existence of a global smooth solution for a degenerate goursat problem of gas dynamics the goursat problem of a mixed type equation p 0 is considered at the ends of its supports we have p 0 which means it is degenerate hyperbolic we prove the global existence of a smooth solution to the degenerate goursat problem up to a boundary where p 0 this problem comes from the expansion of a wedge of gas with constant velocity into vacuum in two dimensional pressure gradient equations in gas dynamics where p is the pressure and p 0 means vacuum

two dimensional riemann problem for a 2 2 system of hyperbolic conservation laws involving three constant states abstract zhang and zheng 1990 conjectured on the structure of a solution for a two dimensional riemann problem for euler equation to resolve this illuminating conjecture many researchers have studied the simplified 2 2 systems in this paper 3 pieces riemann problem for two dimensional 2 2 hyperbolic system is considered without the restriction that each jump of the initial data projects one planar elementary wave we classify twelve topologically distinct solutions and construct analytical and numerical solutions the computed numerical solutions clearly confirm the constructed analytic solutions

interaction of rarefaction waves of the two dimensional self similar euler equations we construct classical self similar solutions to the interaction of two arbitrary planar rarefaction waves for the polytropic euler equations in two space dimensions the binary interaction represents a major type of interaction in the two dimensional riemann problems and includes in particular the classical problem of the expansion of a wedge of gas into vacuum based on the hodograph transformation the method employed here involves the phase space analysis of a second order equation and the inversion back to or development onto the physical space

interaction of three and four rarefaction waves of the pressure gradient system abstract we study the two dimensional pressure gradient system a subsystem of the two dimensional compressible euler system we consider the problem of interaction of four rarefaction waves which is one case of two dimensional riemann problems it is known that when two planar waves interact there exists a smooth solution in the interaction region in this paper we establish the existence of a smooth solution in the hyperbolic domain of determinacy in which we encounter the interaction of simple and planar waves and shock prevention in simple waves

singularity formation for a system of conservation laws in two space dimensions abstract we study singularity formation for the 2 2 system u t u 2 x uv y 0 and v t uv x v 2 y 0 our analysis is based on the argument due to j keller and l ting 1966 comm pure appl math 19 371 420 about the evolution along a characteristic of the compression rate of nearby characteristics this system is one of a class of systems called partially aligned which exhibit a degenerate characteristic structure where a pair of directions replaces the usual cone at every point

asymptotic theory of propeller noise ii supersonic single rotation propeller this paper gives analytical formulas for the far field harmonic components of the acoustic radiation from a b bladed supersonic single rotation propeller in the asymptotic limit b o the dominant source for radiation in a given direction is identified as located at the mach radius along the blade span the radius that approaches the observer at precisely sonic speed once in each rotation favorable agreement is found between the asymptotic prediction and the results of full numerical evaluation of the radiation integrals singular surfaces in the field are identified from the asymptotic formulas which are extended to include chordwise noncompactness effects these show among other things that supersonic propeller noise is an algebraically decreasing function of tip mach number once chordwise noncompactness effects are taken into account in contrast to the exponential increase with tip mach number of subsonic propeller noise

two dimensional riemann problem involving three j s for a hyperbolic system of nonlinear conservation laws in this paper we study the riemann problem for a two dimensional hyperbolic system of conservation laws with initial data projecting exactly three contact discontinuities we use the generalized characteristic analysis method to analyze interactions of contact discontinuities and obtain six analytical solutions and corresponding criteria delta shock waves and spiral structures appear in some solutions we also conduct numerical experiments and numerical results confirm the geometric structures of the constructed solutions

on two dimensional gas expansion for pressure gradient equations of euler system abstract in this paper we establish the existence of global continuous solutions of gas expansion into a vacuum for the two dimensional pressure gradient equations in gas dynamics under irrotational condition by hodograph transformation the flow is governed by the equation p p 2 v p uu 2 p u p v p uv p p 2 u p vv 0 which can be further reduced to a inhomogeneous linearly degenerate system of three equations then the problem of the expansion of a wedge of gas into a vacuum is solved in the same way

similarity solutions for three dimensional euler equations using lie group analysis a similarity analysis of the nonlinear three dimensional unsteady euler equations of gas dynamics is presented using lie group of transformations with commuting infinitesimal operators symmetry groups admitted by the governing system of partial differential equations pdes are obtained and the complete lie algebra of infinitesimal symmetries is established the symmetry generators are used for constructing similarity variables which lead to a reduced system of equations with one independent variable less at each step and eventually to a system of ordinary differential equations odes in some cases it is possible to solve these equations exactly

generalized characteristic analysis and guckenheimer structure abstract we develop a generalized characteristic analysis method using this method we study interactions of elementary waves and construct the riemann solutions for two dimensional scalar conservation laws especially we obtain the necessary and sufficient condition of appearance of guckenheimer structure which is something like mach reflection in gas dynamics

