energy does not exist in the inhomogeneous sobolev space for any short time under some natural assumptions on initial data near the vacuum this implies in particular that the homogeneous sobolev space is as crucial as studying the well posedness for the cauchy problem of compressible navier stokes equations in the presence of a vacuum at far fields even locally in time