In this paper we prove the $C^{\infty }$-well posedness of the Cauchy problem for quasi-linear hyperbolic equations of second order with coefficients non-Lipschitz in t ∈ [0,T] and smooth in x ∈ ℝⁿ.

On considère le problème mixte dans un quadrant pour un opérateur différentiel hyperbolique $P$ en supposant que $P$ et les opérateurs au bord sont homogènes à coefficients constants. On caractérise les conditions au bord pour avoir existence et unicité de la solution du problème mixte, en se plaçant successivement dans le cadre des fonctions $C^{\infty }$, puis, lorsque $P$ est strictement hyperbolique, dans le cadre des espaces de Sobolev. Ces caractérisations s’expriment au moyen d’une condition dite de Lopatinski,...

We consider the Darboux problem for a functional differential equation: $\left(\partial ²u\right)/\left(\partial x\partial y\right)(x,y)=f(x,y,u_{(x,y)},u(x,y),\partial u/\partial x(x,y),\partial u/\partial y(x,y))$ a.e. in [0,a]×[0,b], u(x,y) = ψ(x,y) on [-a₀,a]×[-b₀,b]∖(0,a]×(0,b], where the function $u_{(x,y)}:[-a₀,0]\times [-b₀,0]\to {ℝ}^k$ is defined by $u_{(x,y)}(s,t)=u(s+x,t+y)$ for (s,t) ∈ [-a₀,0]×[-b₀,0]. We give a few theorems about weak and strong inequalities for this problem. We also discuss the case where the right-hand side of the differential equation is linear.

In this paper, we establish Carleman estimates for the two dimensional isotropic non-stationary Lamé system with the zero Dirichlet boundary conditions. Using this estimate, we prove the uniqueness and the stability in determining spatially varying density and two Lamé coefficients by a single measurement of solution over $(0,T)\times \omega$, where $T\>0$ is a sufficiently large time interval and a subdomain $\omega$ satisfies a non-trapping condition.

In this paper, we establish Carleman estimates for the two dimensional isotropic non-stationary Lamé system with the zero Dirichlet boundary conditions. Using this estimate, we prove the uniqueness and the stability in determining spatially varying density and two Lamé coefficients by a single measurement of solution over (0,T) x ω, where T > 0 is a sufficiently large time interval and a subdomain ω satisfies a non-trapping condition.

This article is devoted to incompressible Euler equations (or to Navier-Stokes equations in the vanishing viscosity limit). It describes the propagation of quasi-singularities. The underlying phenomena are consistent with the notion of a cascade of energy.

We study a class of third order hyperbolic operators $P$ in $G=\Omega \cap \{0\le t\le T\},\Omega \subset {ℝ}^{n+1}$ with triple characteristics on $t=0$. We consider the case when the fundamental matrix of the principal symbol for $t=0$ has a couple of non vanishing real eigenvalues and $P$ is strictly hyperbolic for $t\>0.$ We prove that $P$ is strongly hyperbolic, that is the Cauchy problem for $P+Q$ is well posed in $G$ for any lower order terms $Q$.

The existence, uniqueness and regularity of the generalized local solution and the classical local solution to the periodic boundary value problem and Cauchy problem for the multidimensional coupled system of a nonlinear complex Schrödinger equation and a generalized IMBq equation $$i{\epsilon }_t+{\nabla }^2\epsilon -u\epsilon =0,$$$$u_{...}$$

In this work we present a class of partial differential operators with constant coefficients, called multi-quasi-hyperbolic and defined in terms of a complete polyhedron. For them we obtain the well-posedness of the Cauchy problem in generalized Gevrey classes determined by means of the same polyhedron. We present some necessary and sufficient conditions on the operator in order to be multi-quasi-hyperbolic and give some examples.

We introduce a family of new second-order Godunov-type central schemes for one-dimensional systems of conservation laws. They are a less dissipative generalization of the central-upwind schemes, proposed in [A. Kurganov et al., submitted to SIAM J. Sci. Comput.], whose construction is based on the maximal one-sided local speeds of propagation. We also present a recipe, which helps to improve the resolution of contact waves. This is achieved by using the partial characteristic decomposition,...

We present a family of high-order, essentially non-oscillatory, central schemes for approximating solutions of hyperbolic systems of conservation laws. These schemes are based on a new centered version of the Weighed Essentially Non-Oscillatory (WENO) reconstruction of point-values from cell-averages, which is then followed by an accurate approximation of the fluxes via a natural continuous extension of Runge-Kutta solvers. We explicitly construct the third and fourth-order scheme and demonstrate...

We present one- and two-dimensional central-upwind schemes for approximating solutions of the Saint-Venant system with source terms due to bottom topography. The Saint-Venant system has steady-state solutions in which nonzero flux gradients are exactly balanced by the source terms. It is a challenging problem to preserve this delicate balance with numerical schemes. Small perturbations of these states are also very difficult to compute. Our approach is based on extending semi-discrete central schemes...