a complete global solution to the pressure gradient equation abstract we study the domain of existence of a solution to a riemann problem for the pressure gradient equation in two space dimensions the riemann problem is the expansion of a quadrant of gas of constant state into the other three vacuum quadrants the global existence of a smooth solution was established in dai and zhang z dai t zhang existence of a global smooth solution for a degenerate goursat problem of gas dynamics arch ration mech anal 155 2000 277 298 up to the free boundary of vacuum we prove that the vacuum boundary is the coordinate axes

solution of two dimensional riemann problems of gas dynamics by positive schemes the positivity principle and positive schemes to solve multidimensional hyperbolic systems of conservation laws have been introduced in x d liu and p d lax j fluid dynam 5 1996 pp 133 156 some numerical experiments presented there show how well the method works in this paper we use positive schemes to solve riemann problems for two dimensional gas dynamics

structure of riemann solutions for 2 dimensional scalar conservation laws abstract we study the structure of riemann solutions for 2 dimensional scalar conservation laws the riemann data are three constants in three fan domains forming different angles we study the dependence of the structure of the solution upon the value of the constants as well as the angles

concentration and cavitation in the vanishing pressure limit of solutions to the euler equations for nonisentropic fluids numerical simulations 2 d riemann problem in gas dynamics and formation of spiral in nonlinear problems in engineering and science numerical and analytical approach beijing 1991 science press beijing 1992 pp 167 179 discrete contin dyn syst 1 1995 555 584 6 2000 419 430 for the euler equations for gas dynamics in the regime of small pressure showed that for one case the particles seem to be more sticky and tend to concentrate near some shock locations and for the other case in the region of rarefaction waves the particles seem to be far apart and tend to form cavitation in the region in this paper we identify and analyze the phenomena of concentration and cavitation by studying the vanishing pressure limit of solutions of the full euler equations for nonisentropic compressible fluids with a scaled pressure it is rigorously shown that any riemann solution containing two shocks and possibly one contact discontinuity to the euler equations for nonisentropic fluids tends to a shock solution to the corresponding transport equations and the intermediate densities between the two shocks tend to a weighted measure that along with the two shocks and possibly contact discontinuity forms the shock as the pressure vanishes by contrast it is also shown that any riemann solution containing two rarefaction waves and possibly one contact discontinuity to the euler equations for nonisentropic fluids tends to a two contact discontinuity solution to the transport equations and the nonvacuum intermediate states between the two rarefaction waves tend to a vacuum state as the pressure vanishes some numerical results exhibiting the processes of concentration and cavitation are presented as the pressure decreases 2003 elsevier b v all rights reserved msc 35l65 35b30 76e19 35q35 35l67 35b25 65m06

two dimensional riemann problem for a hyperbolic system of nonlinear conservation laws i four j cases abstract the riemann problem for a two dimensional 2 2 hyperbolic system of nonlinear conservation laws has been solved analytically the solutions reveal a variety of geometric structures 57 exact solutions and the corresponding criteria have been obtained a kind of new nonlinear hyperbolic wave which we call delta shock appears in some solutions the delta shock is a discontinuity surface which is the support of a dirac function

new developments of delta shock waves and its applications in systems of conservation laws a theory of delta shock waves with dirac delta functions developing in both state variables for a class of nonstrictly hyperbolic systems of conservation laws is established in this paper we solve constructively the riemann problems for the system under consideration in solutions we find another kind of delta shock waves on which both state variables simultaneously contain the dirac delta functions it is quite different from the previous ones on which only one state variable contains the dirac delta function we strictly prove it satisfies the system in the sense of distributions the generalized rankine hugoniot relation and entropy condition are proposed to solve the delta shock waves furthermore we show all of the existence and stability of solutions including the delta shock waves to reasonable viscous perturbations the generalized rankine hugoniot relation is also confirmed in particular our theory on the delta shock waves possesses the generality and practicability which can be conveniently and successfully applied to those systems investigated by korchinski 1977 tan zhang and zheng 1991 ercole 2000 cheng and yang 2011 etc and we also give a simplified approach to solve a 2 d riemann problem for the system studied by tan and zhang 1990 for the case 2j 2j and obtain the explicit formulae of the delta shock waves finally the numerical simulations completely coinciding with theoretical analysis are presented

zero variance principle for monte carlo algorithms we present a general approach to greatly increase at little cost the efficiency of monte carlo algorithms to each observable to be computed we associate a renormalized observable improved estimator having the same average but a different variance by writing down the zero variance condition a fundamental equation determining the optimal choice for the renormalized observable is derived zero variance principle for each observable separately we show with several examples including classical and quantum monte carlo calculations that the method can be very powerful

