We consider the initial value problem for a pseudodifferential equation with first order hyperbolic part, and an order $γ> 0$ dissipative term. Under an assumption, depending on an integer parameter $L \geq 2$ such that $2 γ< L$, we construct for this initial value problem a parametrix that is a Fourier integral operator of type $ρ= 1 - γ/L$. The assumption implies that where the principal symbol of the dissipative term is zero, the terms of order up to $L-1$ in its Taylor series also vanish.
We construct a semi-classical parametrix for the Laplacian on non-trapping asymptotically hyperbolic manifolds, which generalizes the construction of Melrose, Sa Barreto and Vasy. As applications, we obtain high energy resolvent estimates and resonance free strips. Also, we prove that the Friedlander radiation field decays exponentially for initial data in proper spaces.
In this paper we construct a parametrix for the high-energy asymptotics of the analytic continuation of the resolvent on a Riemannian manifold which is a small perturbation of the Poincaré metric on hyperbolic space. As a result, we obtain non-trapping high energy estimates for this analytic continuation.
We consider the Cauchy problem and the source problem for normally hyperbolic operators on the Minkowski spacetime, and study the determination of solutions from their integrals along null geodesics. For the Cauchy problem, we give a new proof of the stable determination result obtained in Vasy and Wang [12]. For the source problem, we obtain stable determination for sources with space-like singularities. Our proof is based on the microlocal analysis of the normal operator of the light ray transform composed with the parametrix for strictly hyperbolic operators.
We consider mildly degenerate Kirchhoff equations with a small parameter and a weak dissipation term. We prove the existence of global solutions when the parameter is small with respect to the size of initial data. Then we provide global-in-time error estimates on the difference between the solution of our problem and the solution of the corresponding first order problem.
We consider Kirchhoff equations with a small parameter epsilon in front of the second-order time-derivative, and a dissipative term whose coefficient may tend to 0 as t -> + infinity (weak dissipation). In this note we present some recent results concerning existence of global solutions, and their asymptotic behavior both as t -> + infinity and as epsilon -> 0. Since the limit equation is of parabolic type, this is usually referred to as a hyperbolic-parabolic singular perturbation problem. We show in particular that the equation exhibits hyperbolic or parabolic behavior depending on the value…
We consider the hyperbolic-parabolic singular perturbation problem for a nondegenerate quasilinear equation of Kirchhoff type with weak dissipation. This means that the dissipative term is multiplied by a coefficient b(t) which tends to 0 as t tends to +infinity. The case where b(t) behaves like (1+t)^{-p} with p<1 has recently been considered. The result is that the hyperbolic problem has a unique global solution, and the difference between solutions of the hyperbolic problem and the corresponding solutions of the parabolic problem converges to zero both as t tends to +infinity and as epsilon…
We consider a family of Kirchhoff equations with a small parameter epsilon in front of the second-order time-derivative, and a dissipation term with a coefficient which tends to 0 as t -> +infinity. It is well-known that, when the decay of the coefficient is slow enough, solutions behave as solutions of the corresponding parabolic equation, and in particular they decay to 0 as t -> +infinity. In this paper we consider the nondegenerate and coercive case, and we prove optimal decay estimates for the hyperbolic problem, and optimal decay-error estimates for the difference between solutions of th…
We consider an abstract second order linear equation with a strong dissipation, namely a friction term which depends on a power of the "elastic" operator. In the homogeneous case, we investigate the phase spaces in which the initial value problem gives rise to a semigroup, and the further regularity of solutions. In the non-homogeneous case, we study how the regularity of solutions depends on the regularity of forcing terms, and we characterize the spaces where a bounded forcing term yields a bounded solution. What we discover is a variety of different regimes, with completely different behavi…
We consider the hyperbolic-parabolic singular perturbation problem for a degenerate quasilinear Kirchhoff equation with weak dissipation. This means that the coefficient of the dissipative term tends to zero when t tends to +infinity. We prove that the hyperbolic problem has a unique global solution for suitable values of the parameters. We also prove that the solution decays to zero, as t tends to +infinity, with the same rate of the solution of the limit problem of parabolic type.
