transonic shocks in multidimensional divergent nozzles we establish existence uniqueness and stability of transonic shocks for a steady compressible non isentropic potential flow system in a multidimensional divergent nozzle with an arbitrary smooth cross section for a prescribed exit pressure the proof is based on solving a free boundary problem for a system of partial differential equations consisting of an elliptic equation and a transport equation in the process we obtain unique solvability for a class of transport equations with velocity fields of weak regularity non lipschitz an infinite dimensional weak implicit mapping theorem which does not require continuous frechet differentiability and regularity theory for a class of elliptic partial differential equations with discontinuous oblique boundary conditions

exponential energy decay estimate for the solutions of internally damped wave equation in a bounded domain abstract here we are concerned about the stability of the solution of internally damped wave equation y y y with small damping constant 0 in a bounded domain in r n under mixed undamped boundary conditions a uniform exponential energy decay rate e t me t e 0 where m 1 and 0 0 being a quadratic function of is proved for the solution of this type of boundary value problem earlier authors have considered the undamped wave equation with certain forms of damped boundary conditions proving similar and faster energy decay rates

global uniqueness of transonic shocks in two dimensional steady compressible euler flows we prove that for the two dimensional steady complete compressible euler system with given uniform upcoming supersonic flows the following three fundamental flow patterns special solutions in gas dynamics involving transonic shocks are all unique in the class of piecewise c1 smooth functions under appropriate conditions on the downstream subsonic flows i the normal transonic shocks in a straight duct with finite or infinite length after fixing a point the shock front passing through ii the oblique transonic shocks attached to an infinite wedge iii a flat mach configuration containing one supersonic shock two transonic shocks and a contact discontinuity after fixing a point where the four discontinuities intersect these special solutions are constructed traditionally under the assumption that they are piecewise constant and they have played important roles in the studies of mathematical gas dynamics our results show that the assumption of a piecewise constant can be replaced by some weaker assumptions on the downstream subsonic flows which are sufficient to uniquely determine these special solutions mathematically these are uniqueness results on solutions of free boundary problems of a quasi linear system of elliptic hyperbolic composite mixed type in bounded or unbounded planar domains without any assumptions on smallness the proof relies on an elliptic system of pressure p and the tangent of the flow angle w v u obtained by decomposition of the euler system in lagrangian coordinates and a newly developed method for the l estimate that is independent of the free boundaries by combining the maximum principles of elliptic equations and careful analysis of the shock polar applied on the maybe curved shock fronts

backstepping boundary control of a wave pde with spatially distributed time invariant unknown disturbances an observer based boundary control scheme is suggested for an infinite dimensional system modeled as a wave partial differential equation pde with in domain and boundary disturbances we consider a collocated case where the sensors and the actuators are located at the same boundary the aim of the controller is to regulate the rate of change of the opposite boundary around a given reference the unknown in domain disturbance is spatially distributed but it is constant in time same as the unknown boundary disturbance the problem is reformulated as a stabilization of an linear time invariant lti system with simultaneous delays in input and output channels the exponential stability of the equilibrium of the closed loop system is proven the performance of the controller is shown using numerical simulations

a diffusion wave equation with two fractional derivatives of different order we analyse a diffusion wave equation with two fractional derivatives of different order on bounded and unbounded spatial domains thus our model represents a generalized telegraph equation solutions to signalling and cauchy problems in terms of a series and integral representation are given classical wave and heat conduction equations are obtained as limiting cases

ecoulements de fluides parfaits en deux variables independantes de type espace reflexion d un choc plan par un diedre compressif we consider the euler equations of a perfect fluid having only two independent space like variables which account for the stationary 2 dimensional or axisymmetrical 3 dimensional cases as well as the 2 dimensional riemann problem we show that the pressure and the angle between an axis and the velocity field satisfy a first order system which turns out to be elliptic in the subsonic zone in particular the pressure satisfies a maximum principle which has not been stated before to the best of my knowledge using this and the bernouilli law we give various a priori estimates of the pressure the density the enthalpy and the velocity in the problem of the reflection of a shock wave by a wedge we also bound the size of the subsonic region and the force that the fluid applies to the boundary