two dimensional riemann problem for a hyperbolic system of conservation laws in three pieces abstract this paper studies the two dimensional riemann problem for a hyperbolic system of conservation laws with three constant initial data separated by x positive y positive and x negative axes using the generalized characteristic analysis method by considering the interactions of rarefaction waves shock waves and contact discontinuities 13 exact solutions and the corresponding criteria are obtained delta shock waves appear in some solutions moreover the numerical illustrations agree completely with the geometric structures of the exact solutions

two dimensional riemann problems for zero pressure gas dynamics with three constant states abstract the riemann problems for two dimensional zero pressure gas dynamics are solved completely when the initial data take three constant states having discontinuities on x y positive and x negative axes with the help of characteristic analysis by studying interactions among delta shocks vacuums and contact discontinuities the riemann solutions constructed exhibit nine different explicit configurations the mach reflection like configurations appear in some solutions

two dimensional riemann problem involving three contact discontinuities for 2 2 hyperbolic conservation laws in anisotropic media abstract in this study we consider the two dimensional riemann problem for a system of conservation laws which models polymer flooding in an anisotropic oil reservoir the initial data are constants in three sectors centered at the origin which only involve three contact discontinuities by the generalized characteristic analysis method and the phase plane analysis method we find that the solutions are not unique for certain values of the initial data we propose an additional stability condition for the interaction of the contact discontinuities by this stability condition all of the exact solutions and their corresponding criteria can be obtained the solutions exhibit various geometrical structures in particular envelope rarefaction waves develop in some solutions

interaction of four rarefaction waves in the bi symmetric class of the two dimensional euler equations the global existence and structures of solutions to multi dimensional unsteady compressible euler equations are interesting and important open problems in this paper we construct global classical solutions to the interaction of four orthogonal planar rarefaction waves with two axes of symmetry for the euler equations in two space dimensions in the case where the initial rarefaction waves are large the bi symmetric initial data is a basic type of four wave two dimensional riemann problems the solutions in this case are continuous bounded and self similar and we characterize how large the rarefaction waves must be we use the methods of hodograph transformation characteristic decomposition and phase space analysis we resolve binary interactions of simple waves in the process

the riemann problem admitting δ δ shocks and vacuum states the vanishing viscosity approach in this paper using the vanishing viscosity method we construct a solution of the riemann problem for the system of conservation laws ut u2 x 0 vt 2 uv x 0 wt 2 v2 uw x 0 with the initial data u x 0 v x 0 w x 0 u v w x 0 this problem admits shock wave type solutions and vacuum states shock is a new type of singular solutions to systems of conservation laws first introduced in e yu panov v m shelkovich shock waves as a new type of solutions to systems of conservation laws j differential equations 228 2006 49 86 it is a distributional solution of the riemann problem such that for t 0 its second component v may contain dirac measures the third component w may contain a linear combination of dirac measures and their derivatives while the first component u has bounded variation using the above mentioned results we also solve the shock cauchy problem for the first two equations of the above system since shocks can be constructed by the vanishing viscosity method they are natural solutions to systems of conservation laws we describe the formation of the shocks and the vacuum states from smooth solutions of the parabolic problem the results of this paper as well as those of the above mentioned paper show that solutions of systems of conservation laws can develop not only dirac measures as in the case of shocks but their derivatives as well

generalized variational principles global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics we study systems of conservation laws arising in two models of adhesion particle dynamics the first is the system of free particles which stick under collision the second is a system of gravitationally interacting particles which also stick under collision in both cases mass and momentum are conserved at the collisions so the dynamics is described by 2 2 systems of conservations laws we show that for these systems global weak solutions can be constructed explicitly using the initial data by a procedure analogous to the lax oleinik variational principle for scalar conservation laws however this weak solution is not unique among weak solutions satisfying the standard entropy condition we also study a modified gravitational model in which instead of momentum some other weighted velocity is conserved at collisions for this model we prove both existence and uniqueness of global weak solutions we then study the qualitative behavior of the solutions with random initial data we show that for continuous but nowhere differentiable random initial velocities all masses immediately concentrate on points even though they were continuously distributed initially and the set of shock locations is dense

a revisit of the tonal noise of small rotors in this study asymptotic analysis of the frequency domain formulation to compute the tonal noise of the small rotors in the now ubiquitously multi rotor powered drones is conducted simple scaling laws are proposed to evaluate the impacts of the influential parameters such as blade number flow speed rotation speed unsteady motion thrust and observer angle on the tonal noise the rate of noise increment with thrust or rotational speed is determined by orders of blade passing frequency harmonics and the unsteady motion the axial mean flow influence can be approximated by quadratic functions at given thrust the sound decreases rapidly with the radius and blade number as the surface pressure becomes less intensive the higher tonal harmonics are significantly increased if unsteady motions although of small amplitude are existed as indicated by the defined sensitivity function emphasizing that the unsteady motions should be avoided for quiet rotor designs the scaling laws are examined by comparing with the full computations of the rotor noise using the frequency domain method the implementation of which has been validated by comparing with experiments good data collapse is obtained when the proposed scaling laws which highlights the dominant influence of the design parameters are incorporated

