Dirichlet problem for the Laplacian and the Bilaplacian in Lipschitz Domains

Dirichlet problem for the Laplacian and the Bilaplacian in Lipschitz Domains

Abstract: We are interested here in questions related to the maximal regularity of solutions of elliptic problems with Dirichlet boundary condition (see ([1]). For the last 40 years, many works have been concerned with questions when ? is a Lipschitz domain. Some of them contain incorrect results that are corrected in the present work. We give here new proofs and some complements for the case of the Laplacian (see [3]), the Bilaplacian ([2] and [6]) and the operator div (A?) (see ([5]), when A is a matrix or a function. And we extend this study to obtain other regularity results for domains having an adequate regularity. We give also new results for the Dirichlet-to-Neumann operator for Laplacian and Bilaplacian. Using the duality method, we can then revisit the work of Lions-Magenes [4], concerning the so-called very weak solutions, when the data are less regular.
References:
[1] C. Amrouche and M. Moussaoui. The Dirichlet problem for the Laplacian in Lipschitz domains. Submitted. See also the abstract in https://arxiv.org/pdf/2204.02831.pdf
[2] B.E.J. Dahlberg, C.E. Kenig, J. Pipher and G.C. Verchota. Area integral estimates for higher order elliptic equations and systems. Ann. Inst. Fourier, 47-5, 1425 - 1461, (1997).
[3] D. Jerison and C.E. Kenig. The Inhomogeneous Dirichlet Problem in Lipschitz Do mains, J. Funct. Anal. 130, 161 - 219, (1995).
[4] J.L. Lions and E. Magenes. Probl`emes aux limites non-homog`enes et applications, Vol. 1, Dunod, Paris, (1969).
[5] J. Ne?cas. Direct methods in the theory of elliptic equations. Springer Monographs in Mathematics. Springer, Heidelberg, (2012).
[6] G.C. Verchota. The biharmonic Neumann problem in Lipschitz domains. Acta Math. 194-2, 217 - 279, (2005).

Lévy flights in bounded domains

Lévy flights in bounded domains

Abstract:

Regularity results for the Magnetohydrodynamic equations under different boundary conditions for the velocity and the magnetic field.

Regularity results for the Magnetohydrodynamic equations under different boundary conditions for the velocity and the magnetic field.

Abstract: The boundary value problem for the steady-state magnetohydrodynamic (MHD) equations under various boundary conditions for the velocity and the magnetic field is considered in a bounded domain. First, we state the case in which the tangential components of the velocity and the magnetic field are specified on the boundary together with a pressure boundary condition. We give some results of existence, uniqueness and regularity of solutions in Hilbertian case and in Lp theory. Next, we consider the (MHD) equations with Navier type boundary conditions for both the velocity and the magnetic field.

30 years of transferring mathematics to industry

30 years of transferring mathematics to industry

A Lagrangian-Eulerian method for hyperbolic problems: weak asymptotic analysis and applications

A Lagrangian-Eulerian method for hyperbolic problems: weak asymptotic analysis and applications

Abstract: In this talk, we will discuss a new Lagrangian-Eurelian approach for numerical approximation of hyperbolic conservation laws and related Balance law problems. This new methodology is based on an improved concept of no-flow curves as recently discussed in the literature [1,2,3,6]. We analyzed fully-discrete [5] and semi-discrete [7] schemes via a weak asymptotic analysis: convergence for the unique entropy (Kruzhkov) solution to scalar hyperbolic problems and a novel weak positive principle to the more general case of multi-D systems in cartesian. We will discuss how new insights on the improved concepts of No-flow Curves and weak asymptotic methods are good ingredients to study phenomena of nonlinear waves and their inherent properties: nonlinearity, wave-breaking phenomena (shocks) and unique weak-entropy solution. This method has been successfully applied for solving nontrivial (scalar and systems) 1D and Multi-D hyperbolic problems (2D shallow water equations with variable topography, compressible Euler Flows, Orszag-Tang vortex system and a nonstrictly hyperbolic system of conservation law with a resonance effect. We will present several numerical experiments in 1D/Multi-D (e.g., [1,2,3,4,5,6,7]) aiming to discuss the theory and capabilities of the new method.
[1] A fast, robust, and simple Lagrangian-Eulerian solver for balance laws and applications. Computers & Mathematics with Applications, 77(9) (2019) 2310-2336. https://doi.org/10.1016/j.camwa.2018.12.019 joint work with J. Pérez.
[2] A Class of Lagrangian-Eulerian Shock-Capturing Schemes for First-Order Hyperbolic Problems with Forcing Terms. Journal of Scientific Computing 86 (2021) 14 (47 pages). https://link.springer.com/article/10.1007/s10915-020-01392-w (joint work with V. Matos, J. Pérez and P. Bermudez-Rodriguez)
[3] A Class of Positive Semi-discrete Lagrangian-Eulerian Schemes for Multidimensional Systems of Hyperbolic Conservation Laws. Journal of Scientific Computing, v.90 (2022), p.40 (79 pages). https://link.springer.com/article/10.1007/s10915-021-01712-8 (joint work with J. François, W. Lambert and J. Pérez)
[4] A semi-discrete Lagrangian-Eulerian scheme for hyperbolic-transport models. Journal of Computational and Applied Mathematics, v.406 (2022), p.114011. https://www.sciencedirect.com/science/article/abs/pii/S0377042721005963 (joint work with J. François, W. Lambert and J. Pérez)
[5] Convergence, bounded variation properties and Kruzhkov solution of a fully discrete Lagrangian-Eulerian scheme via weak asymptotic analysis for 1D hyperbolic problems. Numerical Methods for Partial Differential Equations, Numerical Methods for Partial Differential Equations 39(3) (2023) 2400-2443. https://doi.org/10.1002/num.22972 (joint work with A. Espírito Santo, W. Lambert and J. Pérez)
[6] A Lagrangian-Eulerian Method on Regular Triangular Grids for Hyperbolic Problems: Error Estimates for the Scalar Case and a Positive Principle for Multidimensional Systems. Journal of Dynamics and Differential Equations (2023). https://doi.org/10.1007/s10884-023-10283-1 (joint work with J. Agudelo, W. Lambert and J. Pérez)
[7] A triangle-based positive semi-discrete Lagrangian-Eulerian scheme via the weak asymptotic method for scalar equations and systems of hyperbolic conservation laws, Journal of Computational and Applied Mathematics, 437 (2024). https://doi.org/10.1016/j.cam.2023.115465 (joint work with J. Agudelo and J. Pérez)

A numerical and analytical approach to a 1D nonlocal porous media equation: blowing up and mass spreads

A numerical and analytical approach to a 1D nonlocal porous media equation: blowing up and mass spreads

Some thermo-hydrodynamic problems in blast furnace steel casts.

Some thermo-hydrodynamic problems in blast furnace steel casts.

Quasi-variational solutions to thick flows.

Quasi-variational solutions to thick flows.

Abstract: We formulate the flow of thick fluids as evolution quasi-variational inequalities, with a variable threshold on the absolute value of the deformation rate tensor depending on the solution itself, and we show the existence of strong and weak solutions in different cases. The results are based on new extensions of the continuous dependence of weak solutions to the variational inequalities for the Navier-Stokes equations with constraints on the derivatives, and on their respective generalised Lagrange multipliers.