Showing papers in "Journal of Hyperbolic Differential Equations in 2004"
TL;DR: This paper showed that the Benjamin-Ono equation is globally well-posed in Hs(R) for s ≥ 1, despite the presence of the derivative in the nonlinearity, which causes the solution map to not be uniformly continuous in H s for any s.
Abstract: We show that the Benjamin–Ono equation is globally well-posed in Hs(R) for s≥1. This is despite the presence of the derivative in the nonlinearity, which causes the solution map to not be uniformly continuous in Hs for any s [18]. The main new ingredient is to perform a global gauge transformation which almost entirely eliminates this derivative.
246 citations
TL;DR: In this article, a review of the present situation with existence and uniqueness theorems for mixed equations and their application to the problems of transonic flow is presented. But it is not shown that this formal limit really holds.
Abstract: This paper reviews the present situation with existence and uniqueness theorems for mixed equations and their application to the problems of transonic flow. Some new problems are introduced and discussed. After a very brief discussion of time-dependent flows (Sec. 1) the steady state and its history is described in Sec. 2. In Secs. 3 and 4, early work on mixed equations and their connection to 2D flow are described and Sec. 5 brings up the problem of shocks, the construction of good airfoils and the relevant boundary value problems. In Sec. 6 we look at what two linear perturbation problems could tell us about the flow. In Sec. 7 we describe other examples of fluid problems giving rise to similar problems. Section 8 is devoted to the uniqueness by a conservation law and Secs. 9–11 to the existence proofs by Friedrichs' multipliers. In Sec. 12 a proof is given of the existence of a steady flow corresponding to some of the previous examples but the equations have been modified to a higher order system with a small parameter which when set to zero yields the equations of transonic flow. It remains to show that this formal limit really holds. Much has been left out especially modern computational results and the text reflects the particular interests of the author.
106 citations
TL;DR: In this article, the collision of two clouds of dust is modeled with pressureless gas equations in the case where the clouds have finite extent and are surrounded by vacuum, and the delta shock that arises in the initial stage of the collision evolves into a delta rarefaction-shock and then into delta double-rarefaction as first one cloud and then the other is fully accreted into the singularity.
Abstract: The equations of isothermal gas dynamics are studied in the limit when the sound speed vanishes, giving the so-called pressureless gas equations. The collision of two clouds of dust is modeled with these equations in the case where the clouds have finite extent and are surrounded by vacuum. The delta shock that arises in the initial stage of the collision evolves into a delta rarefaction-shock and then into a delta double-rarefaction as first one cloud and then the other is fully accreted into the singularity. A high-resolution finite volume method that captures this behavior is also presented and numerical results shown.
72 citations
TL;DR: In this paper, the existence of a scattering operator for field equations is interpreted as the well-posedness of a Goursat problem on null infinity, and the conformal scattering operator is proved to be equivalent to an analytical scattering operator defined in terms of classical wave operators.
Abstract: We work on a class of non-stationary vacuum space-times admitting a conformal compactification that is smooth at null and timelike infinity. Via a conformal transformation, the existence of a scattering operator for field equations is interpreted as the well-posedness of a Goursat problem on null infinity. We solve the Goursat problem in the case of Dirac and Maxwell fields. The case of the wave equation is also discussed and it is shown why the method cannot be applied at present. Then the conformal scattering operator is proved to be equivalent to an analytical scattering operator defined in terms of classical wave operators.
58 citations
TL;DR: In this paper, it was shown that entropy solutions of Dtu+Dxf(u) = 0 belong to SBVloc(Ω) for planar Hamilton-Jacobi PDEs with uniformly convex Hamiltonians.
Abstract: Let Ω⊂ℝ2 be an open set and f∈C2(ℝ) with f" > 0. In this note we prove that entropy solutions of Dtu+Dxf(u) = 0 belong to SBVloc(Ω). As a corollary we prove the same property for gradients of viscosity solutions of planar Hamilton–Jacobi PDEs with uniformly convex Hamiltonians.
