Boundary behavior of solutions of a class of genuinely nonlinear hyperbolic systems
Julian Gevirtz
备注：
43 pages. This second version is largely the same as the first. The changes involve the correction of a number of errors, incorporation of some simplifications and more detailed proofs of a few of the propositions
For a certain class of genuinely nonlinear two-by-two planar hyperbolic systems we show that any classical solution on a smoothly bounded domain has nontangential boundary limits except on a set whose Hausdorff dimension is bounded by some system-dependent constant which is strictly less than 1 and, on the other hand, that for any system of the kind considered there is in fact a solution on a half-plane which fails to have nontangential limits at a set of boundary points of positive Hausdorff dimension.
The paper gives an accurate description of the qualitative structure of an admissible BV solution to a strictly hyperbolic, piecewise genuinely nonlinear system of conservation laws. We prove that there are a countable set $Θ$ which contains all interaction points and a family of countably many Lipschitz curves $\T$ such that outside $\T\cup Θ$ $u$ is continuous, and along the curves in $\T$, u has left and right limit except for points in $Θ$. This extends the corresponding structural result in \cite{BL,Liu1} for admissible solutions. The proof is based on approximate wave-front tracking solu…
We prove that if $t \mapsto u(t) \in \mathrm {BV}(\R)$ is the entropy solution to a $N \times N$ strictly hyperbolic system of conservation laws with genuinely nonlinear characteristic fields \[ u_t + f(u)_x = 0, \] then up to a countable set of times $\{t_n\}_{n \in \mathbb N}$ the function $u(t)$ is in $\mathrm {SBV}$, i.e. its distributional derivative $u_x$ is a measure with no Cantorian part. The proof is based on the decomposition of $u_x(t)$ into waves belonging to the characteristic families \[ u(t) = \sum_{i=1}^N v_i(t) \tilde r_i(t), \quad v_i(t) \in \mathcal M(\R), \ \tilde r_i(t) \…
We provide a precise description of the set of residual boundary conditions generated by the self-similar viscous approximation introduced by Dafermos et al. We then apply our results, valid for both conservative and non conservative systems, to the analysis of the boundary Riemann problem and we show that, under appropriate assumptions, the limits of the self-similar and the classical vanishing viscosity approximation coincide. We require neither genuinely nonlinearity nor linear degeneracy of the characteristic fields.
Existence, uniqueness and stability of kinetic and entropy solutions to the boundary value problem for the Kolmogorov-type genuinely nonlinear ultra-parabolic equation with a smooth source term is established. After this, we consider the case when the source term contains a small positive parameter and collapses to the Dirac delta-function, as this parameter tends to zero. In this case, the limiting passage from the original equation with the smooth source to the impulsive ultra-parabolic equation is fulfilled and rigorously justified. The proofs rely on the method of kinetic equation and on t…
We investigate inverse boundary problems associated with a time-dependent semilinear hyperbolic equation, where both nonlinearity and sources (including initial displacement and initial velocity) are unknown. We establish in several generic scenarios that one can uniquely determine the nonlinearity or the sources by using passive or active boundary observations. In order to exploit the nonlinearity and the sources simultaneously, we develop a new technique, which combines the observability for linear wave equations and a Runge approximation with higher order linearization for the semilinear hy…
This paper is concerned with a class of nonlinear reaction-hyperbolic systems as models for axonal transport in neuroscience. We show the global existence of entropy-satisfying BV-solutions to the initial-value problems by using hyperbolic-type methods. Moreover, we rigorously justify the limit as the biochemical processes are much faster than the transport ones.
We consider a strictly hyperbolic, genuinely nonlinear system of conservation laws in one space dimension. A sharp decay estimate is proved for the positive waves in an entropy weak solution. The result is stated in terms of a partial ordering among positive measures, using symmetric rearrangements and a comparison with a solution of Burgers' equation with impulsive sources.
We study the long time behavior of radial solutions to nonlinear Schrödinger equations on hyperbolic space. We show that the usual distinction between short range and long range nonlinearity is modified: the geometry of the hyperbolic space makes every power-like nonlinearity short range. The proofs rely on weighted Strichartz estimates, which imply Strichartz estimates for a broader family of admissible pairs, and on Morawetz type inequalities. The latter are established without symmetry assumptions.
Here a mixed problem for a nonlinear hyperbolic equation with Neumann boundary value condition is investigated, and a priori estimations for the possible solutions of the considered problem are obtained. These results demonstrate that any solution of this problem possess certain smoothness properties.
We study a class of initial boundary value problems of hyperbolic type. A new topological approach is applied to prove the existence of non-negative classical solutions. The arguments are based upon a recent theoretical result.