In the context of hyperbolic systems of balance laws, the Shizuta-Kawashima coupling condition guarantees that all the variables of the system are dissipative even though the system is not totally dissipative. Hence it plays a crucial role in terms of sufficient conditions for the global in time existence of classical solutions. However, it is easy to find physically based models that do not satisfy this condition, especially in several space dimensions. In this paper, we consider two simple examples of partially dissipative hyperbolic systems violating the Shizuta-Kawashima condition ([SK]) i…
Here we prove a global existence theorem for the solutions of the semi-linear wave equation with critical non-linearity admitting a positive definite Hamiltonian. Formulating a parametrix for the wave equation in a globally hyperbolic curved spacetime, we derive an apriori pointwise bound for the solution of the nonlinear wave equation in terms of the initial energy, from which the global existence follows in a straightforward way. This is accomplished in two steps. First, based on Moncrief's light cone formulation we derive an expression for the scalar field in terms of integrals over the pas…
We study the time-asymptotic behavior of linear hyperbolic systems under partial dissipation which is localized in suitable subsets of the domain. More precisely, we recover the classical decay rates of partially dissipative systems satisfying the stability condition (SK) with a time-delay depending only on the velocity of each component and the size of the undamped region. To quantify this delay, we assume that the undamped region is a bounded space-interval and that the system without space-restriction on the dissipation satisfies the stability condition (SK). The former assumption ensures t…
We investigate the asymptotic behavior, as t goes to infinity, for a semilinear hyperbolic equation with asymptotically smal dissipation and convex potential. We prove that if the damping term behaves like K/t^αfor t large enough, k>0 and 0</alpha<1 then every global solution converges weakly to an equilibrium point. This result is a positive answer to a question left open in the paper [A. Cabot and P. Frankel, Asymptotics for some semilinear hyperbolic equation with non-autonomous damping. J. Differential Equations 252 (2012) 294-322.]
We consider a multi-dimensional scalar wave equation with memory corresponding to the viscoelastic material described by a generalized Zener model. We deduce that this relaxation system is an example of a non-strictly hyperbolic system satisfying Majda's block structure condition. Well-posedness of the associated Cauchy problem is established by showing that the symbol of the spatial derivatives is uniformly diagonalizable with real eigenvalues. A long-time stability result is obtained by plane-wave analysis when the memory term allows for dissipation of energy.
We study the vanishing dissipation limit of the three-dimensional (3D) compressible Navier-Stokes-Fourier equations to the corresponding 3D full Euler equations. Our results are twofold. First, we prove that the 3D compressible Navier-Stokes-Fourier equations admit a family of smooth solutions that converge to the planar rarefaction wave solution of the 3D compressible Euler equations with arbitrary strength. Second, we obtain a uniform convergence rate in terms of the viscosity and heat-conductivity coefficients. For this multi-dimensional problem, we first need to introduce the hyperbolic wa…
Hyperbolic-parabolic systems have spatially homogenous stationary states. When the dissipation is weak, one can derive weakly nonlinear-dissipative approximations that govern perturbations of these constant states. These approximations are quadratically nonlinear. When the original system has an entropy, the approximation is formally dissipative in a natural Hilbert space. We show that when the approximation is strictly dissipative it has global weak solutions for all initial data in that Hilbert space. We also prove a weak-strong uniqueness theorem for it. In addition, we give a Kawashima typ…
