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In this thesis we collect the study of two problems in the Calculus of Variations and Partial Differential Equations. Our first group of results concern the analysis of minimizers in a variational model describing the shape of liquid drops and crystals under the influence of gravity, resting on a horizontal surface. Making use of anisotropic symmetrization techniques and an analysis of fine properties of minimizers within the class of sets of finite perimeter, we establish existence, convexity and symmetry of minimizers. In the case of smooth surface tensions, we obtain uniqueness of minimizers via an ODE characterization. In the second group of results discussed in this thesis, which is joint work with A. Vasseur, we treat a problem in compressible fluid dynamics, establishing a uniform bound from below on the temperature for a variant of the compressible Navier-Stokes-Fourier system under suitable hypotheses. This system of equations forms a mathematical model of the motion of a compressible fluid subject to heat conduction. Building upon the work of (Mellet, Vasseur 2009), we identify a class of weak solutions satisfying a localized form of the entropy inequality (adapted to measure the set where the temperature becomes small) and use a form of the De Giorgi argument for L[superscript infinity] bounds of solutions to elliptic equations with bounded measurable coefficients.

In this thesis we collect the study of two problems in the Calculus of Variations and Partial Differential Equations. Our first group of results concern the analysis of minimizers in a variational model describing the shape of liquid drops and crystals under the influence of gravity, resting on a horizontal surface. Making use of anisotropic symmetrization techniques and an analysis of fine properties of minimizers within the class of sets of finite perimeter, we establish existence, convexity and symmetry of minimizers. In the case of smooth surface tensions, we obtain uniqueness of minimizers via an ODE characterization. In the second group of results discussed in this thesis, which is joint work with A. Vasseur, we treat a problem in compressible fluid dynamics, establishing a uniform bound from below on the temperature for a variant of the compressible Navier-Stokes-Fourier system under suitable hypotheses. This system of equations forms a mathematical model of the motion of a compressible fluid subject to heat conduction. Building upon the work of (Mellet, Vasseur 2009), we identify a class of weak solutions satisfying a localized form of the entropy inequality (adapted to measure the set where the temperature becomes small) and use a form of the De Giorgi argument for L[superscript infinity] bounds of solutions to elliptic equations with bounded measurable coefficients.

The role of compressibility on the convective velocity of large-scale structures in axisymmetric jets is studied using a home-built pulse burst laser system and newly developed high-repetition rate experimental diagnostics. A pulse burst laser system was designed and constructed with the ability to produce a burst of short duration (10 nsec), high energy (order of 10 -100 mJ/pulse) pulses over a ~150 microsecond period with inter-pulse timing as low as 1 microsecond (1 MHz). The application of the pulse burst laser for flow measurements was investigated through the development of MHz rate flow visualization and MHz rate planar Doppler velocimetry (PDV). MHz rate PDV is a spectroscopic technique that produces 28 time-correlated realizations of the velocity over a plane with a maximum repetition rate of up to 1 MHz and accuracies on the order of 5%. Space-time correlations were used to track structures within the flow field and determine their convective velocity. Data produced using flow visualization images agrees with previous research and indicates a strong departure of the convective velocity from theory. Data produced using velocity data, however, shows starkly different trends and does not produce the same measurements of convective velocity. This difference in measurement is attributed to a misinterpretation of the use of space-time correlation for tracking structures. The presence of a distinct boundary between the mixing layer and the jet core as well as the mixing layer and ambient air in the flow visualization data and some of the velocity data leads to a bias in the measurement. The space-time correlation is found to preferentially follow these boundaries, thus leading to faster and/or slower measurements of convective velocity. For the Mach 2.0 jet, velocity data was obtained with seed particles marking the jet core and the mixing layer, but not the ambient air. This lack of velocity measurements on the low-speed side of the jet’s mixing layer biased the space-time correlation results. An algorithm was developed to extrapolate the velocity into unseeded regions of the flow. Space-time correlation results based on the extrapolated data indicate a convective velocity close to the theoretical value.