transonic shock in finitely long nozzle with porous medium boundary condition in this paper we investigate the 2 d euler equations with complex boundary conditions for the entrance and exit of finitely long nozzle we use the supersonic incoming flow condition and the end pressure condition respectively in addition for the nozzle wall the lower one is solid and the upper one is permitted to have leakage we establish the well posedness if appropriate seepage discharge is given we also obtain the shock position by solving a free boundary problem without assuming the shock must go through a fixed point

the transonic shock in a nozzle 2 d and 3 d complete euler systems abstract in this paper we study a transonic shock problem for the euler flows through a class of 2 d or 3 d nozzles the nozzle is assumed to be symmetric in the diverging or converging part if the supersonic incoming flow is symmetric near the divergent or convergent part of the nozzle then as indicated in section 147 of r courant k o friedrichs supersonic flow and shock waves interscience publ new york 1948 there exist two constant pressures p 1 and p 2 with p 1 p 2 such that for given constant exit pressure p e p 1 p 2 a symmetric transonic shock exists uniquely in the nozzle and the position and the strength of the shock are completely determined by p e moreover it is shown in this paper that such a transonic shock solution is unique under the restriction that the shock goes through the fixed point at the wall in the multidimensional setting furthermore we establish the global existence stability and the long time asymptotic behavior of an unsteady symmetric transonic shock under the exit pressure p e when the initial unsteady shock lies in the symmetric diverging part of the 2 d or 3 d nozzle on the other hand it is shown that an unsteady symmetric transonic shock is structurally unstable in a global in time sense if it lies in the symmetric converging part of the nozzle

exact controllability of the euler bernoulli equation with controls in the dirichlet and neumann boundary conditions a nonconservative case this paper considers the euler bernoulli problem 1 1a d with boundary controls g 1 g 2 in the dirichiet and neumann boundary conditions respectively several exact controllability results are shown including the following problem 1 1a d is exactly controllable in an arbitrarily short time t 0 in the space of maximal regularity h 1 omega times v v as in 1 4 i with boundary controls g 1 in l 2 sigma g 2 equiv 0 under some geometrical conditions on omega ii with boundary controls g 1 in l 2 sigma and g 2 in l 2 0 t h 1 gamma without geometrical conditions on omega a direct approach is given based on an operator model for problem 1 la d and on multiplier techniques an additional difficulty of the particular boundary conditions is due to the fact that in the natural norms for the solution arising in the application of multiplier techniques the corresponding homogeneous problem is not conservative this difficulty is overco

global existence and uniform stabilization of a generalized dissipative klein gordon equation type with boundary damping in this paper we prove the existence uniqueness and uniform stability of strong and weak solutions of the nonlinear generalized klein gordon equation 1 1 1 see sec i in bounded domains with nonlinear damped boundary conditions given by 1 1 3 see sec i with some restrictions on function f u h u g ut and b x we prove the existence and uniqueness by means of nonlinear semigroup method and obtain the uniform stabilization by using the multiplier technique

stabilization of wave and plate like equations with nonlinear dissipation on the boundary abstract in this paper we consider second order in time equations of petrovsky type wave and plate like with nonlinear dissipative boundary conditions under certain geometric conditions imposed on the domain the results on asymptotic behavior of the solutions are established

persistence of shocks in ducts abstract we prove local in time existence and stability of shock waves in non isentropic compressible euler flows in a two dimensional straight duct provided the shock satisfies the uniform stability condition and the upcoming supersonic flow and the pressure at the exit of the duct as well as the initial data satisfy certain orders of compatibility and symmetry conditions

some aspects of fractional diffusion equations of single and distributed order abstract the time fractional diffusion equation is obtained from the standard diffusion equation by replacing the first order time derivative with a fractional derivative of order 0 1 the fundamental solution for the cauchy problem is interpreted as a probability density of a self similar non markovian stochastic process related to a phenomenon of sub diffusion the variance grows in time sub linearly a further generalization is obtained by considering a continuous or discrete distribution of fractional time derivatives of order less than one then the fundamental solution is still a probability density of a non markovian process that however is no longer self similar but exhibits a corresponding distribution of time scales

boundary stabilization of nonlinear vibrations of a flexible structure in a bounded domain in rn abstract we study a problem of boundary stabilization of the vibrations of elastic structure governed by the nonlinear integro differential equation u a 2 b u 2 d x u f in a bounded domain in r n with a smooth boundary under mixed boundary conditions to stabilize this system we apply a velocity feedback control only on a part of the boundary we prove that the solution of such system is stable subject to some restriction on the uncertain disturbing force f we also estimate the total energy of the system over any time interval 0 t with a tolerance level of the disturbances finally we establish the uniform decay of solution by a direct method with an explicit form of exponential energy decay estimate when this disturbing force f is insignificant

