the classical obstacle problem for nonlinear variational energies abstract we develop the complete free boundary analysis for solutions to classical obstacle problems related to nondegenerate nonlinear variational energies the key tools are optimal c 1 1 regularity which we review more generally for solutions to variational inequalities driven by nonlinear coercive smooth vector fields and the results in focardi et al 2015 concerning the obstacle problem for quadratic energies with lipschitz coefficients furthermore we highlight similar conclusions for locally coercive vector fields having in mind applications to the area functional or more generally to area type functionals as well

hyperbolic conservation laws with relaxation the effect of relaxation is important in many physical situations it is present in the kinetic theory of gases elasticity with memory gas flow with thermo non equilibrium water waves etc the governing equations often take the form of hyperbolic conservation laws with lower order terms in this article we present and analyze a simple model of hyperbolic conservation laws with relaxation effects dynamic subcharacteristics governing the propagation of disturbances over strong wave forms are identified stability criteria for diffusion waves expansion waves and traveling waves are found and justified nonlinearly time asymptotic expansion and the energy method are used in the analysis for dissipative waves the expansion is similar in spirit to the chapman enskog expansion in the kinetic theory for shock waves however a different approach is needed

asymptotic behavior of solutions to a hyperbolic system with relaxation and boundary effect in this paper we study the initial boundary value problem of the following hyperbolic system with relaxation

lp decay rates for perturbations of degenerate scalar viscous shock waves abstract this contribution shows how to obtain l p decay rates with 2 p for zero mass perturbations of degenerate scalar viscous shock waves using energy methods the proof is based upon previous work by matsumura and nishihara comm math phys 165 1 1994 83 96 by extending their weighted energy estimates to l p norms and by obtaining sharp decay rates for the antiderivative of the perturbation with the aid of basic interpolation inequalities the analysis applies to shocks of all orders of degeneracy and it is elementary

existence of weak backward solutions to a generalized hele shaw flow moving boundary problem in the original hele shaw problem n 2 and u 6 the dirac measure at the origin a couple of essentially equivalent weak formulations of this problem have been given in 4 8 91 and in 3 f or a variant of the above problem in the case n 2 it has been proved in 4 81 that a classical solution this notion been made precise in two different ways in 4 81 also is a weak solution and in 4 8 91 and 3 that a weak solution exists and is unique for the time interval 0 0 for arbitrary given initial domains do it has also been proved 4 81 that at least under some extra hypotheses which probably are fulfilled for almost all f 0 in nonpathological cases ad consists of analytic curves when t 0 if d t 2 0 is a weak solution all the above results except the last one generalize without much trouble to arbitrary n b 2 as noted also in 9 section 31 some variant of the last result probably also holds for n 2 with analytic curve replaced by real analytic hypersurface no results seem to have been published on the existence or uniqueness of weak solutions for time intervals of the kind t 0 t 0 with do d given i e backward solutions the above mentioned results on the analyticity of dd show or indicate that a backward solution

convergence to traveling waves with decay rates for solutions of the initial boundary problem to a relaxation model abstract in this paper we consider a 2 2 relaxation hyperbolic system of conservation laws with a boundary effect and we show that the solutions of this initial boundary problem tend to the traveling wave solutions of the corresponding cauchy problem time asymptotically in particular we give the algebraic and exponential decay rates by using the weighted energy method the location of a shift for the traveling wave to overcome the difficulty in the boundary plays a key role in this paper

almost global existence of classical discontinuous solutions to genuinely nonlinear hyperbolic systems of conservation laws with small bv initial data abstract in the present paper the author investigates the initial problem for genuinely nonlinear quasilinear hyperbolic systems of conservation laws under small bv perturbations of the riemann initial data where the perturbations are in bv but they are assumed to be c 1 smooth with bounded and possibly large c 1 norms combining the techniques employed by li and kong with the modified glimm s functional the author obtains the almost global existence and lifespan of classical discontinuous solutions to the above problem this result is also applied to the system of one dimensional isentropic flow

