Basic notions of functional analysis, functions of complex values, standard properties of classical solutions of semilinear first order equations, heat equation, wave equation, Laplace and Poisson's equations.
Knowledge of mathematical methods that are widely used by researchers in the area of Applied Mathematics, as Sobolev Spaces, distributions. Application of this knowledge to a variety of topics, including the basic equations of mathematical physics and some current research topics about linear and nonlinear partial differential equations.
Distributions. Locally integrable functions. The space of test function D(U). Distributions. Distributions associated to Locally integrable functions. Singular distributions. Examples. Operations on distributions: sum, products times functions, change of variables, restrictions, tensor product. Differentiation and his properties; comparison with classical derivatives. Differentiation of jump functions. Partition of unity. Support of a distribution; compactly supported distributions.
Convolution. Convolution in Lp spaces. Regularity of the convolution. Regularizing sequences and smoothing by means of convolutions. Convolution between distributions and regularization of distributions. Denseness of D(U) in D'(U).
Sobolev spaces. Definition of weak derivatives and his motivation. Sobolev spaces Wk,p(U) and their properties. Interior and global approximation by smooth functions. Extensions. Traces. Embeddings theorems: Gagliardo-Nirenberg-Sobolev inequality and Embedding theorem for p < n. Embedding theorem for p = n. Hölder spaces. Morrey inequality. Embedding theorem for p > n. Sobolev inequalities in the general case. Compact embeddings: Rellich-Kondrachov theorem, Poincaré inequalities. Characterization of the dual space H-1.
Second order parabolic equations. Definition of parabolici operator. Weak solutions for linear parabolici equations. existence of weak solutions: Galerkin approximation, construction of approximating solutions, energy estimates, existence and uniqueness of solutions. Existence of solutions of viscous scalar conservation laws.
First order nonlinear hyperbolic equations. Scalar conservation laws: derivation, examples. Weak solutions, Rankine-Hugoniot conditions, entropy conditions. L1 stability, uniqueness and comparison for weak entropy solutions. Convergence of the vanishing viscosity and existence of the weak, entropy solution. Riemann problem. Definition of hyperbolic system. Quasilinear hyperbolic systems, symmetric and symmetrizable systems. Existence of solutions: approximations, a priori estimate, local existence of classical solutions.
V.S. Vladimirov, Equations of Mathematical Physics. Marcel Dekker, Inc..
C.M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics. Springer.
L.C. Evans, Partial Differential Equations. AMS.
M.E. Taylor, Partial Differential Equations, Nonlinear equations. Springer.
H. Brezis, Sobolev Spaces and Partial Differential Equations. Springer.
Basic notions of functional analysis, functions of complex values, standard properties of the heat equation, wave equation, Laplace and Poisson's equations.
This course is designed to give an overview of fluid dynamics from a mathematical viewpoint and to introduce students to the mathematical modeling of fluid dynamic type. At the end of the course students will be able to perform a qualitative and quantitative analysis of solutions for particular fluid dynamics problems and to use concepts and mathematical techniques learned from this course for analysis of other partial differential equations.
Derivation of the governing equations: Euler and Navier-Stokes.
Eulerian and Lagrangian description of fluid motion; examples of fluid flows.
Vorticity equation in 2D and 3D.
Dimensional analysis: Reynolds number, Mach Number, Frohde number.
From compressible to incompressible models.
Fluid dynamic modeling in various fields: biofluids, atmosphere and ocean, astrophysics.
Existence of solutions for viscid and inviscid fluids.
Linux/Unix OS and tools;
Basic Fortran (or C);
HPC architecture and libraries;
Application (ex ODEs, PDEs, elastodynamics).
Mathematical Analysis, Fourier transform.
This course provides an introduction to the classical kinetic theory of gases and the principles of kinetic modeling.
A special focus is given to the derivation of hydrodynamic equations from kinetic models by means of non-perturbative techniques and to the analysis of numerical schemes for the simulation of fluid flows.
On successful completion of this module the student has the knowledge on the basic principles and the simulation strategies of kinetic models.
Boltzmann equation and the principles of kinetic description.
Kinetic models: BGK,Maxwell molecules, Vlasov equation and Fokker-Planck equation.
The closure problem and methods of reduced description: Chapman-Enskog expansion, Grad's Moment method.
Non-perturbative techniques in kinetic theory: the method of the slow invariant manifold.
Overview on Lattice Boltzmann models.
Monte Carlo simulations of lattice gas models.
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