existence of asymptotically almost periodic solutions for damped wave equations abstract in this paper a class of nonlinear damped wave equations of the form u t u t a u t a u t f t u t t 0 satisfying with prescribed initial conditions are studied some sufficient conditions are established for the existence and uniqueness of an asymptotically almost periodic solution these results have significance in the study of vibrations of flexible structures possessing internal material damping finally an example is presented to illustrate the feasibility and effectiveness of the results

transonic shock in a nozzle i 2d case in this paper we establish the existence and uniqueness of a transonic shock for the steady flow through a general two dimensional nozzle with variable sections the flow is governed by the inviscid potential equation and is supersonic upstream has no flow boundary conditions on the nozzle walls and an appropriate boundary condition at the exit of the exhaust section the transonic shock is a free boundary dividing two regions of c 1 1 0 flow in the nozzle the potential equation is hyperbolic upstream where the flow is supersonic and elliptic in the downstream subsonic region in particular our results show that there exists a solution to the corresponding free boundary problem such that the equation is always subsonic in the downstream region of the nozzle when the pressure in the exit of the exhaustion section is appropriately larger than that in the entry this problem is motivated by the conjecture of courant and friedrichs on the transonic phenomena in a nozzle 10 furthermore the stability of the transonic shock is also proven when the upstream supersonic flow is a small steady perturbation for the uniform supersonic flow and the corresponding pressure at the exit has a small perturbation the main ingredients of our analysis are a generalized hodograph transformation and multiplier methods for elliptic equation with mixed boundary conditions and comer singularities

transonic nozzle flows and free boundary problems for the full euler equations abstract we establish the existence and uniqueness of transonic flows with a transonic shock through a two dimensional nozzle of slowly varying cross sections the transonic flow is governed by the steady full euler equations given an incoming smooth flow that is close to a constant supersonic state i e smooth cauchy data at the entrance and the subsonic condition with nearly horizontal velocity at the exit of the nozzle we prove that there exists a transonic flow whose downstream smooth subsonic region is separated by a smooth transonic shock from the upstream supersonic flow this problem is approached by a one phase free boundary problem in which the transonic shock is formulated as a free boundary the full euler equations are decomposed into an elliptic equation and a system of transport equations for the free boundary problem an iteration scheme is developed and its fixed point is shown to exist which is a solution of the free boundary problem by combining some delicate estimates for the elliptic equation and the system of transport equations with the schauder fixed point argument the uniqueness of transonic nozzle flows is also established by employing the coordinate transformation of euler lagrange type and detailed estimates of the solutions

exact controllability of the euler bernoulli equation with boundary controls for displacement and moment abstract we consider the euler bernoulli problem 1 1 in the solution w t with boundary controls g1 and g2 acting in the dirichlet traces for w and w respectively we show two main exact controllability results both in the spaces of maximal regularity of the solution at t t see i lasiecka and r triggiani regularity theory for a class of euler bernoulli equations a cosine operator approach boll unione mat ital 7 3 b 1989 and both without geometrical conditions on the open bounded domain except for smoothness of its boundary one with g1 l2 0 t l2 and g2 h1 0 t l2 for any t 0 arbitrarily short and one with g1 h01 0 t l2 and g2 l2 0 t l2 for all t 0 sufficiently large an interpolation result between these two cases is also presented a direct approach is given based on two main steps first by means of an operator model i lasiecka and r triggiani regularity theory for a class of euler bernoulli equations a cosine operator approach boll unione mat ital 7 3 b 1989 for problem 1 1 and a functional analytic approach the question of exact controllability is shown to be equivalent to an a priori inequality for the corresponding homogeneous problem next this a priori inequality is proved to hold true by means of multiplier techniques these are inspired by recent progress on the maximal regularity exact controllability and uniform stabilization questions for second order hyperbolic equations

