Results concerning the nature of the continuity of solutions of parabolic equations and some of their applications

Abstract

Theorems are established which deal with the relationship between the moduli of continuity with respect to x and with respect to t (in the spaces C and L l) for the solutions of certain classes of parabolic equations, which are, in general, nonlinear and degenerate. These theorems are then applied to an investigation of the Cauchy problem for quasilinear equations of the first and second order.w Introduction. The structure of parabolic equations entails a certain "disparity" of the independent variable t, corresponding to the time, when comparison is made with the spatial variables x = (x 1 ..... x n) . In particular, this finds its expression in the fact that when methods used to investigate the smoothness with respect to x of the solutions of such equations are applied to investigating smoothness with respect to t, they usually lead to the necessity of introducing certain very objectionable conditions (not fundamentally called for) on the functions defining the equation, especially when the latter is nonlinear. It is then natural, in order to obtain more precise results concerning smoothness of the solutions of parabolic equations, to apply methods which take into account the specific nature of the "disparity" mentioned above between the var{ables t and x, one of the features of this disparity being a definite "subordination" of the properties of the solutions with respect to t to their properties with respect to x.A basic aim of the present paper consists in proving that the moduli of continuity with respect to t (in the spaces C and L1), for bounded solutions of certain classes of parabolic equations, may be estimated in terms of corresponding moduli of continuity with respect to x, under minimal (in a definite sense) assumptions on the functions defining the equations; the equations considered are, in general, nonlinear and degenerate. We devote w167 2 and 4 for this purpose. Analogous results for the case of powers of the moduli of continuity (in C) for the solutions of linear parabolic equations were obtained in [1]; the method used for proving Theorem 2 in w 4 is based on ideas used in [2].In w 3 we consider bounded solutions of the Cauchy problem for the unsteady filtration equationwith the initial conditionUsing Theorem I of w an estimate is obtained of the modulus of continuity with respect to t (in C) for the solution of problem (1), (2); this estimate supplements a recent result of D. Aronson [3], a result whose discussion at O. A. Oleinik's seminar at Moscow State University stimulated to a considerable extent the writing of w w 2 and 3 of the present paper.

Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients

“…To prove that the difference approximations possess some L 1 time continuity, we shall use the following lemma due to Kružkov [33]. Lemma 2.4 (Kružkov's interpolation lemma [33]).…”

“…Lemma 2.4 (Kružkov's interpolation lemma [33]). Let z(x, t) be a bounded measurable function defined on R d × (0, T ).…”

“…The proof of the first part of Theorem 1.1 is based on deriving uniform L ∞ , L 1 , and BV bounds on the approximate solution u ∆t . Equipped with the BV bound, we use the difference scheme itself and Kružkov's interpolation lemma [33] …”

A linearly implicit finite-difference scheme for the one-dimensional porous medium equation

“…Subtracting (2.17) from (2.16) and dividing by Ax, we obtain Lemma 2.3 follows directly from the bound (2.13) and the difference equation (1.7). And Lemma 2.4 can be proved by employing the discrete version of a technique due to Kruzkov [7] for deducing a modulus of continuity in time from a known modulus of continuity in space for solutions of certain parabolic equations. The proofs of Lemmas 2.3 and 2.4 are nearly identical to those of Lemmas 2.6 and 2.7 in [5], and in any case they impose no further constraints upon the mesh parameters.…”

Aronson-Bénilan type estimate and the optimal Hölder continuity of weak solutions for the 1-D degenerate Keller-Segel systems

“…In this direction, see Aronson [2], Kruzhkov [17], Caffarelli-Friedman [7], Gilding-Peletier [14], Gilding [13], DiBenedetto [10], Bénilan [6], Aronson-Caffarelli [5] and DiBenedetto-Friedman [11].…”