We suggest a new model for the dynamics of a suspension bridge through a system of nonlinear nonlocal hyperbolic differential equations. The equations are of second and fourth order in space and describe the behavior of the main components of the bridge: the deck, the sustaining cables and the connecting hangers. We perform a careful energy balance and we derive the equations from a variational principle. We then prove existence and uniqueness for the resulting problem.
We study existence of weak solutions for certain classes of nonlinear Schrödinger equations on the Poincaré ball model $\mathbb{B}^N$, $N\geq 3$. By using the Palais principle of symmetric criticality and suitable group theoretical arguments, we establish the existence of a nontrivial (weak) solution.
The propagation of analyticity for a solution u(t,x) to a nonlinear weakly hyperbolic equation of order m, means that if u, and its time derivatives up to the order m-1, are analytic in the space variables x at the initial time, then they remain analytic for any time. Here we prove that such a property holds for the solutions bounded in C-infinity of a special class of homogeneous equations in one space variable, with time dependent coefficient.
In this paper, we are concerned with a quasi-linear hyperbolic-parabolic system of persistence and endogenous chemotaxis modelling vasculogenesis in $\mathbb{R}$. Under some suitable structural assumption on the pressure function, we first predict the system admits a nonlinear diffusion wave in $\mathbb{R}$ based on the empirical results in the literature. Then we show that the solution of the concerned system will locally and asymptotically converges to this nonlinear diffusion wave if the wave strength is small. By using the time-weighted energy estimates, we further prove that the convergen…
The class of $2\times 2$ nonlinear hyperbolic systems with one genuinely nonlinear field and one linearly degenerate field are considered. Existence of global weak solutions for small initial data in fractional BV spaces $BV^s$ is proved. The exponent $s$ is related to the usual fractional Sobolev derivative. Riemann invariants $w$ and $z$ corresponding respectively to the genuinely nonlinear component and to the linearly degenerate component play different key roles in this work. We obtain the existence of a global weak solution provided that the initial data written in Riemann coordinates $ …
The behavior of sufficiently regular solutions to semilinear hyperbolic equations has attracted a great deal of attention in the past decades, concerning local/global existence, finite time blow-up, critical exponents, and propagation of singularities. Solutions of lower regularity may exhibit unexpected (anomalous) propagation of singularities. The purpose of this paper is to present various striking examples that seemingly have not been addressed in the literature so far. The key issue is the interpretation of the nonlinear operations.
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…
In this paper we deal with diffusive relaxation limits of nonlinear systems of Euler type modeling chemotactic movement of cells toward Keller--Segel type systems. The approximating systems are either hyperbolic--parabolic or hyperbolic--elliptic. They all feature a nonlinear pressure term arising from a \emph{volume filling effect} which takes into account the fact that cells do not interpenetrate. The main convergence result relies on compensated compactness tools and is obtained for large initial data under suitable assumptions on the approximating solutions. In order to justify such assump…
Haar method, averaged matrix, wave cancellations, and L1 stability for hyperbolic systems
Laboratoire Jacques-Louis Lions & Centre National de la Recherche Scientifique (CNRS), Université Pierre et Marie Curie (Paris 6)（雅克-路易·利翁斯实验室 & 法国国家科学研究中心，皮埃尔和玛丽·居里大学）
We consider the inverse problem of determining a general nonlinear term appearing in a semilinear hyperbolic equation on a Riemannian manifold with boundary $(M,g)$ of dimension $n=2,3$. We prove results of unique recovery of the nonlinear term $F(t,x,u)$, appearing in the equation $\partial_t^2u-Δ_gu+F(t,x,u)=0$ on $(0,T)\times M$ with $T>0$, from some partial knowledge of the solutions $u$ on the boundary of the time-space cylindrical manifold $(0,T)\times M$ or on the lateral boundary $(0,T)\times\partial M$. We determine the expression $F(t,x,u)$ both on the boundary $x\in\partial M$ and i…
We study the existence and stability of ground state solutions or solitons to a nonlinear stationary equation on hyperbolic space. The method of concentration compactness applies and shows that the results correlate strongly to those of Euclidean space.
Investigations of spherically symmetric motions of self-gravitating gaseous stars governed by the non-relativistic Newtonian gravitation theory or by the general relativistic theory lead us to a certain type of non-linear hyperbolic equations defined on a finite interval of the space variable. The linearized principal part has regular singularities at the both ends of the interval of the space variable. But the regularity loss caused by the singularities at the boundaries requires application of the Nash-Moser technique. An abstract unified treatment of the problem is presented.