the generalized nonlinear initial boundary riemann problem for linearly degenerate quasilinear hyperbolic systems of conservation laws abstract this work is a continuation of our previous work in the present paper we study the generalized nonlinear initial boundary riemann problem with small bv data for linearly degenerate quasilinear hyperbolic systems of conservation laws with nonlinear boundary conditions in a half space t x t 0 x 0 we prove the global existence and uniqueness of piecewise c 1 solution containing only contact discontinuities to a class of the generalized nonlinear initial boundary riemann problem which can be regarded as a small bv perturbation of the corresponding nonlinear initial boundary riemann problem for general n n linearly degenerate quasilinear hyperbolic system of conservation laws moreover this solution has a global structure similar to the one of the self similar solution u u x t to the corresponding nonlinear initial boundary riemann problem some applications to quasilinear hyperbolic systems of conservation laws arising in the string theory and high energy physics are also given

regularity of local minimizers of the interaction energy via obstacle problems the repulsion strength at the origin for repulsive attractive potentials determines the regularity of local minimizers of the interaction energy in this paper we show that if this repulsion is like newtonian or more singular than newtonian but still locally integrable then the local minimizers must be locally bounded densities and even continuous for more singular than newtonian repulsion we prove this and some other regularity results by first showing that the potential function associated to a local minimizer solves an obstacle problem and then by using classical regularity results for such problems

wave front tracing and asymptotic stability of planar travelling waves for a two dimensional shallow river model abstract the propagation of surface water waves in a frictional channel with a uniformly inclined bed is governed by a two dimensional shallow river model in this paper we consider the time asymptotic stability of weak planar travelling waves for a two dimensional shallow river model with darcy s law we derive an effective parabolic equation to analyze the wave front motion by employing weighted energy estimates we show that weak planar travelling waves are time asymptotically stable under sufficiently small perturbations

nonlinear stability of shock fronts for a relaxation system in several space dimensions it is proposed by jin and xin in 3 to use the following relaxation system u t sum limits i 1 m left upsilon i right x i 0 text u in r n text upsilon i in r n 1 upsilon it a i u x i frac left upsilon i f i left u right right varepsilon text i 1 2 ldots m 2 to approximate multi dimensional hyperbolic conservation laws of the form u t sum limits i 1 m f i left u right x i 0 left x i t right in r 1 times r u in r n 3 here e is a positive constant representing the rate of relaxation and a i a i i a i 0 i 1 2 m with i being the n n identity matrix and a i being positive constants satisfying the subcharacteristic condition see 3 and sc one of the main advantages of the system 1 2 is its form of local relaxation structure and linearity in convection which makes it possible to solve this system quite easily by underresolved stable numerical discretizations using neither riemann solvers spatially nor nonlinear system of algebraic equations solvers temporally a class of numerical schemes solving 1 2 has been proposed in 3 which yields satisfactory numerical solutions to hyperbolic conservation laws 3 3 yet the asymptotic equivalence of the relaxation system 1 2 and the hyperbolic conservation laws 3 as either the rate of relaxation goes to zero or time approaches infinity has been rigorously analysed only in one space dimension see 5 9 and 11

stability of contact discontinuity for jin xin relaxation system abstract in this paper we first construct a contact wave for 1 dimensional jin xin relaxation system s jin z p xin the relaxation schemes for systems of conservation laws in arbitrary space dimensions comm pure appl math 48 1995 235 277 this wave serves as the relaxation version of contact discontinuity of the corresponding hyperbolic system at equilibrium such a contact wave is shown to be nonlinearly stable under small initial perturbation the time decay rate is also obtained by weighted energy estimates

zero relaxation limit to centered rarefaction waves for a rate type viscoelastic system abstract we study a rate type viscoelastic system proposed by i suliciu 1990 internat j engrg sci 28 827 841 which is a 3 3 hyperbolic system with relaxation as the relaxation time tends to zero this system convergences to the well known p system formally in the case where the initial data are the riemann data such that the corresponding solutions of the p system are centered rarefaction waves we show that if the wave strength is suitably small then the solution for the relaxation system exists globally in time and converges to the solution of the corresponding rarefaction waves uniformly as the relaxation time goes to zero except for an initial layer the jump discontinuities in the solutions are decaying exponentially fast as time tends to infinity

