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The purpose of this thesis is to study the phenomenon of singularity formation in large data problems for C^1 solutions to the Cauchy problem of the compressible Euler equations. The classical theory established by P. D. Lax in 1964 (J. Math. Phys. 5: 611-614) shows that, for 2x2 hyperbolic systems, the break-down of C^1 solutions occurs in finite time if initial data contain any compression in some truly nonlinear characteristic field under some additional conditions, which include genuine nonlinearity and the strict positivity of the difference between two corresponding eigenvalues. These strong structural assumptions mean that it is highly non-trivial to apply this theory to archetypal systems of conservation laws, such as the (1+1)-dimensional relativistic Euler equations. Actually, in the (1+1)-dimensional spacetime setting, if the mass-energy density rho does not vanish initially at any finite point, the essential difficulty in considering the possible break-down is coming up with a way to obtain sharp enough control on the lower bound for rho. In Chapter 2 and Chapter 3, we consider the (1+1)-dimensional isentropic Euler equations in both classical and relativistic settings. We will establish a sufficiently sharp lower bound of rho, and show Lax's theory indeed holds for these systems. Furthermore, we will give a better lower bound of rho, at the expense of restricting the class of initial data further. In Chapter 4 and Chapter 5, we consider the (1+1)-dimensional non-isentropic Euler equations in both classical and relativistic settings. As the non-isentropic Euler equations are 3x3 systems, Lax's theory no longer directly applies. Instead, we will establish a sufficiently sharp lower bound of rho, and give a sufficient condition for the solution to break down in finite time. In Chapter 6 and Chapter 7, we consider the multidimensional isentropic Euler equations with spherical symmetry in both classical and relativistic settings. With spherical symmetry, the system will have a singularity at the origin. To bypass this, we consider a specific set of initial data such that the local existence and uniqueness result can be established. Same as in the non-isentropic case, we will give a sufficient condition for the solution to break down in finite time.

The well-posedness theory of the full compressible Navier-Stokes equations (CNS for short) has been an important subject in the development of mathematical physics. For the spherically symmetric flow, even though many favourable regularity properties may be expected for its similarity to the one-dimensional equations, there are still a large number of open questions remaining unsolved. Two central difficulties of spherically symmetric CNS are the coordinate singularity at the centre of symmetry, and the upper/lower bounds of density. In Chapter 1, the Cauchy problems of CNS for the general multidimensional flows, both with and without spherical symmetry, are introduced. Then a heuristic analysis on the Lagrangian coordinate transformation is conducted, which gives insight to the strategies used in the later chapters. Finally, several previous progresses on the well-posedness theory of CNS are highlighted at the end of this chapter. In Chapter 2, the global-in-time existence of a spherically symmetric weak solution to the Cauchy problem in Eulerian coordinate is obtained. The main strategy is to construct a sequence of approximation problems posed in finite annular domains. This circumvents the problem of coordinate singularity at the origin. Moreover, they are also formulated in the Lagrangian coordinates, so that one can obtain a-priori estimates which are uniform with respect to the dimension of annular domains. The weak solution is then constructed as the limit of approximate solutions in the original Eulerian coordinate. The main result of this chapter is produced in collaboration with Gui-Qiang G. Chen and Shengguo Zhu. In Chapter 3, for the Lagrangian approximation problem introduced in Chapter 2, the existence and uniqueness of a local-in-time strong solution are shown. This is done via the Picard iteration scheme, however some major difficulties arise due to the fact that second order differential operators in momentum and energy equations is quasilinear under the Lagrangian formulation. To resolve this, a set of a-priori bounds on each iterative solutions is established so that the quasilinear operators are elliptic at each iterative step. Moreover, the time dependent high regularity estimates for linear iterative solution are carefully derived so that Banach Fixed-point theorem can be applied to obtain the desired solution. In Chapter 4, it is proved that the local-in-time strong solution in Chapter 3 can be extended to all time by bootstrap argument. Moreover, global-in-time strong solutions in the unbounded Lagrangian domain are also obtained by taking the limit of outer and inner radii of annular domains. The main difference of this chapter compared to Chapter 2 is that the limit is taken in the Lagrangian coordinates, and the initial data are assumed to have higher regularities.

This thesis is a study of problems related to isometric immersions of Riemannian manifolds in Euclidean space. We address three main questions: the weak continuity of geometric invariants along sequences of immersions of Riemannian manifolds; the construction of isometric immersions by weak compactness methods; and the validity and regularity of the Gauss equation. First, we investigate the validity of Cartan's equations for W1,p coframes on surfaces, for all 1 ≤ p ≤ ∞, and employ this to derive a version of the Gauß equation valid for W2,p immersed surfaces in R3. Under some additional regularity hypotheses, a distributional formulation of the Gauß equation on immersed surfaces in R3 is proved, and as a corollary, a new local regularity result is established for isometric immersions of positively curved surfaces. Investigating the weak continuity properties of immersions of Riemannian manifolds, we first prove a general weak continuity result for curvatures of connections with Lp bounds on principal bundles. As a corollary, it is proved that when p > 2, curvatures are weakly continuous for the weak W2,p convergence of immersions. In some cases, we also recover the weak continuity of the general Gauss equation when p=2. Finally, we give a general viscosity framework for constructing isometric immersions in prescribed target spaces under natural boundedness assumptions in Lp spaces. Assuming this set-up, we prove a new weak compactness theorem for approximate solutions of the Gauss--Codazzi--Ricci equations.

