Remarks on the Contributions of Constantine M. Dafermos to the Subject of Conservation Laws

Tzavaras, Athanasios E.

doi:10.1016/s0252-9602(12)60002-2

Cited by 2 publications

(3 citation statements)

References 49 publications

(55 reference statements)

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“…We choose c(t) ≥ 0 large enough so that the functions u −→ uf ± (u) are strictly increasing on the interval [min(u 0 (., ))−η, max(u 0 (., ))+ η] ∀ , with η > 0 fixed arbitrary. This is possible as proved in (23)(24)(25) since the interval [min(u 0 (., )) − η, max(u 0 (., )) + η] is bounded therefore (4) suffices to ensure that f is Lipschitz continuous in this interval with fixed Lipschitz constant. We approximate u 0 ∈ L ∞ (T) by continuous functions u 0 (., ) in L ∞ norm.…”

Section: From the Lemma The Assumptions I) And Ii) Are Satisfied Whatmentioning

confidence: 80%

“…. , n, being some nonnegative constants whose possible need is exposed in (23)(24)(25). Equations (35) are endowed with the initial condition…”

Section: The Multidimensional Casementioning

confidence: 99%

“…The seminal survey article [8] and the numerous recent developments by B. Andreianov et al [9]- [13] permit to better understand the issue of admissibility of solutions in relation with specific modeling assumptions. For recent developments in the field see [4,9,12,13,24,52,77], in which are addressed several relevant issues and new conceptual notions of weak entropy solutions, mainly in the scalar case with discontinuous nonlinearities.…”

Section: Theorem 7: Coherence With the Cauchy-kovalevska Solutionmentioning

confidence: 99%

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Weak asymptotic methods for scalar equations and systems

Abreu

Colombeau

Panov

2016

Journal of Mathematical Analysis and Applications

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Please cite this article in press as: E. Abreu et al., Weak asymptotic methods for scalar equations and systems, J. Math. Anal. Appl. (2016), http://dx. AbstractIn this paper we show how one can construct families of continuous functions which satisfy asymptotically scalar equations with discontinuous nonlinearity and systems having irregular solutions. This construction produces weak asymptotic methods which are issued from Maslov asymptotic analysis. We obtain a sequence of functions which tend to satisfy the equation(s) in the weak sense in the space variable and in the strong sense in the time variable. To this end we reduce the partial differential equations to a family of ordinary differential equations in a classical Banach space. For scalar equations we prove that the initial value problem is well posed in the L 1 sense for the approximate solutions we construct. Then we prove that this method gives back the widely accepted solutions when they are known. For systems we obtain existence in the general case and uniqueness in the analytic case using an abstract Cauchy-Kovalevska theorem.

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