57 citations
TL;DR: The concept of strong hyperbolicity was introduced in this article, where it is shown that a system is strongly hyperbranched with respect to a given hypersurface, and that it is also strongly hyperbrochastic in the sense that the maximal propagation speed in any given direction is bounded.
Abstract: We discuss several topics related to the notion of strong hyperbolicity which are of interest in general relativity. After introducing the concept and showing its relevance we provide some covariant definitions of strong hyperbolicity. We then prove that if a system is strongly hyperbolic with respect to a given hypersurface, then it is also strongly hyperbolic with respect to any nearby surface. We then study for how much these hypersurfaces can be deformed and discuss then causality, namely what the maximal propagation speed in any given direction is. In contrast with the symmetric hyperbolic case, for which the proof of causality is geometrical and direct, relaying in energy estimates, the proof for general strongly hyperbolic systems is indirect for it is based in Holmgren's theorem. To show that the concept is needed in the area of general relativity we discuss two results for which the theory of symmetric hyperbolic systems shows to be insufficient. The first deals with the hyperbolicity analysis of systems which are second order in space derivatives; they include certain versions of the ADM and the BSSN families of equations. This analysis is considerably simplified by introducing pseudo-differential first-order evolution equations. Well-posedness for some members of the latter family systems is established by showing they satisfy the strong hyperbolicity property. Furthermore it is shown that many other systems of such families are only weakly hyperbolic, implying they should not be used for numerical modeling. The second result deals with systems having constraints. The question posed is which hyperbolicity properties, if any, are inherited from the original evolution system by the subsidiary system satisfied by the constraint quantities. The answer is that, subject to some condition on the constraints, if the evolution system is strongly hyperbolic then the subsidiary system is also strongly hyperbolic and the causality properties of both are identical.
49 citations
TL;DR: In this article, a new Lp energy method for multi-dimensional viscous conservation laws is introduced, which is useful enough to derive the optimal decay estimates of solutions in the W 1,p space for the Cauchy problem.
Abstract: We introduce a new Lp energy method for multi-dimensional viscous conservation laws. Our energy method is useful enough to derive the optimal decay estimates of solutions in the W1,p space for the Cauchy problem. It is also applicable to the problem for the stability of planar waves in the whole space or in the half space, and gives the optimal convergence rate toward the planar waves as time goes to infinity. This energy method makes use of several special interpolation inequalities.
47 citations
TL;DR: In this paper, a geometric framework for nonlinear hyperbolic systems of partial differential equations in one-space dimension (in either conservative or non-conservative form) is introduced, in which solutions are sought as (continuous) parametrized graphs, satisfying ∂sX ≥ 0, rather than (discontinuous) functions (t, x) ↦ u(t,x).
Abstract: For nonlinear hyperbolic systems of partial differential equations in one-space dimension (in either conservative or non-conservative form) we introduce a geometric framework in which solutions are sought as (continuous) parametrized graphs(t,s) ↦ (X,U)(t,s) satisfying ∂sX ≥ 0, rather than (discontinuous) functions (t,x) ↦ u(t,x). On one hand, we generalize an idea by Dal Maso, LeFloch, and Murat who used a family of traveling wave profiles to define non-conservative products, and we define the notion of graph solution subordinate to a family of Riemann graphs. The latter naturally encodes the graph of the solution to the Riemann problem, which should be determined from an augmented model taking into account small-scale physics and providing an internal structure to the shock waves. In a second definition, we write an evolution equation on the graphs directly and we introduce the notion of graph solution subordinate to a diffusion matrix, which merges together the hyperbolic equations (in the "non-vertical" parts of the graphs) with the traveling wave equation of the augmented model (in the "vertical" parts). We consider the Cauchy problem within the class of graph solutions. The graph solution to the Cauchy problem is constructed by completion of the discontinuities of the entropy solution. The uniqueness is established by applying a general uniqueness theorem due to Baiti, LeFloch, and Piccoli. The proposed geometric framework illustrates the importance of the uniform distance between graphs to deal with solutions of nonlinear hyperbolic problems.