on existence and asymptotic stability of solutions of the degenerate wave equation with nonlinear boundary conditions abstract we study the global existence of solutions of the nonlinear degenerate wave equation 0 x y y 0 in 0 y 0 on 1 0 y y f y g y 0 on 0 0 y x 0 y 0 y x 0 y 1 x in where y denotes the derivative of y with respect to parameter t f s c0 s s and g is a nondecreasing c1 function such that k1 s 2 g s s k2 s 2 for some k1 k2 0 with 0 0 if n 1 2 the existence of solutions is proved by means of the faedo galerkin method furthermore when 0 the uniform decay is obtained by making use of the perturbed energy method

global smooth solutions for the quasilinear wave equation with boundary dissipation abstract we consider the existence of global solutions of the quasilinear wave equation with a boundary dissipation structure of an input output in high dimensions when initial data and boundary inputs are near a given equilibrium of the system our main tool is the geometrical analysis the main interest is to study the effect of the boundary dissipation structure on solutions of the quasilinear system we show that the existence of global solutions depends not only on this dissipation structure but also on a riemannian metric given by the coefficients and the equilibrium of the system some geometrical conditions on this riemannian metric are presented to guarantee the existence of global solutions in particular we prove that the norm of the state of the system decays exponentially if the input stops after a finite time which implies the exponential stabilization of the system by boundary feedback

bounded input bounded output stability of a damped nonlinear string in this paper we prove that the displacement of a damped nonlinear string under bounded distributed forces is bounded the proof is achieved by showing that an energy like lyapunov function corresponding to the string is bounded

space time fractional zener wave equation the space time fractional zener wave equation describing viscoelastic materials obeying the time fractional zener model and the space fractional strain measure is derived and analysed this model includes waves with finite speed as well as non propagating disturbances the existence and the uniqueness of the solution to the generalized cauchy problem are proved special cases are investigated and numerical examples are presented

an approximate algorithm for a kirchhoff wave equation the initial boundary value problem for a hyperbolic type quasilinear equation w tt varphi int 0 pi w x 2dx w xx is considered its solution is derived by means of a numerical algorithm consisting of the projection method and the difference method for approximation with respect to spatial and time variables while the resulting discrete system is solved by an iteration process the accuracy of the proposed algorithm is estimated

existence and exponential decay for a kirchhoff carrier model with viscosity abstract in this work we study the existence of global solutions and exponential decay for the following nonlinear problem formula where m is a c 1 function m 0 0 0

a note on the boundary stabilization of the wave equation we study the energy decay rates of the wave equation in a domain where boundary damping is present we generalize the geometrical conditions obtained earlier in j math purer appl 58 1979 pp 249 273 by using some more general multipliers of strauss comm pure appl math 28 1975 pp 265 278 the interaction between distributed damping and boundary damping is discussed a regulator problem is also formally discussed by the synthesis method

exact controllability stabilization and perturbations for distributed systems exact controllability is studied for distributed systems of hyperbolic type or for petrowsky systems like plate equations the control is a boundary control or a local distributed control exact controllability consists in trying to drive the system to rest in a given finite time the solution of the problems depends on the function spaces where the initial data are taken and also depends on the function space where the control can be chosen a systematic method named hum for hilbert uniqueness method is introduced as the terminology indicates it is based on uniqueness results classical or new and on hilbert spaces constructed in infinitely many ways by using uniqueness a number of applications are indicated having a general method for exact controllability implies having a general method for stabilization this leads to new and complicated nonlinear riccati s type pdes to be compared with direct methods when available in the last part of the paper we consider how all this behaves fo

on transonic shocks in a conic divergent nozzle with axi symmetric exit pressures abstract in this paper we establish the existence and stability of a 3 d transonic shock solution to the full steady compressible euler system in a class of de laval nozzles with a conic divergent part when a given variable axi symmetric exit pressure lies in a suitable scope thus for this class of nozzles we have solved such a transonic shock problem in the axi symmetric case described by courant and friedrichs 1948 in section 147 of 8 given the appropriately large exit pressure p e x if the upstream flow is still supersonic behind the throat of the nozzle then at a certain place in the diverging part of the nozzle a shock front intervenes and the gas is compressed and slowed down to subsonic speed so that the position and the strength of the shock front are automatically adjusted such that the end pressure at the exit becomes p e x