asymptotic stability of planar rarefaction waves for the relaxation approximation of conservation laws in several dimensions abstract this paper concerns the large time behavior toward planar rarefaction waves of solutions for the relaxation approximation of conservation laws in several dimensions it is shown that a planar rarefaction wave is nonlinear stable in the sense that it is an asymptotic attractor for the relaxation approximation of conservation laws

weakly nonlinear limit for the relaxation approximation of conservation laws in several space dimensions abstract in this paper the weakly nonlinear limit for the relaxation approximation of conservation laws in several space dimensions is derived through asymptotic expansions and justified by employing the energy estimates compared with the work of g q chen c d levermore and t p liu 1994 comm pure appl math 47 787 830 the main difficulty we confront in our analyzes lies in the fact that the problem of the existence of an convex entropy which was essential in the proof of their estimate 5 29 for the system under our consideration is still an open problem since their theorem 3 2 was proved only for the 2 2 case

stability of viscous scalar shock fronts in several dimensions we prove nonlinear stability of planar shock front solutions for viscous scalar conservation laws in two or more space dimensions the proof uses the integrated equation and an effective equation for the motion of the front itself we derive energy estimates that balance terms from the integrated equation with terms from the front motion equation in this paper we prove that viscous shock profiles for scalar conservation laws are stable in two or more space dimensions these multidimensional stability questions are separate from their one dimensional analogues because of the possibility of transverse instabilities such as those that occur in combustion fronts lu and in shock waves with phase changes the proof here is a rigorous version of arguments that are used to derive effective equations such as the kuramoto sivashinsky equation to describe the behavior of fronts the one dimensional stability for scalar conservation laws was proven by ii in and oleinik io using the integrated equation see below and a max imum principle another proof based on weighted norms and spectral theory for the linearized problem was given by sattinger s the multidimensional stability proof below has more in common with the stability proofs for systems of conservation laws in one space dimension begun by kawashima and mat sumura km and goodman go and completed by liu li these proofs use l2 energy estimates for the integrated equation we consider equations of the form ut f u x g u y uxx uyy where f u is a strictly convex function of u

lifespan of classical discontinuous solutions to general quasilinear hyperbolic systems of conservation laws with small bv initial data shocks and contact discontinuities abstract this work is a continuation of our previous work z q shao global structure stability of riemann solutions for linearly degenerate hyperbolic conservation laws under small bv perturbations of the initial data nonlinear anal real world appl 11 2010 3791 3808 in the present paper we investigate the global structure stability of riemann solutions for general quasilinear hyperbolic systems of conservation laws under small bv perturbations of the initial data where the riemann solution only contains the shocks and contact discontinuities and at least a shock wave the perturbations are in bv but they are assumed to be c 1 smooth with bounded and possibly large c 1 norms we get a lower bound of the lifespan of the piecewise c 1 solution to a class of the generalized riemann problem which can be regarded as a small bv perturbation of the corresponding riemann problem some applications to quasilinear hyperbolic systems of conservation laws arising in physics particularly to one dimensional euler equations of gas dynamics for a compressible inviscid non heat conducting gas in eulerian coordinates are also given

stability of traveling waves for hamilton jacobi equations with finite speed perturbations abstract we prove nonlinear stability of planar shock for general hamilton jacobi equations with finite speed perturbation here we use energy estimates it is shown that the solution connecting a weak shock is asymptotically stable under small perturbations

relaxation limit of interacting shock waves for non isentropic gas dynamics abstract in this paper we study the relaxation limit of interacting shock waves for non isentropic gas dynamics such a model is introduced by jin and xin for numerical analysis and named as jin xin relaxation system given any entropy solution of non isentropic gas dynamics which consists of two different families of shocks interacting at some positive time we show that such solution is the relaxation limit for a family of smooth global solutions of the jin xin relaxation system we remark that after the interacting time not only shocks but also contact discontinuity is generated

