Diffusion-Wave Type Solutions to the Second-Order Evolutionary Equation with Power Nonlinearities

The paper deals with a nonlinear second-order one-dimensional evolutionary equation related to applications and describes various diffusion, filtration, convection, and other processes. The particular cases of this equation are the well-known porous medium equation and its generalizations. We construct solutions that describe perturbations propagating over a zero background with a finite velocity. Such effects are known to be atypical for parabolic equations and appear as a consequence of the degeneration of the equation at the points where the desired function vanishes. Previously, we have constructed it, but here the case of power nonlinearity is considered. It allows for conducting a more detailed analysis. We prove a new theorem for the existence of solutions of this type in the class of piecewise analytical functions, which generalizes and specifies the earlier statements. We find and study exact solutions having the diffusion wave type, the construction of which is reduced to the second-order Cauchy problem for an ordinary differential equation (ODE) that inherits singularities from the original formulation. Statements that ensure the existence of global continuously differentiable solutions are proved for the Cauchy problems. The properties of the constructed solutions are studied by the methods of the qualitative theory of differential equations. Phase portraits are obtained, and quantitative estimates are determined by constructing and analyzing finite difference schemes. The most significant result is that we have shown that all the special cases for incomplete equations take place for the complete equation, and other configurations of diffusion waves do not arise.

A Note on a Meshless Method for Fractional Laplacian at Arbitrary Irregular Meshes

The existence and uniqueness of the discrete solutions of a porous medium equation with diffusion are demonstrated. The Cauchy problem contains a fractional Laplacian and it is equivalent to the extension formulation in the sense of trace and harmonic extension operators. By using the generalized finite difference method, we obtain the convergence of the numerical solution to the classical/theoretical solution of the equation for nonnegative initial data sufficiently smooth and bounded. This procedure allows us to use meshes with complicated geometry (more realistic) or with an irregular distribution of nodes (providing more accurate solutions where needed). Some numerical results are presented in arbitrary irregular meshes to illustrate the potential of the method.

A simple proof of the generalized Leibniz rule on bounded Euclidean domains

This paper is devoted to a simple proof of the generalized Leibniz rule in bounded domains. The operators under consideration are the so-called spectral Laplacian and the restricted Laplacian. Equations involving such operators have lately been considered by Constantin and Ignatova in the framework of the SQG equation [P. Constantin and M. Ignatova, Critical SQG in bounded domains, Ann. PDE 2 2016, 2, Article ID 8] in bounded domains, and by two of the authors [Q.-H. Nguyen and J. L. Vázquez, Porous medium equation with nonlocal pressure in a bounded domain, Comm. Partial Differential Equations 43 2018, 10, 1502–1539] in the framework of the porous medium with nonlocal pressure in bounded domains. We will use the estimates in this work in a forthcoming paper on the study of porous medium equations with pressure given by Riesz-type potentials.

Porous Medium Equation in Graphene Oxide Membrane: Nonlinear Dependence of Permeability on Pressure Gradient Explained

Membrane performance in gas separation is quantified by its selectivity, determined as a ratio of measured gas permeabilities of given gases at fixed pressure difference. In this manuscript a nonlinear dependence of gas permeability on pressure difference observed in the measurements of gas permeability of graphene oxide membrane on a manometric integral permeameter is reported. We show that after reasoned assumptions and simplifications in the mathematical description of the experiment, only static properties of any proposed governing equation can be studied, in order to analyze the permeation rate for different pressure differences. Porous Medium Equation is proposed as a suitable governing equation for the gas permeation, as it manages to predict a nonlinear behavior which is consistent with the measured data. A coefficient responsible for the nonlinearity, the polytropic exponent, is determined to be gas-specific—implications on selectivity are discussed, alongside possible hints to a deeper physical interpretation of its actual value.

Bounded $$H^\infty $$-calculus for a degenerate elliptic boundary value problem

On a manifold X with boundary and bounded geometry we consider a strongly elliptic second order operator A together with a degenerate boundary operator T of the form $$T=\varphi _0\gamma _0 + \varphi _1\gamma _1$$ T = φ 0 γ 0 + φ 1 γ 1 . Here $$\gamma _0$$ γ 0 and $$\gamma _1$$ γ 1 denote the evaluation of a function and its exterior normal derivative, respectively, at the boundary. We assume that $$\varphi _0, \varphi _1\ge 0$$ φ 0 , φ 1 ≥ 0 , and $$\varphi _0+\varphi _1\ge c$$ φ 0 + φ 1 ≥ c , for some $$c>0$$ c > 0 , where either $$\varphi _0,\varphi _1\in C^{\infty }_b(\partial X)$$ φ 0 , φ 1 ∈ C b ∞ ( ∂ X ) or $$\varphi _0=1 $$ φ 0 = 1 and $$\varphi _1=\varphi ^2$$ φ 1 = φ 2 for some $$\varphi \in C^{2+\tau }(\partial X)$$ φ ∈ C 2 + τ ( ∂ X ) , $$\tau >0$$ τ > 0 . We also assume that the highest order coefficients of A belong to $$C^\tau (X)$$ C τ ( X ) and the lower order coefficients are in $$L_\infty (X)$$ L ∞ ( X ) . We show that the $$L_p(X)$$ L p ( X ) -realization of A with respect to the boundary operator T has a bounded $$H^\infty $$ H ∞ -calculus. We then obtain the unique solvability of the associated boundary value problem in adapted spaces. As an application, we show the short time existence of solutions to the porous medium equation.