OYEKAN, O. Modeling of nonlinear diffusion [online]. Brno: Vysoké učení technické v Brně. Fakulta strojního inženýrství. 2019.

Posudek vedoucího

Franců, Jan

Diffusion is a phenomenon which appears in technology, physics and other branches. Modelling of linear diffusion, namely heat conduction, ranks to basic equations of mathematical physics, is included in all textbooks on P.D.E. and it is deeply studied also in several monographs. On the other hand there are only few monographs on nonlinear diffusion, some of them are cited in the thesis. According to exponent m in the equation u_t=Delta(u^m) we classify the nonlinear diffusion, for 0 1 we obtain slow diffusion equation also called porous media equation. On contrary to linear and fast diffusion, solutions of the slow diffusion equation exhibit some different properties. The first part of the thesis deals with classification, derivation of the equation and derivation and analysis of the Barenblatt solution. This theoretical part is mostly well written. The second part should be devoted to particular solutions, their comparison and visualization. The author used MAPLE system for drawing the graphs. But the thesis contains only very few examples and graphs, they have often uncomplete description and the Maple source file of the pictures is missing. The author started intensive work on his thesis in the last days. I saw the final version only couple of days before the deadline and thus there was not enough time to make all corrections. Thus many errors, mistakes remained in the text. Since the topic is difficult in spite of all mentioned (and many unmentioned) imperfections I recommend the work for defence.

Posudek oponenta

Hájek, Jiří

The thesis provides a broad summary on the so-called nonlinear diffusion, collected from literature and equipped by illustrations and some detailed derivations, prepared by the author. The work was clearly quite demanding. It includes both theoretical and more practical parts. Structure of the text is clear and logical; the amount of information is quite extensive. More detailed comments follow. The theoretical part of the text in chapters 3 and 4 contains numerous complex, far-reaching statements. Although the purpose of including such information is clear, it is not clear, where from come the statements. On the whole, contributions adopted from literature or created by the author are identified only scarcely, which gives rise to some doubts about originality and also makes it difficult for the reader to compare with original sources. Of a total 14 references included in the list of sources, 5 have not been mentioned in the text: [2, 3, 6, 7, 9]. Other factual deficiencies: • The term „blow up“, used first on page 19 and then on several other occasions is not clear, as its meaning in the context of this work has been left unexplained. • On page 20, lines 2 and 3, two identical expressions are presented as different. • On page 20, lines 4 through 6, sentences do not make much sense, possibly due to multiple grammar errors. • Explanation below equation (4.1) repeats the explanation above it, but confuses the statement by a mention of “blow up of energy”, while the equation does not contain any energy term or variable. In reality, the meaning of the PME is mass conservation and the variable is defined as “concentration” (ambiguously). • The whole paragraph above heading 4.1 is deeply confusing, only partly due to grammar errors. The PME is in fact a form of continuity equation, as explained in section 3.2. It contains no energy, while here “energy” is used to define initial and boundary conditions, without explanation. The text further says that “energy is constant” and implies that this is equivalent to “energy is conserved”, which is not true. • The requirement in the first equation below (4.9) is left unexplained, but several lines below again a notion of constant energy is repeated and probably is the equivalent to the integral condition. This is what makes the discussion confusing to anyone for whom mass, heat, energy, concentration, (molar) amount of substance or indeed any physical variable is not just a word. • On page 27, part of a long equation is outside of the page. • Quite arbitrarily, the author switches the meaning of the unknown in PME to temperature in a Remark on page 31 and makes some contradictory conclusions based on this capricious assumption. Such a change in physical meaning is however not possible without physical arguments and clear explanation of the meaning of such a modified equation. Here the author probably forgot for a moment about the existence of Fourier law… The grammar of the text is largely satisfactory and existing errors mostly do not impede understanding. Some errors are however occurring repeatedly, like: • Plurals and singulars in subject and verb do not agree, e.g. above equation 4.16 “these condition”. • Words are missing or completely confusing, e.g. in the Remark on top of page 26, where “We” should be replaced by “With”. • Sentences are badly constructed, e.g. below Figure 2 “Thus the derivatives are discontinues at the interface.”. In summary, I attest that the author has fulfilled the goals of the thesis. In spite of rather extensive list of deficiencies, I recommend the work for defence.

eVSKP id 116681