time distributed order diffusion wave equation ii applications of laplace and fourier transformations a cauchy problem for a time distributed order multi dimensional diffusion wave equation containing a forcing term is reinterpreted in the space of tempered distributions and a distributional diffusion wave equation is obtained the distributional equation is solved in the general case of weight function or distribution solutions are given in terms of solution kernels green s functions which are studied separately for two cases the first case is when the order of the fractional derivative is in the interval 0 1 while in the second case the order of the fractional derivative is in the interval 0 2 solutions of fractional diffusionwave and fractional telegraph equations are obtained as special cases numerical experiments are also performed an analogue of the maximum principle is also presented

distributed order calculus and equations of ultraslow diffusion abstract we consider equations of the form d u t x u t x f t x t 0 x r n where d is a distributed order derivative that is d t 0 1 d t d d is the caputo dzhrbashyan fractional derivative of order is a positive weight function the above equation is used in physical literature for modeling diffusion with a logarithmic growth of the mean square displacement in this work we develop a mathematical theory of such equations study the derivatives and integrals of distributed order

multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type we are concerned with the existence and stability of multidimensional transonic shocks for the euler equations for steady potential compressible fluids the euler equations consisting of the conservation law of mass and the bernoulli law for the velocity can be written as the following second order nonlinear equation of mixed elliptic hyperbolic type for the velocity potential r r 1 1 div d d 0 where the density function q is

exponential energy decay estimates for the solutions of n dimensional kirchhoff type wave equation here we study uniform exponential stabilization of the vibrations of kirchhoff type wave equation in a bounded domain in rn with a smooth boundary under mixed boundary conditions to stabilize the system we incorporate separately the passive viscous damping and the internal material damping of kelvin voigt type in the model explicit forms of exponential energy decay rates are obtained by a direct method for the solution of such boundary value problems

global existence for the wave equation with nonlinear boundary damping and source terms abstract the paper deals with local and global existence for the solutions of the wave equation in bounded domains with nonlinear boundary damping and source terms the typical problem studied is u tt u 0 in 0 u 0 on 0 0 u u t m 2 u t u p 2 u on 0 1 u 0 x u 0 x u t 0 x u 1 x on where r n n 1 is a regular and bounded domain 0 1 m 1 2 p r where r 2 n 1 n 2 when n 3 r when n 1 2 and the initial data are in the energy space we prove local existence of the solutions in the energy space when m r r 1 p or n 1 2 and global existence when p m or the initial data are inside the potential well associated to the stationary problem

time distributed order diffusion wave equation i volterra type equation a single order time fractional diffusion wave equation is generalized by introducing a time distributed order fractional derivative and forcing term while a laplacian is replaced by a general linear multi dimensional spatial differential operator the obtained equation is in the case of the laplacian called a time distributed order diffusion wave equation we analyse a cauchy problem for such an equation by means of the theory of an abstract volterra equation the weight distribution occurring in the distributed order fractional derivative is specified as the sum of the dirac distributions and the existence and uniqueness of solutions to the cauchy problem and the corresponding volterra type equation were proven for a general linear spatial differential operator as well as in the special case when the operator is laplacian

boundary control quiet boundaries super stability and super instability this paper deals with the boundary feedback control of a bar undergoing axial vibrations the feedback force is applied to one end of the bar and it is proportional to its velocity the problem is of fundamental interest in control theory structural dynamics and the development of quiet boundaries in fields like earthquake engineering and computational mechanics the system is not self adjoint and exhibits a variety of interesting behaviors which are explained through a combination of several inter twined strands of thought using mathematics physical interpretations and numerical simulations besides providing a rigorous mathematical solution to the problem the paper explains the physical origin of super stable behavior introduces the new concept of super unstable behavior and points out that these regimes of behavior are intricately connected with the continuum model that is used it is shown that any finite dimensional approximation of the system no matter how finely discretized can not qualitatively depict superstable behavior

steady transonic shocks and free boundary problems in infinite cylinders for the euler equations we establish the existence and stability of multidimensional transonic shocks hyperbolic elliptic shocks for the euler equations for steady compressible potential fluids in infinite cylinders the euler equations consisting of the conservation law of mass and the bernoulli law for velocity can be written as a second order nonlinear equation of mixed elliptic hyperbolic type for the velocity potential the transonic shock problem in an infinite cylinder can be formulated into the following free boundary problem the free boundary is the location of the multidimensional transonic shock which divides two regions of c1 flow in the infinite cylinder and the equation is hyperbolic in the upstream region where the c1 perturbed flow is supersonic we develop a nonlinear approach to deal with such a free boundary problem in order to solve the transonic shock problem in unbounded domains our results indicate that there exists a solution of the free boundary problem such that the equation is always elliptic in the unbounded downstream region the uniform velocity state at infinity in the downstream direction is uniquely determined by the given hyperbolic phase and the free boundary is c1 provided that the hyperbolic phase is close in c1 to a uniform flow we further prove that if the steady perturbation of the hyperbolic phase is c2 the free boundary is c2 and stable under the steady perturbation 2003 wiley periodicals inc