In this paper, a nonlinear inverse boundary value problem for the second-order hyperbolic equation with nonlocal conditions is studied. To investigate the solvability of the original problem, we first consider an auxiliary inverse boundary value problem and prove its equivalence (in a certain sense) to the original problem. Then using the Fourier method and contraction map pings principle, the existence and uniqueness theorem for auxiliary problem is proved. Further, on the basis of the equivalency of these problems the existence and uniqueness theorem for the classical solution of the conside…
This paper is devoted to the study of the well-posedness of a singular nonlinear fractional pseudo-hyperbolic system. The fractional derivative is described in Caputo sense. The equations are supplemented by classical and nonlocal boundary conditions. Upon some a priori estimates and density arguments, we establish the existence and uniqueness of the strongly generalized solution for the associated linear fractional system in some Sobolev fractional spaces. On the basis of the obtained results for the linear fractional system, we apply an iterative process in order to establish the well-posedn…
Consider the hyperbolic nonlinear Schrödinger equation (HNLS) over $\mathbb{R}^d$ $$ iu_t + u_{xx} - Δ_{\textbf{y}} u + λ|u|^σu=0. $$ We deduce the conservation laws associated with (HNLS) and observe the lack of information given by the conserved quantities. We build several classes of particular solutions, including \textit{spatial plane waves} and \textit{spatial standing waves}, which never lie in $H^1$. Motivated by this, we build suitable functional spaces that include both $H^1$ solutions and these particular classes, and prove local well-posedness on these spaces. Moreover, we prove a …
We analyse dissipative boundary conditions for nonlinear hyperbolic systems in one space dimension. We show that a previous known sufficient condition for exponential stability with respect to the C^1-norm is optimal. In particular a known weaker sufficient condition for exponential stability with respect to the H^2-norm is not sufficient for the exponential stability with respect to the C^1-norm. Hence, due to the nonlinearity, even in the case of classical solutions, the exponential stability depends strongly on the norm considered. We also give a new sufficient condition for the exponential…
We address the questions (P1), (P2) asked in Kirchheim-Müller-Šverák (2003) concerning the structure of the Rank-$1$ convex hull of a submanifold $\mathcal{K}_1\subset M^{3\times 2}$ that is related to weak solutions of the two by two system of Lagrangian equations of elasticity studied by DiPerna (1985) with one entropy augmented. This system serves as a model problem for higher order systems for which there are only finitely many entropies. The Rank-$1$ convex hull is of interest in the study of solutions via convex integration: the Rank-$1$ convex hull needs to be sufficiently non-trivial f…
In this article we study some aspects of dispersive and concentration phenomena for the Schrödinger equation posed on hyperbolic space $\mathbb{H}^n$, in order to see if the negative curvature of the manifold gets the dynamics more stable than in the Euclidean case. It is indeed the case for the dispersive properties : we prove that the dispersion inequality is valid, in a stronger form than the one on $\mathbb{R}^n$. However, the geometry does not have enough of an effect to avoid the concentration phenomena and the picture is actually worse than expected. The critical nonlinearity power for …
We consider semilinear evolution equations of the form $a(t)\partial_{tt}u + b(t) \partial_t u + Lu = f(x,u)$ and $b(t) \partial_t u + Lu = f(x,u),$ with possibly unbounded $a(t)$ and possibly sign-changing damping coefficient $b(t)$, and determine precise conditions for which linear instability of the steady state solutions implies nonlinear instability. More specifically, we prove that linear instability with an eigenfunction of fixed sign gives rise to nonlinear instability by either exponential growth or finite-time blow-up. We then discuss a few examples to which our main theorem is immed…
We establish necessary conditions for the existence of solutions to a class of semilinear hyperbolic problems on complete noncompact Riemannian manifolds, extending some nonexistence results for the wave operator with power nonlinearity on the whole Euclidean space. A general weight function depending on spacetime is allowed in front of the power nonlinearity.
