boundedness of inhomogeneous journe s type operators on multi parameter local hardy spaces abstract in this paper we establish a boundedness criterion of a class of inhomogeneous journe s type of multi parameter singular integral operators on multi parameter local hardy spaces h p r n 1 r n k recently introduced in ding et al 2019 the consideration of such inhomogeneous journe s type of multi parameter singular integral operators is motivated by the study of the multi parameter pseudo differential operators on multi parameter local hardy spaces the lack of the vanishing moments of both the corresponding functions in the multi parameter local calderon reproducing formula and the multi parameter local littlewood paley stein functions as well as the lack of vanishing moments of atoms in the multi parameter local hardy spaces h p r n 1 r n k with large size all create substantial difficulties in our proofs

stability theorem and truncation error analysis for the glimm scheme and for a front tracking method for flows with strong discontinuities typical nonlinear wave interaction problems involve strong waves moving through a background of weak disturbance previous existence theorems and error analysis for computations are usually restricted to more idealized situations such as small data or single equations we consider here the problem of a single strong discontinuity interacting with a weak background for general hyperbolic systems of conservation laws we obtain the stability consistency theorems and upper bounds of the truncation errors for the glimm scheme and for a front tracking method the major error in the glimm scheme is the error generated by the strong discontinuity this error is reduced when a front tracking method is applied to follow the location of the strong discontinuity this demonstrates an advantage of front tracking methods in one space dimension

the convergence rate of approximate solutions for nonlinear scalar conservation laws let v varepsilon x t varepsilon 0 be a family of approximate solutions for the nonlinear scalar conservation law u t f u x 0 with c 0 1 initial data assume that v varepsilon x t are textit lip stable in the sense that they satisfy oleinik s e entropy condition it is shown that if these approximate solutions are textit lip consistent i e if v varepsilon cdot 0 u cdot 0 textit lip x v t varepsilon f v varepsilon x textit lip x t mathcal o varepsilon then they converge to the entropy solution and the convergence rate estimate v varepsilon cdot t u cdot t textit lip x mathcal o varepsilon holds consequently the familiar l p type and new pointwise error estimates are derived these convergence rate results are demonstrated in the context of entropy satisfying finite difference and glimm s schemes

radon measure solutions for steady compressible euler equations of hypersonic limit conical flows and newton s sine squared law abstract we formulate a mathematical problem on hypersonic limit of three dimensional steady uniform non isentropic compressible euler flows of polytropic gases passing a straight cone with arbitrary cross section and attacking angle which is to study radon measure solutions of a nonlinear hyperbolic system of conservation laws on the unit 2 sphere the construction of a measure solution with density containing dirac measures supported on the surface of the cone is reduced to find a regular periodic solution of highly nonlinear and singular ordinary differential equations ode for a circular cone with zero attacking angle we then proved the newton s sine squared law by obtaining such a measure solution this provides a mathematical foundation for the newton s theory of pressure distribution on three dimensional bodies in hypersonic flows

on the atomic decomposition for hardy spaces on lipschitz domains of rn let be a special lipschitz domain on rn and l be a second order elliptic self adjoint operator in divergence form l div a on lipschitz domain subject to neumann boundary condition in this paper we give a simple proof of the atomic decomposition for hardy spaces hnp of for a range of p by means of nontangential maximal function associated with the poisson semigroup of l

boundedness of multi parameter fourier multiplier operators on triebel lizorkin and besov lipschitz spaces abstract the main purpose of this paper is three fold first we prove that under the limited smoothness conditions multi parameter fourier multiplier operators are bounded on multi parameter triebel lizorkin and besov lipschitz spaces by the littlewood paley decomposition and the strong maximal operator second we offer a different and more direct method to deal with the boundedness instead of transforming fourier multiplier operators into multi parameter calderon zygmund operators third we also prove the boundedness of multi parameter fourier multiplier operators on weighted multi parameter triebel lizorkin and besov lipschitz spaces when the fourier multiplier is only assumed with limited smoothness

a remark on abstract multiplier conditions for robustness problems abstract this paper presents a generalization of recent results of acikmese and corless b acikmese m corless stability analysis with quadratic lyapunov functions some necessary and sufficient multiplier conditions systems control letters 57 2008 78 94 concerning multiplier conditions of quadratic stability of uncertain nonlinear systems abstract full block s procedure results are formulated extending the results of scherer c w scherer lpv control and full block multipliers automatica 37 2001 361 375 for such cases when the sets representing the uncertainties do not necessarily have subspace structure the main contribution of the present work is the investigation of the conditions under which the results of scherer and acikmese and corless can be formulated in a unified framework