We consider a restricted Dirichlet-to-Neumann map associated to a wave type operator on a Riemannian manifold with boundary. The restriction corresponds to the case where the Dirichlet traces are supported on one subset of the boundary and the Neumann traces are restricted on another subset. We show that the restricted Dirichlet-to-Neumann map determines the lower order terms in the wave equation, up the natural gauge invariances, along a convex foliation of the manifold. We allow the lower order terms to be non-self-adjoint, and in particular, the corresponding physical system may have dissip…
This paper is concerned with quasilinear systems of partial differential equations consisting of two hyperbolic operators interacting dissipatively. Its main theorem establishes global-in-time existence and asymptotic stability of strong solutions to the Cauchy problem close to homogeneous reference states. Notably, the operators are not required to be symmetric hyperbolic, instead merely the existence of symbolic symmetrizers is assumed. The dissipation is characterized by conditions equivalent to the uniform decay of all Fourier modes at the reference state. On a technical level, the theory …
The global dynamics and regularity of parabolic-hyperbolic systems is an interesting topic in PDEs due to the coupling of competing dissipation and hyperbolic effects. This paper is concerned with the Cauchy problem of a parabolic-hyperbolic system derived from a chemotaxis model describing the dynamics of the initiation of tumor angiogenesis. It is shown that, as time tends to infinity, the Cauchy problem with large-amplitude discontinuous data admit global weak solutions which converge to a constant state (resp. a viscous shock wave) if the asymptotic states of initial values at far field ar…
In the context of some bidimensionnal Navier-Stokes model, we exhibit a family of exact oscillating solutions $\{u_ε\}_ε$ defined on some strip $[0,T]\times\R^2$ which does not depend on $ε\in]0,1]$. The exact solutions is described thanks to a complete expansions which reveal a boundary layer in time $t=0$. The interactions of the various scales (1, $1/ε$ and $1/ε^2$) produce a macroscopic effect given by the addition of a diffusion. To justify the existence of $\{u_ε\}_ε$, we need to perform various Sobolev estimates that rely on a refined balance between the informations coming from the hyp…
This article deals with relaxation approximations of nonlinear systems of hyperbolic balance laws. We introduce a class of relaxation schemes and establish their stability and convergence to the solution of hyperbolic balance laws before the formation of shocks, provided that we are within the framework of the compensated compactness method. Our analysis treats systems of hyperbolic balance laws with source terms satisfying a special mechanism which induces weak dissipation in the spirit of Dafermos [C.M. Dafermos J. Hyp. Diff. Equations, 3, 505-527, 2006], as well as hyperbolic balance laws w…
In the present article, we consider a thermoelastic plate of Reissner-Mindlin-Timoshenko type with the hyperbolic heat conduction arising from Cattaneo's law. In the absense of any additional mechanical dissipations, the system is often not even strongly stable unless restricted to the rotationally symmetric case, etc. We present a well-posedness result for the linear problem under general mixed boundary conditions for the elastic and thermal parts. For the case of a clamped, thermally isolated plate, we show an exponential energy decay rate under a full damping for all elastic variables. Rest…
We consider a degenerate abstract wave equation with a time-dependent propagation speed. We investigate the influence of a strong dissipation, namely a friction term that depends on a power of the elastic operator. We discover a threshold effect. If the propagation speed is regular enough, then the damping prevails, and therefore the initial value problem is well-posed in Sobolev spaces. Solutions also exhibit a regularizing effect analogous to parabolic problems. As expected, the stronger is the damping, the lower is the required regularity. On the contrary, if the propagation speed is not re…
We consider an abstract second order evolution equation with damping. The "elastic" term is represented by a self-adjoint nonnegative operator A with discrete spectrum, and the nonlinear term has order greater than one at the origin. We investigate the asymptotic behavior of solutions. We prove the coexistence of slow solutions and fast solutions. Slow solutions live close to the kernel of A, and decay as negative powers of t as solutions of the first order equation obtained by neglecting the operator A and the second order time-derivatives in the original equation. Fast solutions live close t…
