Existence of solutions for quasilinear problem with Neumann boundary conditions

My abstract is:This paper is devoted to the study of the existence of weak solutionsfor quasilinear systems of a partial dierential equations which are the combinationof the Perona-Malik equation and the Heat equation. The proof of the main resultsare based on the compactness method and the motonocity arguments.

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.

Boundary layer solutions to singularly perturbed quasilinear systems

<p style='text-indent:20px;'>We consider weak boundary layer solutions to the singularly perturbed ODE systems of the type <inline-formula><tex-math id="M1">\begin{document}$ \varepsilon^2\left(A(x, u(x), \varepsilon)u'(x)\right)' = f(x, u(x), \varepsilon) $\end{document}</tex-math></inline-formula>. The new features are that we do not consider one scalar equation, but systems, that the systems are allowed to be quasilinear, and that the systems are spatially non-smooth. Although the results about existence, asymptotic behavior, local uniqueness and stability of boundary layer solutions are similar to those known for semilinear, scalar and smooth problems, there are at least three essential differences. First, the asymptotic convergence rates valid for smooth problems are not true anymore, in general, in the non-smooth case. Second, a specific local uniqueness condition from the scalar case is not sufficient anymore in the vectorial case. And third, the monotonicity condition, which is sufficient for stability of boundary layers in the scalar case, must be adjusted to the vectorial case.</p>

A study of a quasilinear model of the particles of a suspension that are aggregated and settled in an inhomogeneous field

The mathematical model of the sedimentation process of suspension particles is usually a quasilinear hyperbolic system of partial differential equations, supplemented by initial and boundary conditions. In this work, we study a complex model that takes into account the aggregation of particles and the inhomogeneity of the field of external mass forces. The case of homogeneous initial conditions is considered, when all the parameters of the arising motion depend on only one spatial Cartesian coordinate x and on time t. In contrast to the known formulations for quasilinear systems of equations (for example, as in gas dynamics), the solutions of which contain discontinuities, in the studied formulation the basic system of equations occurs only on one side of the discontinuity line in the plane of variables (t; x). On the opposite side of the discontinuity surface, the equations have a different form in general. We will restrict ourselves to considering the case when there is no motion in a compact zone occupied by settled particles, i.e. all velocities are equal to zero and the volumetric contents of all phases do not change over time. The problem of erythrocyte sedimentation in the field of centrifugal forces in a centrifuge, with its uniform rotation with angular velocity ω = const is considered. We have studied the conditions for the existence of various types of solutions. One of the main problems is the evolution (stability) problem of the emerging discontinuities. The solution of this problem is related to the analysis of the relationships for the characteristic velocities and the velocity of the discontinuity surface. The answer depends on the number of characteristics that come to the jump, and the number of additional conditions set on the interface. The discontinuity at the lower boundary of the area occupied by pure plasma is always stable. But for the surface separating the zones of settled and of moving particles, the condition of evolution may be violated. In this case, it is necessary to adjust the original mathematical model.