on the global existence and stability of a multi dimensional supersonic conic shock wave we establish the global existence and stability of a three dimensional supersonic conic shock wave for a compactly perturbed steady supersonic flow past an infinitely long circular cone with a sharp angle the flow is described by a 3 d steady potential equation which is multi dimensional quasilinear and hyperbolic with respect to the supersonic direction making use of the geometric properties of the pointed shock surface together with the rankine hugoniot conditions on the conic shock surface and the boundary condition on the surface of the cone we obtain a global uniform weighted energy estimate for the nonlinear problem by finding an appropriate multiplier and establishing a new hardy type inequality on the shock surface based on this we prove that a multi dimensional conic shock attached to the vertex of the cone exists globally when the mach number of the incoming supersonic flow is sufficiently large moreover the asymptotic behavior of the 3 d supersonic conic shock solution which is shown to approach the corresponding background shock solution in the downstream domain for the uniform supersonic constant flow past the sharp cone is also explicitly given

supersonic flow onto a solid wedge we consider the problem of two dimensional supersonicflow onto a solid wedge or equivalently in a concave corner formed by two solid walls for mild corners there are two possible steady state solutions one with a strong and one with a weak shock emanating from the corner the weak shock is observed in supersonic flights a longstanding natural conjecture is that the strong shock is unstable in some sense we resolve this issue by showing that a sharp wedge will eventually produce weak shocks at the tip when accelerated to a supersonic speed more precisely we prove that for upstream state as initial data in the entire domain the timedependent solution is self similar with a weak shock at the tip of the wedge we construct analytic solutions for self similar potential flow both isothermal and isentropic with arbitrary 1 in the process of constructing the self similar solution we develop a large number of theoretical tools for these elliptic regions these tools allow us to establish large data results rather than a small perturbation we show that the wave pattern persists as long as the weak shock is supersonic supersonic when this is no longer true numerics show a physical change of behavior in addition we obtain rather detailed information about the elliptic region including analyticity as well as bounds for velocity components and shock tangents c 2007 wiley periodicals inc

stability analysis with quadratic lyapunov functions some necessary and sufficient multiplier conditions in this paper we present several conditions which are both necessary and sufficient for quadratic stability of an uncertain nonlinear system these conditions involve multiplier matrices which characterize the uncertain nonlinear terms in the system description it is known that some of these conditions are sufficient for quadratic stability one of the main contributions of this paper is to demonstrate that these conditions are also necessary conditions hence they are not conservative conditions for quadratic stability by presenting multiplier matrices for many common types of uncertain nonlinear terms the paper also demonstrates the usefulness of the multiplier matrix approach in the analysis and control of nonlinear time varying uncertain systems

stability of the e h type regular shock refraction abstract in this paper the refraction of shocks on the interface for 2 d steady compressible flow is discussed particularly the e h type regular refraction is defined and its global stability of the wave structure is proved the 2 d steady potential flow equation is employed to describe the motion of the fluid the stability of the e h type regular refraction can be reduced to a free boundary problem of a nonlinear mixed type equation in an unbounded domain the corresponding linearized problem is a generalized tricomi problem of the lavrentiev bitsadze mixed type equation and it can be reduced to a nonlocal boundary value problem of an elliptic system the later is finally solved by establishing the bijection of a corresponding nonlocal operator in a weighted holder space the result obtained in this paper develops the theory of nonlinear mixed type equations and gives an application to a significant physical problem

a discrete transform and decompositions of distribution spaces abstract we study a representation formula of the form q q q for a distribution on r n this formula is obtained by discretizing and localizing a standard littlewood paley decomposition the map taking to the sequence q q with q running over the dyadic cubes in r n is called the transform the functions q and q have a particularly simple form moreover most of the familiar distribution spaces lp spaces 1 r n we also consider pointwise multipliers for the characteristic function of a domain we obtain boundedness results for a general class of domains which properly includes lipschitz domains several interpolation methods are easily analyzed via the sequence spaces for real interpolation we obtain among other things an extension to the case p 0 this in turn gives a new approach to the traditional atomic decomposition of hardy spaces

boundedness of singular integrals on the flag hardy spaces on heisenberg group abstract we prove that the classical one parameter convolution singular integrals on the heisenberg group are bounded on multiparameter flag hardy spaces which satisfy the intermediate dilation between the one parameter anisotropic dilation and the product dilation on c n r implicitly