delta shock waves as limits of vanishing viscosity for hyperbolic systems of conservation laws abstract for simple models of hyperbolic systems of conservation laws we study a new type of nonlinear hyperbolic wave a delta shock wave which is a dirac delta function supported on a shock we prove that delta shock waves are w limits in l1 of solutions to some reasonable viscous perturbations as the viscosity vanishes further we solve the multiplication problem of a delta function with a discontinuous function to show that delta shock waves satisfy the equations in the sense of distributions under suitable generalized rankine hugoniot and entropy conditions we establish the existence and uniqueness of solutions involving delta shock waves for the riemann problems the existence of solutions to the cauchy problem is also investigated

a new aerodynamic integral equation based on an acoustic formula in the time domain an aerodynamic integral equation for bodies moving at transonic and supersonic speeds is presented based on a time dependent acoustic formula for calculating the noise emanating from the outer portion of a propeller blade travelling at high speed the ffowcs williams hawking formulation the loading terms and a conventional thickness source terms are retained two surface and three line integrals are employed to solve an equation for the loading noise the near field term is regularized using the collapsing sphere approach to obtain semiconvergence on the blade surface a singular integral equation is thereby derived for the unknown surface pressure and is amenable to numerical solutions using galerkin or collocation methods the technique is useful for studying the nonuniform inflow to the propeller

compressible helicoidal surface theory for propeller aerodynamics and noise acceleration potential techniques from three dimensional thin wing theory have been generalized for propeller and prop fan analysis helicoidal reference surfaces take the place of the planar surface in wing theories otherwise the theories are equivalent the acoustic branch of the theory including nonlinear source terms extends and unifies frequency domain noise theories dating back to gutin for aerodynamic applications it is shown that prop fans satisfy well known criteria for use of linearized theory at transonic speeds by virtue of small aspect ratio and small thickness ratio the results are in the form of integral equations for downwash as functions of thickness and steady or unsteady loading distributions for the case of no rotation the kernel functions reduce to well known kernels of wing theory the analysis within the restrictions of linearization treats rigorously any planform and any flight condition including the combination of subsonic roots and supersonic tips typical of prop fans the effects of thickness camber angle of attack sweep offset blade interference and tip relief are all treated without approximation

two dimensional riemann problem for scalar conservation laws abstract using the generalized characteristic analysis method we study the two dimensional riemann problem for scalar conservation laws which is nonconvex along the y direction and interactions of its elementary waves give the classification of initial discontinuities and construct all riemann solutions which riemann data are two or three pieces of constants all kinds of guckenheimer structure appear in the solutions and the necessary and sufficient condition of appearance of it is given

dynamics of propagation and interaction of δ shock waves in conservation law systems abstract we introduce a new definition of a shock wave type solution for a class of systems of conservation laws in the one dimensional case the weak asymptotics method developed by the authors is used to construct formulas describing the propagation and interaction of shock waves the dynamics of merging two shocks is described by explicit formulas continuously in time

riemann problems for a class of coupled hyperbolic systems of conservation laws abstract we study a class of coupled hyperbolic systems of conservation laws which contain the one dimensional zero pressure gas dynamics as a prototypical example the riemann problems are solved constructively the riemann solutions exactly include two kinds delta shock wave solutions and vacuum solutions under the generalized rankine hugoniot relation and entropy condition all of the existence uniqueness and stability of solutions to viscous perturbations are proved two typical examples are presented finally

interactions between two rarefaction waves for the pressure gradient equations in the gas dynamics in this paper we consider the interactions of forward and backward centered rarefaction waves for pressure gradient equations for the goursat problem developed on the domain of interactions using characteristic analysis it is proved that there never appears vacuum in this field when there is no vacuum on characteristic boundary with the help of uniform a priori estimate the existence and uniqueness of smooth solution is obtained the results of interactions between two rarefaction waves are presented moreover the numerical illustrations agree completely with the theoretical analysis

global solutions to the two dimensional riemann problem for a system of conservation laws we study the global solutions to the two dimensional riemann problem for a system of conservation laws the initial data are three constant states separated by three rays emanating from the origin under the assumption that each ray in the initial data outside of the origin projects exactly one planar contact discontinuity this problem is classified into five cases by the self similar transformation the reduced system changes type from being elliptic near the origin to being hyperbolic far away in self similar plane then in hyperbolic region applying the generalized characteristic analysis method a goursat problem is solved to describe the interactions of planar contact discontinuities while in elliptic region a boundary value problem arises it is proved that this boundary value problem admits a unique solution based on these preparations five explicit solutions and their corresponding criteria can be obtained in self similar plane