transonic shock wave in an infinite nozzle asymptotically converging to a cylinder abstract we construct a single transonic shock wave pattern in an infinite nozzle asymptotically converging to a cylinder which is close to a uniform transonic shock wave in other words suppose there is a uniform transonic shock wave in an infinite cylinder nozzle which can be constructed easily if we perturbed the supersonic incoming flow and the infinite nozzle a little bit we can obtain a transonic wave near the uniform one as a consequence we can show that the uniform transonic wave is stable with respect to the perturbation of the incoming flow and nozzle wall based on the theory of g q chen m feldman existence and stability of multi dimensional transonic flows through an infinite nozzle of arbitrary cross sections arch ration mech anal 184 2007 185 242 the crucial parts of this paper are to derive the uniform schauder estimates of the linear elliptic equation for the infinite nozzle asymptotically converging to a cylinder

multi dimensional solutions of space time fractional diffusion equations green9s functions and propagators for the multi dimensional anisotropic space time fractional diffusion equation are obtained in integral form uniqueness of the solutions is studied in the framework of abstract volterra equations unimodality of propagator functions is demonstrated in the centrally symmetric case the transition from diffusion to wave propagation is discussed numerical examples are constructed

transonic shocks for the full compressible euler system in a general two dimensional de laval nozzle in this paper we study the transonic shock problem for the full compressible euler system in a general two dimensional de laval nozzle as proposed in courant and friedrichs supersonic flow and shock waves interscience new york 1948 given the appropriately large exit pressure pe x if the upstream flow is still supersonic behind the throat of the nozzle then at a certain place in the diverging part of the nozzle a shock front intervenes and the gas is compressed and slowed down to subsonic speed so that the position and the strength of the shock front are automatically adjusted such that the end pressure at the exit becomes pe x we solve this problem completely for a general class of de laval nozzles whose divergent parts are small and arbitrary perturbations of divergent angular domains for the full steady compressible euler system the problem can be reduced to solve a nonlinear free boundary value problem for a mixed hyperbolic elliptic system one of the key ingredients in the analysis is to solve a nonlinear free boundary value problem in a weighted holder space with low regularities for a second order quasilinear elliptic equation with a free parameter the position of the shock curve at one wall of the nozzle and non local terms involving the trace on the shock of the first order derivatives of the unknown function

energy decay rates for the semilinear wave equation with nonlinear localized damping and source terms abstract in this paper we develop an intrinsic approach to derivation of energy decay rates for the semilinear wave equation with localized interior nonlinear monotone damping g u t and a source term f u the proposed approach allows to consider in an unified way much more general classes of hyperbolic problems than addressed before in the literature these generalizations refer to both geometric and topological aspects of the problem the method leads to optimal decay rates for solutions of semilinear hyperbolic equations driven by a source of critical exponent and subjected to a nonlinear damping localized in a small region adjacent to a portion of the boundary the distinct features of the model include i neumann boundary conditions are assumed and ii no growth conditions are imposed on the damping g s it is well known that neumann boundary does not satisfy lopatinski condition and therefore the analysis of propagation of energy in the absence of the damping on the neumann part of the boundary requires special geometric considerations in addition the sole conditions assumed on g s are monotonicity continuity and g 0 0 in particular no differentiability and no growth conditions are imposed on the damping both at the origin and at the infinity the asymptotic decay rates for the energy function are obtained from an intrinsic algorithm driven by solutions of simple ode several examples illustrate the theory by exhibiting various decay rates exponential algebraic rational logarithmic etc for the energy functional an important corollary of our energy decay theorem is a stability result which shows that under certain conditions when dissipation is sublinear at infinity the solution of the system remains uniformly bounded for all time in the norms above the finite energy level even in the presence of a nonlinear source term