42 citations
TL;DR: In this article, the authors proposed a new stability condition, the reduced stability condition which is weaker than the previous one, but still has the property to imply the subcharacteristic or interlacing conditions, and the dissipativity of the leading term in the Chapman-Enskog expansion.
Abstract: We consider multidimensional hyperbolic systems of conservation laws with relaxation, together with their associated limit systems. A strong stability condition for such asymptotics has been introduced by Chen, Levermore and Liu, namely the existence of an entropy extension. We propose here a new stability condition, the reduced stability condition, which is weaker than the previous one, but still has the property to imply the subcharacteristic or interlacing conditions, and the dissipativity of the leading term in the Chapman–Enskog expansion. This reduced stability condition has the advantage of involving only the submanifold of equilibria, or maxwellians, so that it is much easier to check than the entropy extension condition. Our condition generalizes the one introduced by the author in the case of kinetic, i.e. diagonal semilinear relaxation. We provide an adapted stability analysis in the context of approximate Riemann solvers obtained via relaxation systems.
42 citations
TL;DR: In this paper, the Coulomb Hamiltonian with ultraviolet and infrared cutoffs was shown to be self-adjoint and has a ground state for sufficiently small coupling constants, and it was shown that the Hamiltonian is selfadjoint with respect to the current density with transversal photons.
Abstract: We consider a Hamiltonian with ultraviolet and infrared cutoffs, describing the interaction of relativistic electrons and positrons in the Coulomb potential with photons in Coulomb gauge. The interaction includes both interaction of the current density with transversal photons and the Coulomb interaction of charge density with itself. We prove that the Hamiltonian is self-adjoint and has a ground state for sufficiently small coupling constants.
41 citations
TL;DR: The FORCE scheme is shown to be consistent with the Lax entropy inequality, that is, the limit functions of the FORCE approximate solutions are entropy solutions.
Abstract: A hierarchy of centered (non-upwind) difference schemes is identified for solving hyperbolic equations. The bottom of the hierarchy is the classical Lax–Friedrichs scheme, which is the least accurate in computation, and the top of the hierarchy is the FORCE scheme, which is the optimal scheme in the family. The FORCE scheme is optimal in the sense that it is monotone, has the optimal stability condition for explicit methods, and has the smallest numerical viscosity. It is shown that the FORCE scheme is consistent with the Lax entropy inequality, that is, the limit functions of the FORCE approximate solutions are entropy solutions. The convergence of the FORCE scheme is also established for the isentropic Euler equations and the shallow water equations. Some related centered difference schemes are also surveyed and discussed.
TL;DR: In this paper, the kinematics of one-dimensional motion have been applied to construct an evolution equation for strong cylindrical and spherical shock waves propagating into a low density gas at rest.
Abstract: Converging shock waves in an almost ideal medium are considered. The kinematics of one-dimensional motion have been applied to construct an evolution equation for strong cylindrical and spherical shock waves propagating into a low density gas at rest. The approximate value of the similarity parameter obtained from there is compared with those derived from Whitham's Rule and the exact similarity solution at the instant of collapse of the shock wave. The above computation is carried out for different values of the parameter α, which depends on the internal volume of the gas molecules.
TL;DR: In this article, the Ricci defects of microlocalized solutions of the Einstein vacuum equations were proven and used in the proof of the crucial Asymptotics Theorem in [1].
Abstract: This is the third and last in our series of papers concerning rough solutions of the Einstein vacuum equations expressed relative to wave coordinates. In this paper we prove an important result, concerning Ricci defects of microlocalized solutions, stated and used in the proof of the crucial Asymptotics Theorem in [1]. The result is of independent interest in situations when one has to smooth out the given spacetime metric in order to achieve good causality properties.
TL;DR: In this article, the authors consider an n×n system of hyperbolic balance laws with coinciding shock and rarefaction curves and prove the well-posedness of this system, provided there exists a domain that is invariant both with respect to the homogeneous conservation law and to the ordinary differential system generated by the right-hand side.