We consider a nonlinear damped hyperbolic reaction-diffusion system in a bounded interval of the real line with homogeneous Neumann boundary conditions and we study the metastable dynamics of the solutions. Using an "energy approach" introduced by Bronsard and Kohn [CPAM 1990] to study slow motion for Allen-Cahn equation and improved by Grant [SIAM J. Math. Anal. 1995] in the study of Cahn-Morral systems, we improve and extend to the case of systems the results valid for the hyperbolic Allen-Cahn equation. In particular, we study the limiting behavior of the solutions as $\varepsilon\to0^+$, w…
This paper deals with a hyperbolic system of two nonlinear conservation laws, where the phase space contains two contact manifolds. The governing equations are modelling bidisperse suspensions, which consist of two types of small particles that are dispersed in a viscous fluid and differ in size and viscosity. For certain parameter choices quasi-umbilic points and a contact manifold in the interior of the phase space are detected. The dependance of the solutions structure on this contact manifold is examined. The elementary waves that start in the origin of the phase space are classified. Prot…
We establish the well-posedness of a strongly damped semilinear wave equation equipped with nonlinear hyperbolic dynamic boundary conditions. Results are carried out with the presence of a parameter distinguishing whether the underlying operator is analytic, $α>0$, or only of Gevrey class, $α=0$. We establish the existence of a global attractor for each $α\in[0,1],$ and we show that the family of global attractors is upper-semicontinuous as $α\rightarrow0.$ Furthermore, for each $α\in[0,1]$, we show the existence of a weak exponential attractor. A weak exponential attractor is a finite dimensi…
In this paper, we characterized resonant interaction of weakly nonlinear hyperbolic waves in gas dynamics with a real gas background. An asymptotic approach is used to study the interaction between waves, governed by the Euler equations of gas dynamics, supplemented by a van der Waal equation of state; the evolution equation is an integro-differential equation composed of a Burgers type nonlinear term and a convolution term with a known kernel. A one parameter family of traveling wave solutions for various values of van der Waals parameter is studied analytically and numerically. Effect of the…
In the first part of this series, an augmented PDE system was introduced in order to couple two nonlinear hyperbolic equations together. This formulation allowed the authors, based on Dafermos's self-similar viscosity method, to establish the existence of self-similar solutions to the coupled Riemann problem. We continue here this analysis in the {restricted} case of one-dimensional scalar equations and investigate the internal structure of the interface in order to derive a selection criterion associated with the underlying regularization mechanism and, in turn, to characterize the nonconserv…
In this talk we present an overview on the extensions of the De Giorgi approach to general second order nonlinear hyperbolic equations. We start with an introduction to the original conjecture by E. De Giorgi (De Giorgi '96) and to its solution by E. Serra and P. Tilli (Serra&Tilli '12). Then, we discuss a first extension of this idea (Serra&Tilli '16) aimed at investigating a wide class of homogeneous equations. Finally, we announce a further extension to nonhomogeneous equations, obtained by the author in collaboration with P. Tilli (Tentarelli&Tilli '17).
This paper is concerned with the existence and stability of phase transition steady states to a quasi-linear hyperbolic-parabolic system of chemotactic aggregation, which was proposed in \cite{ambrosi2005review, gamba2003percolation} to describe the coherent vascular network formation observed {\it in vitro} experiment. Considering the system in the half line $ \mathbb{R}_{+}=(0,\infty)$ with Dirichlet boundary conditions, we first prove the existence \textcolor{black}{and uniqueness of non-constant phase transition steady states} under some structure conditions on the pressure function. Then …
In this paper, we consider a class of nonlinear reaction-hyperbolic systems with relaxation terms as models for axonal transport in neuroscience. We show the Kruzkov entropy-satisfying BV-solutions of the systems converge towards the solution of an equilibrium model at the rate of $O(\sqrtδ)$ in L1 norm as the relaxation time $δ$ tends to zero. But we don't make sure the rate is optimal.
An averaging principle is derived for the abstract nonlinear evolution equation where the almost periodic right hand-side is a continuous perturbation of the time-dependent family of linear operators determining a linear evolution system. It generalizes classical Henry's results for perturbations of sectorial operators on fractional spaces. It is also proved that the main hypothesis of the nonlinear averaging principle is satisfied for general hyperbolic evolution equations introduced by Kato.
In this paper we prove global and almost global existence theorems for nonlinear wave equations with quadratic nonlinearities in infinite homogeneous waveguides. We can handle both the case of Dirichlet boundary conditions and Neumann boundary conditions. In the case of Neumann boundary conditions we need to assume a natural nonlinear Neumann condition on the quasilinear terms. The results that we obtain are sharp in terms of the assumptions on the dimensions for the global existence results and in terms of the lifespan for the almost global results. For nonlinear wave equations, in the case w…
The paper recalls two of the regularity results for Burgers' equation, and discusses what happens in the case of genuinely nonlinear, strictly hyperbolic systems of conservation laws. The first regularity result which is considered is Oleinik-Ambroso-De Lellis SBV estimate: it provides bounds on the x-derivative of u when u is an entropy solution of the Cauchy problem for Burgers' equation with bounded initial data. Its extensions to the case of systems is then mentioned. The second regularity result of debate is Schaeffer's theorem: entropy solutions to Burgers' equation with smooth and gener…
We consider a degenerate hyperbolic equation of Kirchhoff type with a small parameter epsilon in front of the second-order time-derivative. In a recent paper, under a suitable assumption on initial data, we proved decay-error estimates for the difference between solutions of the hyperbolic problem and the corresponding solutions of the limit parabolic problem. These estimates show in the same time that the difference tends to zero both as epsilon -> 0, and as t -> +infinity. In particular, in that case the difference decays faster than the two terms separately. In this paper we consider the co…