discrete kinetic schemes for multidimensional systems of conservation laws we present here some numerical schemes for general multidimensional systems of conservation laws based on a class of discrete kinetic approximations which includes the relaxation schemes by s jin and z xin these schemes have a simple formulation even in the multidimensional case and do not need the solution of the local riemann problems for these approximations we give a suitable multidimensional generalization of the whitham s stability subcharacteristic condition in the scalar multidimensional case we establish the rigorous convergence of the approximated solutions to the unique entropy solution of the equilibrium cauchy problem

initial boundary value problem for a class of linear relaxation systems in arbitrary space dimensions abstract in this paper we consider the characteristic initial boundary value problem ibvp for the multi dimensional jin xin relaxation model in a half space with arbitrary space dimension n 2 as in the one dimensional case n 1 see j differential equations 167 2000 388 437 our main interest is on the precise structural stability conditions on the relaxation system particularly the formulation of boundary conditions such that the relaxation ibvp is stiffly well posed that is uniformly well posed independent of the relaxation parameter e 0 and the solution of the relaxation ibvp converges as e 0 to that of the corresponding limiting equilibrium system except for a sharp transition layer near the boundary our main result can be roughly stated as stiff kreiss condition uniform kreiss condition for the relaxation ibvp we consider in this paper which is in sharp contrast to the one dimensional case z xin and w q xu j differential equations 167 2000 388 437 more precisely we show that the uniform kreiss condition which is necessary and sufficient for the well posedness of the relaxation ibvp for each fixed e together with the subcharacteristic condition which is necessary and sufficient for the stiff well posedness of the corresponding cauchy problem also guarantees the stiff well posedness of our relaxation ibvp and the asymptotic convergence to the corresponding equilibrium system in the limit of small relaxation rate optimal convergence rates are obtained and various boundary layer behaviors are also rigorously justified

nonlinear stability of strong planar rarefaction waves for the relaxation approximation of conservation laws in several space dimensions abstract in this paper we show that a strong planar rarefaction wave is nonlinear stable namely it is an attractor for the relaxation approximation of the scalar conservation laws in several space dimensions compared with former results obtained by t p liu 1987 comm math phys 108 153 175 and t luo 1997 j differential equations 133 255 279 our main novelty lies in the fact that the planar rarefaction waves do not need to be small and in the one dimensional case the initial disturbance can also be chosen arbitrarily large

the large time energy concentration in solutions to the navier stokes equations with nonzero external forces abstract it is known that the turbulent solutions to the unforced navier stokes equations exhibit a large time energy concentration in the frequency space if we suppose the existence of a nonzero time dependent external force f l 1 0 l 2 in the equations the situation is more complicated it is possible to construct simple examples of solutions in which the energy is asymptotically distributed over the entire spectrum of the stokes operator further we will show as the main result of this paper that there still exist wide classes of initial conditions and external forces yielding solutions with the large time energy concentration analogical to the situation in the unforced navier stokes equations we will also show that the frequency spectrum of the solution does not generally relate to the frequency spectrum of the external force

application of variational inequalities to the moving boundary problem in a fluid model for biofilm growth vi betraktar ett problem med rorlig rand motsvarande ett gransfall i en fluid modell for biofilmstillvaxt som foreslagits av j dockery och i klapper siam j appl math 62 3 2001

zero relaxation limit to centered rarefaction waves for jin xin relaxation system abstract in this paper we study the zero relaxation limit problem for the following jin xin relaxation system e u t v x 0 v t a 2 u x 1 f u v with initial data i u v x 0 u 0 x v 0 x u v u 0 as x this system was proposed by jin and xin 1995 1 with an interesting numerical origin as the relaxation time tends to zero this system converges to the equilibrium conservation law formally our interest is to study the case where the initial data are allowed to have jump discontinuities such that the corresponding solutions to the equilibrium conservation law contain centered rarefaction waves and the limits u v of the initial data at x do not satisfy the equilibrium equation i e v f u in particular riemann data connected by rarefaction curves are included we show that if the wave strength is sufficiently small then the solution for the relaxation system exists globally in time and converges to the solution of the corresponding rarefaction waves uniformly as the relaxation time goes to zero except for an initial layer

