2013
DOI: 10.1002/mma.2759
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Well posedness of sudden directional diffusion equations

Abstract: Our goal is to establish existence with suitable initial data of solutions to general parabolic equation in one dimension, ut = L(ux)x, where L is merely a monotone function. We also expose the basic properties of solutions, concentrating on maximal possible regularity. Analysis of solutions with convex initial data explains why we may call them almost classical. Some qualitative aspects of solutions, such as facets (flat regions of solutions), are studied too. Copyright © 2013 John Wiley & Sons, Ltd.

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Cited by 13 publications
(30 citation statements)
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“…We may repeat the argument of [10], [12] to claim that u ǫ converges to u in L p (0, T ; L q (Ω)), p, q ∈ (1, ∞), (4.8)…”
Section: Integrability Of the Derivative Of Solutions To The Evolutiomentioning
confidence: 99%
See 1 more Smart Citation
“…We may repeat the argument of [10], [12] to claim that u ǫ converges to u in L p (0, T ; L q (Ω)), p, q ∈ (1, ∞), (4.8)…”
Section: Integrability Of the Derivative Of Solutions To The Evolutiomentioning
confidence: 99%
“…We know from [10] that u ǫ,γ x converges to u ǫ x strongly in L p (0, T ; L q (T)), p ≥ 1 and a.e. in Q T , (4.5) thus G(u ǫ x (·, t)) ≤ G(u ǫ 0,x ).…”
Section: Integrability Of the Derivative Of Solutions To The Evolutiomentioning
confidence: 99%
“…Here, by a weak solution u to (1.5) we mean such a function that u x ∈ L ∞ (0, T ; BV (I )), (1.7) and the following identity holds We do not present the proof of the existence here, it will appear in [20]. We apply there the same technique as used in [19,Sect.…”
mentioning
confidence: 90%
“…We stress that we relegate the technical issues of the existence of a solution to another paper, see [20]. A stationary version of (1.5) has been considered in [17].…”
mentioning
confidence: 99%
“…In case of (1.3), we have W (p) = |p|, this is why it is called the total variation flow. Equation (1.3) has been studied quite extensively, [15], [19], [24], [25], [30], [32], [35]. These authors did not allow jumps in the initial datum u 0 .…”
Section: Introductionmentioning
confidence: 99%