weak solutions
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2022 ◽  
Vol 40 ◽  
pp. 1-11
Ghasem A. Afrouzi ◽  
Z. Naghizadeh ◽  
Nguyen Thanh Chung

In this paper, we are interested in a class of bi-nonlocal problems with nonlinear Neumann boundary conditions and sublinear terms at infinity. Using $(S_+)$ mapping theory and variational methods, we establish the existence of at least two non-trivial weak solutions for the problem provied that the parameters are large enough. Our result complements and improves some previous ones for the superlinear case when the Ambrosetti-Rabinowitz type conditions are imposed on the nonlinearities.

David P. Bourne ◽  
Charlie P. Egan ◽  
Beatrice Pelloni ◽  
Mark Wilkinson

AbstractWe give a new and constructive proof of the existence of global-in-time weak solutions of the 3-dimensional incompressible semi-geostrophic equations (SG) in geostrophic coordinates, for arbitrary initial measures with compact support. This new proof, based on semi-discrete optimal transport techniques, works by characterising discrete solutions of SG in geostrophic coordinates in terms of trajectories satisfying an ordinary differential equation. It is advantageous in its simplicity and its explicit relation to Eulerian coordinates through the use of Laguerre tessellations. Using our method, we obtain improved time-regularity for a large class of discrete initial measures, and we compute explicitly two discrete solutions. The method naturally gives rise to an efficient numerical method, which we illustrate by presenting simulations of a 2-dimensional semi-geostrophic flow in geostrophic coordinates generated using a numerical solver for the semi-discrete optimal transport problem coupled with an ordinary differential equation solver.

Abdelaziz Hellal

This paper is concerned with the study of the nonlinear elliptic equations in a bounded subset Ω ⊂ RN Au = f, where A is an operator of Leray-Lions type acted from the space W1,p(·)0(Ω) into its dual. when the second term f belongs to Lm(·), with m(·) > 1 being small. we prove existence and regularity of weak solutions for this class of problems p(x)-growth conditions. The functional framework involves Sobolev spaces with variable exponents as well as Lebesgue spaces with variable exponents.

2022 ◽  
Vol 40 ◽  
pp. 1-19
Hamid EL Bahja

In this paper, we discuss a class of degenerate parabolic equations with variable exponents. By  using the Steklov average and Young's inequality, we establish energy and logarithmicestimates for solutions to these equations. Then based on the intrinsic scaling method, we provethat local weak solutions are locally continuous.

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