Global existence for systems of parabolic conservation laws in several space variables

We prove the global existence of solutions of the Cauchy problem for certain systems of conservation laws with artificial viscosity terms added. The system is assumed to admit a quadratic entropy which is consistent with the viscosity matrix, and the initial data is assumed to be close to a constant in L² ∩ L^∞. In particular, our result applies to the equations of compressible fluid flow in two and three space variables.     

Source-type solutions of heat equations with convection in several variables space

In this paper we study the source-type solution for the heat equation with convection: ut = △u + ■· ▽un for (x,t) ∈ ST→ RN × (0,T] and u(x,0) = δ(x) for x ∈ RN, where δ(x) denotes Dirac measure in = RN,N 2,n 0 and b = (b1,...,bN) ∈ RN is a vector. It is shown that there exists a critical number pc = N+2 such that the source-type solution to the above problem exists and is unique if 0 N n < pc and there exists a unique similarity source-type solution in the case n = N+1 , while such a solution does not exist...     

Global classical solutions to a kind of quasilinear hyperbolic systems in several space variables

For a kind of quasilinear hyperbolic systems in several space variables whose coefficient matrices commute each other, by means of normalized coordinates, formulas of wave decomposition and pointwise decay estimates, the global existence of classical solution to the Cauchy problem for small and decaying initial data is obtained, under hypotheses of weak linear degeneracy and weakly strict hyperbolicity.     

Lipschitz continuity theorems for free discontinuity problem in several variables

In this paper, following the method in [S. Solimini, Simplified excision techniques for free discontinuity problems in several variables, J. Funct. Anal. 151 (1997) 1-34], we prove a regularity of the function in minimizer for free discontinuity problem. Namely, we prove that the function is globally Lipschitz continuous out of a small neighborhood of the singular set.     

A characterization of the blow-up time for the solution of a conservation law in several space variable

We derive a necessary and sufficient condition on the L^∞ Cauchy data for a conservation law in several space variables under which the solution will be locally Lipschitz continuous up to time T . The largesf such T is therefore the “blow-up” time. Roughly, our condition is that the data can be approximated by smoother functions satisfying uniformly a certain estimate. We present an example which shows that the existence of the approximations is crucial: it is not sufficient that the data itself satisfy this estimate.     

Locally bounded solutions of one-dimensional conservation laws

A one-dimensional conservation law with a power-law flux function and an exponential initial condition is considered. We construct a generalized entropy solution with countably many shock waves. This solution is sign-alternating and one-sided periodic.     

Asymptotic spatial homogeneity of solutions to conservation laws with memory in several space dimensions

We prove that any solution of the problem (u+K*u)_t+∑a_i(u)u=0 on (0,∞)×ℝ^N with spatially periodic initial data converges to a constant provided some non-degeneracy conditions on the kernel K and the non-linear functions a_i,i=1,…,N are imposed. © 1997 B. G. Teubner Stuttgart-John Wiley & Sons Ltd.     

A generalized Stefan problem in several space variables

A multidimensional, multiphase problem of Stefan type, involving quasilinear parabolic equations and nonlinear boundary conditions is considered. Regularization techniques and monotonicity methods are exploited. Existence and uniqueness of a weak solution to the problem, as well as continuous and monotone dependence of the solution upon data are shown.     

Exact controllability to trajectories for entropy solutions to scalar conservation laws in several space dimensions

We describe a new method that allows us to obtain a result of exact controllability to trajectories of multidimensional conservation laws in the context of entropy solutions and under a mere non-degeneracy assumption on the flux and a natural geometric condition.