littlewood paley characterization and duality of weighted anisotropic product hardy spaces the authors study anisotropic product hardy spaces hwp a associated with a pair a a1 a2 of expansive dilations and a class of product muckenhoupt weights a a on rn rm this article is a continuation of their earlier work in bownik et al 2010 8 the authors establish the littlewood paley g function characterization and the transform characterization of hwp a 0 p 1 the authors also introduce the weighted anisotropic product campanato space lp w a and establish its transform characterization as an application the authors identify the dual space of hwp a with lp w a the results of this article improve the existing results for weighted product hardy spaces on r r and are new even in the weighted isotropic setting

application of a multiplier method to uncertain lur e like systems abstract this paper investigates the conditions under which an abstract matrix multiplier method can be applied to determine guaranteeing cost controls for systems containing nonlinear uncertain elements via linear matrix inequalities lmis quadratically constrained uncertainties and nonlinearities are considered which comprehend the cases of norm bounded positive real and sector bounded uncertainties nonlinearities both the discrete time and the continuous time cases are discussed necessary and sufficient conditions are formulated in the case of unstructured uncertainty the conditions are sufficient in the structured case the cost guaranteeing controls can be determined by solving lmis numerical examples illustrate the effectiveness of the proposed method

error bounds for glimm difference approximations for scalar conservation laws on deduit des bornes d erreur pour l approximation aux differences de glimm pour la solution d une loi de conservation scalaire non lineaire

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

global structure stability of riemann solutions for general hyperbolic systems of conservation laws in the presence of a boundary abstract this work is a continuation of our previous work z q shao global structure instability of riemann solutions for general quasilinear hyperbolic systems of conservation laws in the presence of a boundary j math anal appl 330 1 2007 511 540 in the present paper we study the global structure stability of the riemann solution u u x t containing only shocks and contact discontinuities of general n n quasilinear hyperbolic systems of conservation laws in the presence of a boundary we prove the existence and uniqueness of global piecewise c 1 solution containing only shocks and contact discontinuities to a class of the mixed initial boundary value problem for general n n quasilinear hyperbolic systems of conservation laws in the half space t x t 0 x 0 our result indicates that this kind of riemann solution u u x t mentioned above for general n n quasilinear hyperbolic systems of conservation laws in a half space possesses a global structural stability some applications to quasilinear hyperbolic systems of conservation laws arising in physics and other disciplines are also given

global supersonic conic shock wave for the steady supersonic flow past a cone polytropic gas abstract in this paper we establish the global existence and stability of a steady conic shock wave for the symmetrically perturbed supersonic flow past an infinitely long conic body as long as the vertex angle is less than a critical value the flow is assumed to be polytropic isentropic and described by a steady potential equation based on the delicate asymptotic expansion of the background solution one can verify that the boundary conditions on the shock and the conic surface satisfy the dissipative property from this property by use of the reflected characteristics method and the special form of the shock equation we show that the conic shock attached at the vertex of the cone exists globally in the whole space when the speed of the supersonic coming flow is appropriately large on the other hand we remove the smallness restriction on the sharp vertex angle in order to establish the global existence of a shock or a global weak solution moreover our proof approach is different from that in shuxing chen zhouping xin huicheng yin global shock wave for the supersonic flow past a perturbed cone comm math phys 228 2002 47 84 and zhouping xin huicheng yin global multidimensional shock wave for the steady supersonic flow past a three dimensional curved cone anal appl 4 2 2006 101 132

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

and variants of multiparameter calderon zygmund theory we consider calderon zygmund operators on product domains under certain weak conditions on the kernel a singular integral operator can be proved to be bounded on h p r r r 0 p 1 if its behaviour on l 2 and on certain scalar valued and vector valued rectangle atoms is known another result concerns an extension of the authors results on l p variants of calderon zygmund theory 1 23 to the product domain setting as an application one obtains estimates for fourier multipliers and pseudo differential operators

product hardy spaces associated to operators with heat kernel bounds on spaces of homogeneous type the aim of this article is to develop the theory of product hardy spaces associated with operators which possess the weak assumption of davies gaffney heat kernel estimates in the setting of spaces of homogeneous type we also establish a calderon zygmund decomposition on product spaces which is of independent and use it to study the interpolation of these product hardy spaces we then show that under the assumption of generalized gaussian estimates the product hardy spaces coincide with the lebesgue spaces for an appropriate range of p

differentiation of the integral hardy spaces and calderon zygmund operators in the product setting abstract this article concerns strong differentiation and operators on product hardy spaces we first show that an example created by papoulis of a function on r 2 whose integral is strongly differentiable almost everywhere but the integral of its absolute value fails to be strongly differentiable on a set of positive measure belongs to the product hardy space h 1 r r the methods that we develop enable us to present a relaxed version of chang fefferman p atoms with a lower number of required vanishing moments and no smoothness needed on the elementary particles in analogy with the proof of this result we show a generalization of a theorem of r fefferman which concludes h p l p 0 p 1 boundedness of multiparameter operators from their behavior on rectangle atoms in addition we extend a result of pipher concerning boundedness of multiparameter calderon zygmund operators from h p to l p