bilinear optimal control of a kirchhoff plate abstract we consider the problem of optimal control of a kirchhoff plate a bilinear control is used as a force to make the plate close to a desired profile taking into the account a quadratic cost of control we prove the existence of an optimal control and characterize it uniquely through the solution of an optimality system

a causal and fractional all frequency wave equation for lossy media this work presents a lossy partial differential acoustic wave equation including fractional derivative terms it is derived from first principles of physics mass and momentum conservation and an equation of state given by the fractional zener stress strain constitutive relation for a derivative order in the fractional zener relation the resulting absorption k obeys frequency power laws as k 1 in a low frequency regime k 1 2 in an intermediate frequency regime and k 1 in a high frequency regime the value 1 corresponds to the case of a single relaxation process the wave equation is causal for all frequencies in addition the sound speed does not diverge as the frequency approaches infinity this is an improvement over a previously published wave equation building on the fractional kelvin voigt constitutive relation

note on boundary stabilization of wave equations an energy decay rate is obtained for solutions of wave type equations in a bounded region in mathbb r n whose boundary consists partly of a nontrapping reflecting surface and partly of an energy absorbing surface unlike most previous results on this subject the results presented here are potentially valid for regions having connected boundaries

stability of transonic shock fronts in two dimensional euler systems we study the stability of stationary transonic shock fronts under two dimensional perturbation in gas dynamics the motion of the gas is described by the full euler system the system is hyperbolic ahead of the shock front and is a hyperbolic elliptic composed system behind the shock front the stability of the shock front and the downstream flow under two dimensional perturbation of the upstream flow can be reduced to a free boundary value problem of the hyperbolic elliptic composed system we develop a method to deal with boundary value problems for such systems the crucial point is to decompose the system to a canonical form in which the hyperbolic part and the elliptic part are only weakly coupled in their coefficients by several sophisticated iterative processes we establish the existence and uniqueness of the solution to the described free boundary value problem our result indicates the stability of the transonic shock front and the flow field behind the shock

waves in fractional zener type viscoelastic media classical wave equation is generalized for the case of viscoelastic materials obeying fractional zener model instead of hooke s law cauchy problem for such an equation is studied existence and uniqueness of the fundamental solution is proven and solution is calculated

stabilization of quasi linear vibrations of an inhomogeneous beam in this note we study the stabilization of a vibrating inhomogeneous beam under distributed disturbing force we prove that the amplitude of such vibration is bounded subject to some restriction on the uncertain disturbing force we also estimate the total energy of the system over time interval 0 t for any time t 0 together with a tolerance level of the disturbance finally we establish the uniform decay of solution with an explicit form of exponential energy decay estimate when the uncertain disturbances are insignificant

stability of spherically symmetric subsonic flows and transonic shocks under multidimensional perturbations abstract we develop a method that works in general product riemannian manifold to decompose the three dimensional steady full compressible euler system which is of elliptic hyperbolic composite mixed type for subsonic flows the method is applied to show stability of spherically symmetric subsonic flows and transonic shocks in space r 3 under multidimensional perturbations of boundary conditions

transonic shocks in 3 d compressible flow passing a duct with a general section for euler systems this paper is devoted to the study of a transonic shock in three dimensional steady compressible flow passing a duct with a general section the flow is described by the steady full euler system which is purely hyperbolic in the supersonic region and is of elliptic hyperbolic type in the subsonic region the upstream flow at the entrance of the duct is a uniform supersonic one adding a three dimensional perturbation while the pressure of the downstream flow at the exit of the duct is assigned apart from a constant difference the problem to determine the transonic shock and the flow behind the shock is reduced to a free boundary value problem of an elliptic hyperbolic system the new ingredients of our paper contain the decomposition of the elliptic hyperbolic system the determination of the shock front by a pair of partial differential equations coupled with the three dimensional euler system and the regularity analysis of solutions to the boundary value problems introduced in our discussion

dynamical boundary control of two dimensional wave equations vibrating membrane on general domain dynamical boundary control of a vibrating membrane with edge mass and of general shape is mathematically formulated as an abstract evolutionary system the approximate controllability and the strong stabilization by linear boundary damping feedback are established by an approach based on lasalle s invariance principle the mathematical model and the reduced system can be used to investigate some optimization problems

discussions of a galerkin method for a nonlinear integro differential wave system by i christie and j m sanz serna comput meth appl mech engrg 44 1984 229 237