Abstract: Consider an n×n system of hyperbolic balance laws with coinciding shock and rarefaction curves. This note proves the well-posedness in the large of this system, provided there exists a domain that is invariant both with respect to the homogeneous conservation law and to the ordinary differential system generated by the right-hand side. No "non-resonance" hypothesis is assumed.
TL;DR: Canic et al. as discussed by the authors proposed a method for solving free boundary problems for quasilinear degenerate elliptic equations which arise when shocks interact with the subsonic (nonhyperbolic) part of the solution.
Abstract: Self-similar reduction of an important class of two-dimensional conservation laws leads to boundary value problems for equations which change type. We have established a method for solving free boundary problems for quasilinear degenerate elliptic equations which arise when shocks interact with the subsonic (nonhyperbolic) part of the solution. This paper summarizes the principal features of the method. A preliminary version of these notes formed the basis of a series of three lectures at the Newton Institute in April, 2003. They are a report of research carried out jointly with Suncica Canic, Eun Heui Kim and Gray Lieberman.
TL;DR: In this paper, the axially symmetric piston problem for compressible fluids was studied under the assumption that both the velocity of the piston and the density of the gas outside the piston are small, and the global existence of a shock front solution was proved by using a modified Glimm scheme.
Abstract: In this paper, we study the axially symmetric piston problem for compressible fluids when the velocity of the piston is a perturbation of a constant. Under the assumptions that both the velocity of the piston and the density of the gas outside the piston are small, we prove the global existence of a shock front solution by using a modified Glimm scheme.
TL;DR: In this paper, a local in time existence theorem and continuation criteria for cosmological solutions of the Einstein-Vlasov-scalar field system, with the sources generated by a distribution function and a scalar field, subject to the Vlasov and wave equations respectively, were proved for spherical, plane and hyperbolic symmetry.
Abstract: We prove in the cases of spherical, plane and hyperbolic symmetry a local in time existence theorem and continuation criteria for cosmological solutions of the Einstein–Vlasov-scalar field system, with the sources generated by a distribution function and a scalar field, subject to the Vlasov and wave equations respectively. This system describes the evolution of self-gravitating collisionless matter and scalar waves within the context of general relativity. In the case where the only source is a scalar field it is shown that a global existence result can be deduced from the general theorem.
TL;DR: In this paper, the existence of a canonical foliation on a null incoming hypersurface was proved and it was used for the construction of asymptotically flat global Einstein spacetimes, with appropriate initial conditions.
Abstract: We prove the existence of a "canonical foliation" on a null incoming hypersurface. Here "canonical" denotes a foliation whose level function is a solution of a given nonlinear system of equations. The existence of this foliation is an important ingredient for the construction of asymptotically flat global Einstein spacetimes, with appropriate initial conditions.
TL;DR: In this paper, a front tracking method for scalar conservation laws with source term is proposed and proved to converge to the solution of the Riemann problem in the front tracking procedure.
Abstract: We propose and prove convergence of a front tracking method for scalar conservation laws with source term. The method is based on writing the single conservation law as a 2×2 quasilinear system without a source term, and employ the solution of the Riemann problem for this system in the front tracking procedure. In this way the source term is processed in the Riemann solver, and one avoids using operator splitting. Since we want to treat the resonant regime, classical arguments for bounding the total variation of numerical solutions do not apply here. Instead compactness of a sequence of front tracking solutions is achieved using a variant of the singular mapping technique invented by Temple [69]. The front tracking method has no CFL-condition associated with it, and it does not discriminate between stiff and non-stiff source terms. This makes it an attractive approach for stiff problems, as is demonstrated in numerical examples. In addition, the numerical examples show that the front tracking method is able to preserve steady-state solutions (or achieving them in the long time limit) with good accuracy.
TL;DR: In this article, two new classes of examples of sup-norm blowup in finite time for strictly hyperbolic systems of conservation laws are considered, where the explosive growth in amplitude is caused either by a gradient catastrophe or by a singularity in the flux function.
