We develop the convergence analysis of discontinuous Galerkin finite element approximations to second-order quasilinear elliptic and hyperbolic systems of partial differential equations of the form, respectively, − ∑d α=1 ∂xαSiα(∇u(x)) = fi(x), i = 1, . . . , d, and ∂2 t ui− ∑d α=1 ∂xαSiα(∇u(t, x)) = fi(t, x), i = 1, . . . , d, with ∂xα = ∂/∂xα, in a bounded spatial domain in R d, subject to mi...

We establish the existence and uniqueness of the solution to some inner obstacle problems for a coupling of a multidimensional quasilinear firstorder hyperbolic equation set in a region Ωh with a quasilinear parabolic one set in the complementary Ωp = Ω\Ωh. We start by providing the definition of a weak solution through an entropy inequality on the whole domain. Since the interface ∂Ωp ∩ ∂Ωh co...

Based on our recent work on quasilinear parabolic evolution equations and maximal regularity we prove a general result for quasilinear evolution equations with memory. It is then applied to the study of quasilinear parabolic differential equations in weak settings. We prove that they generate Lipschitz semiflows on natural history spaces. The new feature is that delays can occur in the highest ...

We consider in this paper a hyperbolic quasilinear version of the Navier-Stokes equations three space dimensions, obtained by using Cattaneo type law instead Fourier law. In our earlier work [2], we proved global existence and uniqueness solutions for initial data small enough H4(R3)3 ? H3(R3)3. paper, refine previous result establish under significantly lower regularity. first prove local solu...

Summary We consider the coupling between equations of motion a compressible fluid in two and three space dimensions with Christov’s equation for heat flux. is frame indifferent formulation classical model Cattaneo that allows flux to be eliminated obtain single hyperbolic temperature. The obtained system written quasilinear form state variables density, velocity, temperature It then shown this ...

By means of a direct and constructive method based on the theory of semi-global C solution, the local exact boundary observability is established for one-dimensional first order quasilinear hyperbolic systems with general nonlinear boundary conditions. An implicit duality between the exact boundary controllability and the exact boundary observability is then shown in the quasilinear case. Mathe...

We present a control design for semilinear and quasilinear 2 × hyperbolic partial differential equations with the input at one boundary nonlinear ordinary equation coupled to other. The controller can be designed asymptotically stabilize system an equilibrium or relative reference signal. Two related but different controllers general systems are presented additional challenges in discussed. Mor...

A quasilinear system of elliptic-hyperbolic partial differential equations arising from the uniformly asymptotic approximation of solutions to the Helmholtz equation is discussed. These equations produce a model for high-frequency waves which is stable on both sides of a smooth, convex caustic. We prove the existence of strong solutions to a class of inhomogeneous boundary-value problems for an...

A quasilinear system of elliptic-hyperbolic partial differential equations arising from the uniformly asymptotic approximation of solutions to the Helmholtz equation is discussed. These equations produce a model for high-frequency waves which is stable on both sides of a smooth, convex caustic. We prove the existence of strong solutions to a class of inhomogeneous boundary-value problems for an...