existence and stability of multidimensional transonic flows through an infinite nozzle of arbitrary cross sections we establish the existence and stability of multidimensional steady transonic flows with transonic shocks through an infinite nozzle of arbitrary cross sections including a slowly varying de laval nozzle the transonic flow is governed by the inviscid potential flow equation with supersonic upstream flow at the entrance uniform subsonic downstream flow at the exit at infinity and the slip boundary condition on the nozzle boundary our results indicate that if the supersonic upstream flow at the entrance is sufficiently close to a uniform flow there exists a solution that consists of a c1 subsonic flow in the unbounded downstream region converging to a uniform velocity state at infinity and a c1 multidimensional transonic shock separating the subsonic flow from the supersonic upstream flow the uniform velocity state at the exit at infinity in the downstream direction is uniquely determined by the supersonic upstream flow and the shock is orthogonal to the nozzle boundary at every point of their intersection in order to construct such a transonic flow we reformulate the multidimensional transonic nozzle problem into a free boundary problem for the subsonic phase in which the equation is elliptic and the free boundary is a transonic shock the free boundary conditions are determined by the rankine hugoniot conditions along the shock we further develop a nonlinear iteration approach and employ its advantages to deal with such a free boundary problem in the unbounded domain we also prove that the transonic flow with a transonic shock is unique and stable with respect to the nozzle boundary and the smooth supersonic upstream flow at the entrance

compressible flow and transonic shock in a diverging nozzle the paper is devoted to the study of compressible flows and transonic shocks in diverging nozzles in the framework of the full compressible euler system consider a nozzle having a shape as a diverging truncated sector with generic opening angle if the upstream flow at the entrance is supersonic and is near to an axial symmetric flow and if all parameters of the upstream flow and the receiver pressure at the exit are suitably assigned then a transonic shock appears in the nozzle to determine the transonic shock and the flow in the nozzle leads to a free boundary value problem for a nonlinear partial differential equation we prove that the receiver pressure can uniquely determine the location of the transonic shock as well as the flow behind the shock such a conclusion was conjectured by courant and friedrichs and is confirmed theoretically in this paper for the divergent nozzles

uniform decay rates for the wave equation with nonlinear damping locally distributed in unbounded domains with finite measure this paper is concerned with the study of the uniform decay rates of the energy associated with the wave equation with nonlinear damping locally distributed u tt delta u a x g u t 0 hbox in omega times 0 infty subject to dirichlet boundary conditions where omega subset mathbb r n n geq 2 is an unbounded open set with finite measure and unbounded smooth boundary partial omega gamma the function a x responsible for the localized effect of the dissipative mechanism is assumed to be nonnegative and essentially bounded and in addition a x geq a 0 0 hbox a e in omega where omega omega cup x in omega x r r 0 and omega is a neighborhood in omega of the closure of partial omega cap b r where b r x in omega x r

nonlinear boundary feedback stabilization for a semilinear kirchhoff plate with dissipation acting only via moments limiting behavior abstract a semilinear kirchhoff plate with a nonlinear dissipation acting via moments only is considered it is shown that the plate is uniformly stabilizable with uniform energy decay rates with respect to the parameter 0 representing rotational forces assumed small moreover it is shown that the solutions of the kirchhoff plate converge to solutions of the semilinear euler bernoulli plate when 0 which is also uniformly stable in finite energy norm

transonic shocks in compressible flow passing a duct for three dimensional euler systems in this paper we study the transonic shock in steady compressible flow passing a duct the flow is a given supersonic one at the entrance of the duct and becomes subsonic across a shock front which passes through a given point on the wall of the duct the flow is governed by the three dimensional steady full euler system which is purely hyperbolic ahead of the shock and is of elliptic hyperbolic composed type behind the shock the upstream flow is a uniform supersonic one with the addition of a three dimensional perturbation while the pressure of the downstream flow at the exit of the duct is assigned apart from a constant difference the problem of determining the transonic shock and the flow behind the shock is reduced to a free boundary value problem in order to solve the free boundary problem of the elliptic hyperbolic system one crucial point is to decompose the whole system to a canonical form in which the elliptic part and the hyperbolic part are separated at the level of the principal part due to the complexity of the characteristic varieties for the three dimensional euler system the calculus of symbols is employed to complete the decomposition the new ingredient of our analysis also contains the process of determining the shock front governed by a pair of partial differential equations which are coupled with the three dimensional euler system