multiparameter hardy space theory on carnot caratheodory spaces and product spaces of homogeneous type this paper is inspired by the work of nagel and stein in which the lp 1 p theory has been developed in the setting of the product carnot caratheodory spaces m m1 mn formed by vector fields satisfying hormander s finite rank condition the main purpose of this paper is to provide a unified approach to develop the multiparameter hardy space theory on product spaces of homogeneous type this theory includes the product hardy space its dual the product bmo space the boundedness of singular integral operators and calderon zygmund decomposition and interpolation of operators as a consequence we obtain the endpoint estimates for those singular integral operators considered by nagel and stein 2004 in fact we will develop most of our theory in the framework of product spaces of homogeneous type which only satisfy the doubling condition and some regularity assumption on the metric all of our results are established by introducing certain banach spaces of test functions and distributions developing discrete calderon identity and discrete littlewood paley stein theory our methods do not rely on the journe type covering lemma which was the main tool to prove the boundedness of singular integrals on the classical product hardy spaces

on the instability problem of a 3 d transonic oblique shock wave abstract in this paper we are concerned with the instability problem of a 3 d transonic oblique shock wave for the steady supersonic flow past an infinitely long sharp wedge the flow is assumed to be isentropic and irrotational it was indicated on p 317 of 7 that if a steady supersonic flow comes from minus infinity and hits a sharp symmetric wedge then it follows from the rankine hugoniot conditions and the physical entropy condition that there possibly appears a weak shock or a strong shock attached at the edge of the sharp wedge which corresponds to a supersonic shock or a transonic shock respectively the question arises which of the two actually occurs it has frequently been stated that the strong one is unstable and that therefore only the weak one could occur however a convincing proof of this instability has apparently never been given the aim of this paper is to understand such a longstanding open question we will show that the attached 3 d transonic oblique shock problem is overdetermined with respect to the periodic perturbation which implies that the 3 d transonic shock is unstable in general

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

steady supersonic flow past an almost straight wedge with large vertex angle abstract this paper studies the problem on the steady supersonic flow at the constant speed past an almost straight wedge with a piecewise smooth boundary it is well known that if each vertex angle of the straight wedge is less than an extreme angle determined by the shock polar the shock wave is attached to the tip of the wedge and constant states on both side of the shock are supersonic this paper is devoted to generalizing this result under the hypotheses that each vertex angle is less than the extreme angle and the total variation of tangent angle along each edge is sufficiently small a sequence of approximate solutions constructed by a modified glimm scheme is proved to be convergent to a global weak solution of the steady problem a sequence of the corresponding approximate leading shock fronts issuing from the tip is shown to be convergent to the leading shock front of the obtained solution the regularity of the leading shock front is established and the asymptotic behaviour of the obtained solution at infinity is also studied

the euler equations in planar nonsmooth convex domains abstract as a model problem for the barotropic mode of the primitive equations of the oceans and atmosphere we consider the euler system on a bounded convex planar domain endowed with non penetrating boundary conditions for 4 3 p 2 and initial and forcing data with l p vorticity we show the existence of a weak solution enriching and extending the results of taylor 2000 32 in the physical case of a rectangular domain 0 l 1 0 l 2 a similar result holds for all 2 p as well moreover by means of a new bmo type regularity estimate for the dirichlet problem on a planar domain with corners we prove uniqueness of solutions with bounded initial vorticity

nonlinear stability of a self similar 3 dimensional gas flow we show that the 3 dimensional supersonic gas flow past an infinite cone is nonlinear staple upon the perturbation of the obstacle the perturbed flow exists globally in space and tends to the self similar flow downstream there is a thin layer of concentration of vorticities and entropy variation our analysis is based on an approximation scheme using local self similar solutions as building blocks this enables us to obtain global estimates of the nonlinear interactions of waves needed for the stability analysis

global shock waves for the supersonic flow past a perturbed cone we prove the global existence of a shock wave for the stationary supersonic gas flow past an infinite curved and symmetric cone the flow is governed by the potential equation as well as the boundary conditions on the shock and the surface of the body it is shown that the solution to this problem exists globally in the whole space with a pointed shock attached at the tip of the cone and tends to a self similar solution under some suitable conditions our analysis is based on a global uniform weighted energy estimate for the linearized problem combining this with the local existence result of chen li 1 we establish the global existence and decay rate of the solution to the nonlinear problem