In this thesis we contribute to the study of two large data problems within the realm of Hyperbolic Partial Differential Equations. The first result concerns the Relativistic Euler equations, which adequately describe fluid motion in the context of special relativity. The main result is the provision of a both necessary and sufficient condition on the initial data, uniformly away from vacuum, for the formation of singularities in finite time under the evolution of (1) in the case d = 1. This is joint work with Shengguo Zhu. The second project developed in this thesis shifts focus from the area of conservation laws to that of mathematical General Relativity. We provide, in joint work with Xinliang An, a trapped surface formation criterion for the Einstein-Maxwell system where the right-hand side corresponds to the Maxwell energy-momentum tensor that models electromagnetic effects.

This thesis comprises an introduction and three subsequent chapters; each focusing on a particular problem, and containing novel results that complement the existing theory. Throughout, we are concerned with the existence of solutions of the governing equations of inviscid compressible fluid mechanics; the Euler equations. As these equations form an archetypal system of conservation laws, we study them using approaches from the theory of hyperbolic differential equations. Specifically, we implement methods of compensated compactness developed by Tartar and Murat. To begin with, in Chapter 2, we prove the existence of a finite relative energy entropy solution of the one-dimensional compressible Euler equations under the assumption of an approximately isothermal pressure law. In particular, we obtain this solution as the vanishing viscosity limit of solutions of the one-dimensional compressible Navier–Stokes equations. This procedure can in fact be carried out for a more general class of pressure laws, which we call asymptotically isothermal. This is the subject of Chapter 3, where we show the existence of a finite relative energy entropy solution of the Euler equations in this new setting. The focus of Chapter 4 is a related two-dimensional stationary problem. Therein, we consider the existence of bounded entropy solutions of the steady compressible potential flow equations in two dimensions; the Morawetz problem for transonic flow. We provide partial results for a γ-law gas of index γ ∈ [3,∞). This involves a detailed analysis of the singular ordinary differential equations that arise when considering the Lax entropy pairs of the system in the presence of a vacuum, which had yet to be performed.

The study of this thesis is motivated by the stochastic Lagrangian representations of solutions to the Navier-Stokes equations. The stochastic Lagrangian formulation to the Navier-Stokes equations is described by stochastic differential equations, which essentially represent the diffusions under divergence-free velocity fields. The associated stochastic differential equations are closely related to a class of parabolic equations and these two types of equations are the central objects of this thesis. The difficulty of the problem mainly comes from the low regularity of the velocity field. The key point is that we use the divergence-free condition to relax the regularity assumptions. The thesis is divided into two parts. The first part is the Aronson-type estimate which is an a priori estimate on the fundamental solutions (transition probability) independent of the smoothness of the coefficients. In the critical case, we obtain the Aronson estimate in its classical form, while in supercritical cases we obtain a weaker Aronson-type estimate. In the second part, we use approximation arguments to apply the Aronson estimate to the construction of solutions to the parabolic equations and the stochastic differential equations, and further regularity theory of the solutions is obtained for the critical case. Under the supercritical conditions, we will focus on the uniqueness of solutions to the parabolic equations and their relation to the construction of the diffusion processes.

In this thesis, we present a collection of newly obtained results concerning the existence of vanishing viscosity solutions to the one-dimensional compressible Euler equations of gas dynamics, with and without geometric structure. We demonstrate the existence of such vanishing viscosity solutions, which we show to be entropy solutions, to the transonic nozzle problem and spherically symmetric Euler equations in Chapter 4, in both cases under the simple and natural assumption of relative finite-energy. In Chapter 5, we show that the viscous solutions of the one-dimensional compressible Navier-Stokes equations converge, as the viscosity tends to zero, to an entropy solution of the Euler equations, again under the assumption of relative finite-energy. In so doing, we develop a compactness framework for the solutions and approximate solutions to the Euler equations under the assumption of a physical pressure law. Finally, in Chapter 6, we consider the Euler equations in special relativity, and show the existence of bounded entropy solutions to these equations. In the process, we also construct fundamental solutions to the entropy equations and develop a compactness framework for the solutions and approximate solutions to the relativistic Euler equations.

