Order Quasilinear Systems Research Articles

Conservation laws for even order systems of polyharmonic map type

Following Rivière’s study of conservation laws for second order quasilinear systems with critical nonlinearity and Lamm/Rivière’s generalization to fourth order, we consider similar systems of order 2m. Typical examples are m-polyharmonic maps. Under natural conditions, we find a conservation law for weak solutions on 2m-dimensional domains. This implies continuity of weak solutions.

Classical solutions to quasilinear parabolic problems with dynamic boundary conditions

We study linear nonautonomous parabolic systems with dynamic boundary conditions. Next, we apply these results to show a theorem of local existence and uniqueness of a classical solution to a second order quasilinear system with nonlinear dynamic boundary conditions.

Hyperbolic Navier-Stokes equations I: Local well-posedness

We replace a Fourier type law by a Cattaneo type law in the derivation of the fundamental equations of fluid mechanics. This leads to hyperbolicly perturbed quasilinear Navier-Stokes equations. For this problem the standard approach by means of quasilinear symmetric hyperbolic systems seems to fail by the fact that finite propagation speed might not be expected. Therefore a somewhat different approach via viscosity solutions is developed in order to prove higher regularity energy estimates for the linearized system. Surprisingly, this method yields stronger results than previous methods, by the fact that we can relax the regularity assumptions on the coefficients to a minimum. This leads to a short and elegant proof of a local-in-time existence result for the corresponding first order quasilinear system, hence also for the original hyperbolicly perturbed Navier-Stokes equations.

A product formula for semigroups of Lipschitz operators associated with abstract quasilinear evolution equations

AbstractThe notion of semigroups of Lipschitz operators associated with abstract quasilinear evolution equations is introduced and a product formula for such semigroups is established. The product formula obtained in the paper is applied to the solvability of the Cauchy problem for a first order quasilinear system through a finite difference scheme of the Lax‐Friedrichs type.

Exact solutions of Euler equations of ideal gasdynamics via Lie group analysis

In this paper, we explicitly characterize a class of solutions to the first order quasilinear system of partial differential equations (PDEs), governing one dimensional unsteady planar and radially symmetric flows of an adiabatic gas involving shock waves. For this, Lie group analysis is used to identify a finite number of generators that leave the given system of PDEs invariant. Out of these generators, two commuting generators are constructed involving some arbitrary constants. With the help of canonical variables associated with these two generators, the assigned system of PDEs is reduced to an autonomous system, whose simple solutions provide non trivial solutions of the original system. It is interesting to remark that one of the special solutions obtained here, using this approach, is precisely the blast wave solution known in the literature.

Weakly non-linear high frequency waves for a first order quasi-linear system involving source-like terms

Making use of the method of asymptotic expansion of multiple scales, a study of weakly non-linear, high frequency waves, through “homogeneous” media characterized by dissipative or dispersive hyperbolic systems of partial differential equations, is proposed.

When nonautonomous equations are equivalent to autonomopus ones

We consider nonlinear systems of first order partial differential equations admitting at least two one-parameter Lie groups of transformations with commuting infinitesimal operators. Under suitable conditions it is possible to introduce a variable transformation based on canonical variables which reduces the model in point to autonomous form. Remarkably, the transformed system may admit constant solutions to which there correspond non-constant solutions of the original model. The results are specialized to the case of first order quasilinear systems admitting either dilatation or spiral groups of transformations and a systematic procedure to characterize special exact solutions is given. At the end of the paper the equations of axi-symmetric gas dynamics are considered.

A Singular Singularly Perturbed Boundary Value Problem of the Second Order Quasilinear Systems

The singular singularly perturbed boundary value problem ϵ d 2x/dt 2 = A(x, t)dx/dt + B(x, t), x(0, ϵ) = α(ϵ), ϵ[a dx(0, ϵ)/dt + b dx(1, ϵ)/dt] = β(ϵ) for an m-dimensional system of quasilinear differential equations is considered under the assumption that there is a vector-valued function F( x, t) such that ∇ x F( x, t) = A( x, t). The asymptotic solution is constructed by a modified Vasil′eva method where there are 3 boundary layer corrections, i.e., the left boundary layer contains 2 different stretched variables t/√ϵ and t/ϵ. The existence and uniqueness of the exact solution and the uniform validity of the formal asymptotic solution for the boundary value problem are proved by using the Banach-Picard fixed-point theorem.

Boundary and angular layer behavior in singular perturbed quasilinear systems

In this paper, using the method of differential inequalities, we study the existence of solutions and their asymptotic behavior, as ɛ→0+, of Dirichlt problem for second order quasilinear systems. Depending on whether the reduced solutionu(t) has or does not have a continuous first-derivative in (a, b), we study two types of asymptotic behaviour, thus leading to the phenomena of boundary and angular layers.

Non-equilibrium effects on the breakdown of weak solutions in non-linear wave propagation

The weak solutions of the first order quasi-linear system governing a non-equilibrium flow of a gas are studied in the characteristic plane. It is shown that a linear solution in the characteristic plane can exhibit non-linear behaviour in the physical plane. As an application of the theory the law of propagation and the growth equation of weak discontinuities are obtained and the breakdown of weak solutions is discussed. It is also shown that non-equilibrium effects play an important role in the process of steepening and flattening of weak non-linear waves with planar, cylindrical and spherical symmetry. The critical timet c for the breakdown of weak solutions is obtained. The critical amplitudea c of the initial compressive disturbance has been determined such that any compressive weak wave with an initial amplitude greater thana c terminates into a shock wave, while an initial amplitude less than the critical one results in a decay of the disturbance. A non-linear steepening and non-equilibrium effects provide a particular answer to the substantial question as for when a shock wave will be formed on the breakdown of weak solutions, when the characteristics will pile up on the wave front. The non-equilibrium effects have a stabilizing influence on the wave propagation in the sense that not all compressive disturbances will grow into shock waves, whereas in ordinary gas flows all compressive waves will grow into shock waves.