Here we develop a method for investigating global strong solutions of partially dissipative hyperbolic systems in the critical regularity setting. Compared to the recent works by Kawashima and Xu, we use hybrid Besov spaces with different regularity exponent in low and high frequency. This allows to consider more general data and to track the exact dependency on the dissipation parameter for the solution. Our approach enables us to go beyond the L^2 framework in the treatment of the low frequencies of the solution, which is totally new, to the best of our knowledge. Focus is on the one-dimensi…
We consider an initial-boundary-value problem for a thermoelastic Kirchhoff & Love plate, thermally insulated and simply supported on the boundary, incorporating rotational inertia and a quasilinear hypoelastic response, while the heat effects are modeled using the hyperbolic Maxwell-Cattaneo-Vernotte law giving rise to a 'second sound' effect. We study the local well-posedness of the resulting quasilinear mixed-order hyperbolic system in a suitable solution class of smooth functions mapping into Sobolev $H^{k}$-spaces. Exploiting the sole source of energy dissipation entering the system throu…
In this paper, we study the hypocoercivity for a class of linear kinetic equations with both transport and degenerately dissipative terms. As concrete examples, the relaxation operator, Fokker-Planck operator and linearized Boltzmann operator are considered. By constructing equivalent temporal energy functionals, time rates of the solution approaching equilibrium in some Hilbert spaces are obtained when the spatial domain takes the whole space or torus and when there is a confining force or not. The main tool of the proof is the macro-micro decomposition combined with Kawashima's argument on d…
We establish the following well-posedness results on Ericksen-Leslie's parabolic-hyperbolic liquid crystal model: 1, if the dissipation coefficients β= μ_4 - 4 μ_6 > 0, and the size of the initial energy E^{in} is small enough, then the life span of the solution is at least -O(\ln E^{in}); 2, for the special case that the coefficients μ_1 = μ_2 = μ_3 = μ_5 = μ_6 = 0, for which the model is the Navier-Stokes equations coupled with the wave map from \mathbb{R}^n to \mathbb{S}^2, the same existence result holds but without the smallness restriction on the size of the initial data; 3, with further…
Numerical methods with controlled dissipation for small-scale dependent shocks
Laboratoire Jacques-Louis Lions and Centre National de la Recherche Scientifique Université Pierre et Marie Curie（雅克-路易·利翁斯实验室与法国国家科学研究中心皮埃尔和玛丽·居里大学）；Seminar for Applied Mathematics D-Math Eidgenössische Technische Hochschule HG（苏黎世联邦理工学院应用数学研讨会）
In this paper we study the zero inertia limit that is from the hyperbolic to parabolic Ericksen-Leslie's liquid crystal flow. By introducing an initial layer and constructing an energy norm and energy dissipation functional depending on the solutions of the limiting system, we derive a global in time uniform energy bound to the remainder system under the small size of the initial data.
This paper shows that in second-order hyperbolic systems of partial differential equations proposed in authors' earlier paper (J. Math. Phys. 59 (2018)) for modelling the relativistic dynamics of barotropic fluids in the presence of viscosity and heat conduction, shock waves of arbitrary strength have smooth, monotone dissipation profiles. The results and arguments extend classical considerations of Weyl (Comm. Pure Appl. Math. 2 (1949)) and Gilbarg (Amer. J. Math. 73 (1951)) to the relativistic setting.
Coupling techniques for nonlinear hyperbolic equations. III. The well-balanced approximation of thick interfaces
Institut de Recherche Mathématiques de Rennes, Université de Rennes I（雷恩数学研究所，雷恩第一大学）；Centre de Mathématiques Appliquées & Centre National de la Recherche Scientifique, Ecole Polytechnique（应用数学中心与国家科学研究中心，巴黎综合理工学院）；Laboratoire Jacques–Louis Lions & Centre National de la Recherche Scientifique, Université Pierre et Marie Curie (Paris 6)（雅克-路易·利翁斯实验室与国家科学研究中心，皮埃尔和玛丽·居里大学（巴黎第六大学））