Abstract: We consider two new classes of examples of sup-norm blowup in finite time for strictly hyperbolic systems of conservation laws. The explosive growth in amplitude is caused either by a gradient catastrophe or by a singularity in the flux function. The examples show that solutions of uniformly strictly hyperbolic systems can remain as smooth as the initial data until the time of blowup. Consequently, blowup in amplitude is not necessarily strictly preceded by shock formation.
TL;DR: For non-conservative hyperbolic systems several definitions of shock waves have been introduced in the literature as mentioned in this paper, and all of them agree to third order near a given state, which is a generalization of the result of Bianchini and Bressan.
Abstract: For non-conservative hyperbolic systems several definitions of shock waves have been introduced in the literature. In this paper, we propose a new and simple definition in the case of genuinely nonlinear fields. Relying on a vanishing viscosity process we prove the existence of shock curves for viscosity matrix commuting with the matrix of the hyperbolic system. This setting generalizes a recent result by Bianchini and Bressan. Furthermore we prove that all definitions agree to third order near a given state.
TL;DR: In this paper, the authors studied the asymptotic behavior of entropy solutions of the Cauchy problem for multi-dimensional systems of conservation laws of the form, where the gα are real smooth functions defined in [0,+∞], and when the initial data are perturbations of two-state nonplanar Riemann data.
Abstract: We study the asymptotic behavior of entropy solutions of the Cauchy problem for multi-dimensional systems of conservation laws of the form , where the gα are real smooth functions defined in [0,+∞), and when the initial data are perturbations of two-state nonplanar Riemann data. Specifically, if R0(x) is such Riemann data and ψ∈L∞(ℝd)n satisfies ψ(Tx)→0 in , as T→∞, then an entropy solution, u(x,t), of the Cauchy problem with u(x,0)=R0(x)+ψ(x) satisfies u(ξt,t)→R(ξ) in , as t→∞, where R(x/t) turns out to be the unique self-similar entropy solution of the corresponding Riemann problem.
TL;DR: In this article, the limiting behavior of the solution of a scalar conservation law with slow diffusion and fast bistable reaction is considered, and it is proved that solutions with monotone initial data approach the shock layer or rarefaction layer waves as time goes to infinity.
Abstract: The limiting behavior of the solution of a scalar conservation law with slow diffusion and fast bistable reaction is considered. In a short time the solution develops transition patterns connected by shock layers and rarefaction layers, when the initial data has finitely many monotone pieces. The existence and uniqueness of the front profiles for both shock layers and rarefaction layers are established. A variational characterization of wave speeds of these profiles is derived. These profiles are shown to be stable. Furthermore, it is proved that solutions with monotone initial data approach the shock layer or rarefaction layer waves as time goes to infinity.
TL;DR: In this paper, a thermodynamically consistent Galilean-invariant formalization of the elasticity equations is proposed, where conservation laws are valid only on a solution with special initial values.
Abstract: A thermodynamically consistent Galilean-invariant formalization of the elasticity equations is proposed. The equations under consideration form a symmetric hyperbolic system but they are non-conservative. The conservation laws are valid only on a solution with special initial values. Difficulties arising while introducing the dissipative terms are briefly discussed.
TL;DR: In this article, the authors studied the evolution of the energy density of a sequence of solutions to the Kelvin-Voigt viscoelasticity equation and proved that, in the zone where the viscosity matrix is invertible, this term prevents propagation of concentation and oscillation effects contrary to what happens in the wave equation.
Abstract: In this paper, we study the evolution of the energy density of a sequence of solutions to the Kelvin–Voigt viscoelasticity equation. We do not suppose lower bounds on the non-negative viscosity matrix. We prove that, in the zone where the viscosity matrix is invertible, this term prevents propagation of concentation and oscillation effects contrary to what happens in the wave equation. We calculate precisely the weak limit of the energy density in terms of microlocal defect measures associated with the initial data under the assumption that the oscillations of the data are not microlocally localized on directions which are in the kernel of the viscosity matrix.