the jin xin relaxation approximation of scalar conservation laws in several dimensions with large initial perturbation abstract this paper is concerned with nonlinear stability of strong planar rarefaction waves for the jin xin relaxation approximation of scalar conservation laws in several dimensions for such a problem local stability of weak or strong planar rarefaction waves have been obtained in luo 1997 20 and zhao 2000 43 respectively for the global stability results to the best of our knowledge the only result available now is on the one dimensional case cf zhao 2000 43 which is based on the maximum principle established in natalini 1996 30 the main purpose of this paper is try to deduce some nonlinear stability results with large initial perturbation our analysis is based on the elementary energy method and the continuation argument

unidirectional evolution equations of diffusion type abstract this paper is concerned with the uniqueness existence partial smoothing effect comparison principle and long time behavior of solutions to the initial boundary value problem for a unidirectional diffusion equation the unidirectional evolution often appears in damage mechanics due to the strong irreversibility of crack propagation or damage evolution the existence of solutions is proved in an l 2 framework by employing a backward euler scheme and by introducing a new method of a priori estimates based on a reduction of discretized equations to variational inequalities of obstacle type and by developing a regularity theory for such obstacle problems the novel discretization argument will be also applied to prove the comparison principle as well as to investigate the long time behavior of solutions

global structure stability of riemann solutions of quasilinear hyperbolic systems of conservation laws shocks and contact discontinuities abstract in this paper the author proves the global structure stability of the lax s riemann solution u u x t containing only shocks and contact discontinuities of general n n quasilinear hyperbolic system of conservation laws more precisely the author proves the global existence and uniqueness of the piecewise c1 solution u u t x of a class of generalized riemann problem which can be regarded as a perturbation of the corresponding riemann problem for the quasilinear hyperbolic system of conservation laws moreover this solution has a global structure similar to that of the solution u u x t combining the results in kong global structure instability of riemann solutions of quasilinear hyperbolic systems of conservation laws rarefaction waves to appear the author proves that the lax s riemann solution of general n n quasilinear hyperbolic system of conservation laws is globally structurally stable if and only if it contains only non degenerate shocks and contact discontinuities but no rarefaction waves and other weak discontinuities

asymptotic stability of planar rarefaction waves for viscous conservation laws in several dimensions this paper concerns the large time behavior toward planar rarefaction waves of the solutions for scalar viscous conservation laws in several dimensions it is shown that a planar rarefaction wave is nonlinearly stable in the sense that it is an asymptotic attractor for the viscous conservation law this is proved by using a stability result of rarefaction wave for scalar viscous conservation laws in one dimension and an elementary l 2 energy method 0 introduction we will establish the asymptotic stability of planar rarefaction waves for scalar viscous conservation laws in two or more space dimensions we consider n dimensional scalar viscous conservation laws of the form n n 1 ut el f u x e aijux1u x e r t 0 i 1 i j l where u e r1 a aij called the viscosity matrix is a constant positive definite matrix and we assume that all the flux functions are smooth say in cn and equation 1 is genuinely nonlinear in the x1 direction 8 i e for a fixed constant a 0 2 jf u a the initial data for equation 1 is 3 u x o uo x satisfying 4 lim u x1 u ilx rn i 0 where u u u are two constants a planar rarefaction wave in x1 direction ur xl t is a solution of the following initial value problem for the received by the editors september 27 1988 1980 mathematics subject classification 1985 revision primary 35q99 35k35

lp convergence rate to diffusion waves for p system with relaxation in this paper we consider the asymptotic behavior of solution for the cauchy problem for p system with relaxation vt ux 0 ut p v x 1 f v u e with initial data v u x 0 v0 x u0 x v0 u as x i by applying the elementary energy method and the green function method for the parabolic equation we obtain the l p convergence rate to the diffusion waves for the cauchy