a feasible moving horizon cal h _ infty control scheme for constrained uncertain linear systems this note presents a robust moving horizon hinfin control scheme that achieves disturbance attenuation with performance adaptation or for a prescribed level the on line solved optimization problem is a lagrange relaxation of the constrained minimax problem with the benefit of achieving feasibility in the moving horizon implementation uncertainties are incorporated by the use of the full block multiplier technique simulation and comparison results are given

the atomic decomposition of weighted hardy spaces associated to self adjoint operators on product spaces abstract let l be a non negative self adjoint operator acting on l 2 r n satisfying a pointwise gaussian estimate for its heat kernel let w be an a p weight on r n r n 1 p in this article we obtain a weighted q atomic decomposition with q p for the weighted hardy space h l w 1 r n r n associated to l

global transonic conic shock wave for the symmetrically perturbed supersonic flow past a cone in this paper we are concerned with the global existence and stability of a steady transonic conic shock wave for the symmetrically perturbed supersonic flow past an infinitely long conic body the flow is assumed to be polytropic isentropic and described by a steady potential equation theoretically as indicated in r courant k o friedrichs supersonic flow and shock waves interscience publishers inc new york 1948 it follows from the rankine hugoniot conditions and the entropy condition that there will appear a weak shock or a strong shock attached at the vertex of the sharp cone in terms of the different pressure states at infinity behind the shock surface which correspond to the supersonic shock and the transonic shock respectively in the references shuxing chen zhouping xin huicheng yin global shock wave for the supersonic flow past a perturbed cone comm math phys 228 2002 47 84 dacheng cui huicheng yin global conic shock wave for the steady supersonic flow past a cone polytropic case preprint 2006 dacheng cui huicheng yin global conic shock wave for the steady supersonic flow past a cone isothermal case pacific j math 233 2 2007 257 289 and zhouping xin huicheng yin global multidimensional shock wave for the steady supersonic flow past a three dimensional curved cone anal appl 4 2 2006 101 132 the authors have established the global existence and stability of a supersonic shock for the perturbed hypersonic incoming flow past a sharp cone when the pressure at infinity is appropriately smaller than that of the incoming flow at present for the supersonic symmetric incoming flow we will study the global transonic shock problem when the pressure at infinity is appropriately large

lpv control and full block multipliers a novel construction of linear parameter varying controllers is presented for arbitrarily fast parameter variations it is shown how full block multipliers allow to considerably reduce conservatism various auxiliary results of independent interest are included

steady supersonic flow past a curved cone abstract this paper studies the steady supersonic flow past a lipschitz curved cone under the assumptions that the cone has an opening angle less than a critical value and has sufficiently small total variation of the tangent of the perturbation and that the mach number of incoming flow is sufficiently large the global weak solution is constructed via glimm scheme for 1 3

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

multi parameter local hardy spaces abstract though multi parameter hardy space theory has been well developed in the past half century not much has been studied for a local hardy space theory in the multi parameter settings such multi parameter local hardy spaces can play an important role in studying the boundedness of multi parameter pseudo differential operators multi parameter singular integrals of non convolution type and applications to partial differential equations etc by establishing a bi parameter local reproducing formula bi parameter local hardy space h p r n 1 r n 2 is introduced in this paper this space coincides with l p r n 1 n 2 when p 1 while p 1 this space is substantially different from the classical bi parameter hardy spaces h p r n 1 r n 2 we will establish its atomic decomposition of the bi parameter local hardy spaces and as an application we prove the boundedness from h p r n 1 r n 2 to l p r n 1 n 2 of the bi parameter singular integral operators in inhomogeneous journe class for p near 1 for the simplicity we have chosen to present all the results in the bi parameter setting nevertheless all of them hold for arbitrary number of parameters the multi parameter local theory developed in this paper can serve as a model case for similar theory in other multi parameter settings

global weakly discontinuous solutions for hyperbolic conservation laws in the presence of a boundary abstract this work is a continuation of our previous work in the present paper we study the mixed initial boundary value problem for general n n quasilinear hyperbolic systems of conservation laws with non linear boundary conditions in the half space t x t 0 x 0 under the assumption that each characteristic with positive velocity is linearly degenerate we prove the existence and uniqueness of global weakly discontinuous solution u u t x with small amplitude and this solution possesses a global structure similar to that of the self similar solution u u x t of the corresponding riemann problem some applications to quasilinear hyperbolic systems of conservation laws arising in physics and other disciplines particularly to the system describing the motion of the relativistic string in minkowski space r 1 n are also given

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