TL;DR: This work proves the convergence of numerical approximations of compressive solutions for scalar conservation laws with convex flux with fully discrete approach and proves a rate of convergence in O(Δx) uniformly in time, if the initial data is a shock, or asymptotically after the compression of the initial profile.
Abstract: We prove the convergence of numerical approximations of compressive solutions for scalar conservation laws with convex flux. This new proof of convergence is fully discrete and does not use Kuznetsov's approach. We recover the well-known rate of convergence in O(Δx½). With the same fully discrete approach, we also prove a rate of convergence in O(Δx) uniformly in time, if the initial data is a shock, or asymptotically after the compression of the initial profile. Numerical experiments confirm the theoretical analysis.
TL;DR: In this article, the limit energy density of solutions of the linear wave equation in a thin three-dimensional domain, if the wavelength of the Cauchy data is bounded from below by the thickness of the domain, was derived.
Abstract: We compute the limit energy density of solutions of the linear wave equation in a thin three-dimensional domain, if the wavelength of the Cauchy data is bounded from below by the thickness of the domain. As an application, we obtain a geometric criterion for the uniform observability of solutions of a damped wave equation on such a domain.
TL;DR: In this paper, an advanced hyperbolic divergence cleaning scheme based on generalized Lagrange multiplier (GLM) for the equations of shallow water magnetohydrodynamics (SMHD) is presented.
Abstract: An advanced hyperbolic divergence cleaning scheme based on "generalized Lagrange multiplier" (GLM) for the equations of "shallow water magnetohydrodynamics" (SMHD) is presented. This scheme is based on the two-step method which is comprised of the standard finite-volume updating step for the nonlinear genuine SMHD system and the divergence cleaning step for the linear GLM-based Maxwell subsystem. The divergence cleaning step can be applied several times per each computational time step, in order to accelerate the transports of the divergence error out of the computational domain. The presented two-step method is compared with the standard GLM method based on operator splitting. It is shown that the standard operator-splitting based method has the shock dissipation problem, particularly when the multiple subcycles of the divergence cleaning step is performed per each time step. On the contrary, the introduction of the multiple subcycles for the new GLM–Maxwell subsystem does not suffer from the dissipation of the shocks, and produces better shock resolution. The presented method can be further applied to the full magnetohydrodynamics equations.
TL;DR: In this paper, the authors prove the global existence and uniqueness of piecewise C1 solution containing n small amplitude waves (shocks corresponding to genuinely nonlinear fields and contact discontinuities corresponding to linearly degenerate fields) for the generalized Riemann problem associated with quasilinear hyperbolic systems of conservation laws.
Abstract: Assuming that each characteristic field is either genuinely nonlinear or linearly degenerate in the sense of Lax, we prove the global existence and uniqueness of piecewise C1 solution containing n small amplitude waves (shocks corresponding to genuinely nonlinear fields and contact discontinuities corresponding to linearly degenerate fields) for the generalized Riemann problem associated with quasilinear hyperbolic systems of conservation laws with small, decaying, piecewise C1 initial data. The solution possesses a global structure similar to that of the similarity solution to the corresponding Riemann problem.
TL;DR: In this article, the authors construct approximate solutions of the initial value problem for dynamical phase transition problems via a variational scheme in one space dimension, and prove the convergence of approximate solutions to a weak solution which satisfies the natural dissipation inequality.
Abstract: We construct approximate solutions of the initial value problem for dynamical phase transition problems via a variational scheme in one space dimension. First, we deal with a local model of phase transition dynamics which contains second and third order spatial derivatives modeling the effects of viscosity and surface tension. Assuming that the initial data are periodic, we prove the convergence of approximate solutions to a weak solution which satisfies the natural dissipation inequality. We note that this result still holds for non-periodic initial data. Second, we consider a model of phase transition dynamics with only Lipschitz continuous stress–strain function which contains a non-local convolution term to take account of surface tension. We also establish the existence of weak solutions. In both cases the proof relies on implicit time discretization and the analysis of a minimization problem at each time step.