In this Thesis, we study the problem of Shock Reflection-Diffraction by a Wedge for the Nonlinear Wave System. Shock Reflection-Diffraction problems in Fluid Dynamics, and the corresponding wave patterns have been studied by many authors, via many different analytical, experimental and numerical methods. The Nonlinear Wave System is obtained from the Full Euler System, by making the assumption that the inertial terms are small. It has many similar properties (such as the structure of the characteristics) in common with it, so is a useful model for understanding the Full Euler System, and also plays an important role in some operator splitting schemes in numerical analysis. The solutions to the problem of Shock Reflection-Diffraction by a wedge that we seek are Self-Similar Regular Shock Reflection-Diffraction configurations. The first necessary ingredient in the structure of these solutions is the existence of local reflection at the point where the incident shock hits the wedge. We find that there exists sonic and detachment angles, depending only on the initial data, determining the existence of the local reflection, and the pattern which appears. With the local theory in hand, we use a similar approach to that used in [1, 26], by using a nonlinear method of continuity that relies on the Leray- Schauder degree theory from [87], to obtain existence in all cases where the local reflection exists. There are three key difficulties in the approach. The first is the changing elliptic or degenerate elliptic structure as the wedge angle passes through the sonic angle, and the corresponding change in wave patterns. In particular, we find that for some subsonic wedge angles, we require a new technique to attain suitable regularity near the reflection point, from the limited structure estimates available. The second is the degenerate obliqueness of the free boundary condition where the shock wave meets the flat part of the wedge. Because obliqueness is a property that relies not just on the values of the coeficients, but on the shape of the boundary, this is handled via a new technique involving imposing a cutoff on the boundary itself. The third is ensuring that, in the iteration problem, we obtain a gainin- regularity so that compactness of the iteration map holds. This involves careful analysis of the regularity of both the density and boundary.

This dissertation consists chiefly of three parts, which tell different facets in the development of one topic. The first part is an exploration of continuous dependence estimates of stochastically driven degenerate parabolic equations. The second is an extension of work done by Debussche and Vovelle on first order stochastic conservation laws - we extend their results to degenerate parabolic-hyperbolic conservation laws with additive noise, and derive results on the existence and uniqueness of invariant measures. In the third part we explore the long time behaviour of solutions to stochastic degenerate parabolic-hyperbolic conservation laws with multiplicative noise, depending non-linearly on the solution itself.

In the thesis we investigate two problems on Partial Differential Equations (PDEs) in differential geometry and fluid mechanics. First, we prove the weak L p continuity of the Gauss-Codazzi-Ricci (GCR) equations, which serve as a compatibility condition for the isometric immersions of Riemannian and semi-Riemannian manifolds. Our arguments, based on the generalised compensated compactness theorems established via functional and micro-local analytic methods, are intrinsic and global. Second, we prove the vanishing viscosity limit of an incompressible fluid in three-dimensional smooth, curved domains, with the kinematic and Navier boundary conditions. It is shown that the strong solution of the Navier-Stokes equation in H r+1 (r > 5/2) converges to the strong solution of the Euler equation with the kinematic boundary condition in H r, as the viscosity tends to zero. For the proof, we derive energy estimates using the special geometric structure of the Navier boundary conditions; in particular, the second fundamental form of the fluid boundary and the vorticity thereon play a crucial role. In these projects we emphasise the linkages between the techniques in differential geometry and mathematical hydrodynamics.

We study the hydrodynamic limit for a one dimensional isothermal anharmonic finite chain in Lagrangian coordinates with hyperbolic space-time scaling. The temperature is kept constant by putting the chain in contact with a heath bath, realised via the addition of a stochastic momentum-preserving noise to the dynamics of the chain. The noise is designed to be large at the microscopic level, but vanishing in the hydrodynamic limit. Boundary conditions are also considered: one end of the chain is kept fixed, while a time-variable tension is applied to the other end. We show that the microscopic deformation and momentum converge (in an appropriate sense) to solutions of a system of hyperbolic conservation laws (isothermal Euler equations in Lagrangian coordinates) with boundary conditions. Since these solutions may develop shocks in a finite time, they are obtained in a weak sense. This is done by adapting the theory of compensated compactness to a stochastic setting. Finally, the external tension allows us to define thermodynamic transformations between equilibrium states. We use this to deduce the first and the second principle of Thermodynamics for our model.

The main purpose of this work is to prove short-time structural stability of compressible vortex sheets with surface tension. The main result can be summarised as follows. Assume we start with an initial vortex-sheet configuration which consists of two inviscid fluids with density bounded below flowing smoothly past each other, where a strictly positive fixed coefficient of surface tension produces a surface tension force across the common interface, balanced by the pressure jump. We assume the fluids are modelled by the compressible Euler equations in three space dimensions with a very general equation of state relating the pressure, entropy and density in each fluid such that the sound speed is positive. Then, for a short time, which may depend on the initial configuration, there exists a unique solution of the equations with the same structure, that is, two fluids with density bounded below flowing smoothly past each other, where the surface tension force across the common interface balances the pressure jump. The mathematical approach consists of introducing a carefully chosen artificial viscosity-type regularisation which allows one to linearise the system so as to obtain a collection of transport equations for the entropy, pressure and curl together with a parabolic-type equation for the velocity. We prove a high order energy estimate for the non-linear equations that is independent of the artificial viscosity parameter which allows us to send it to zero. This approach loosely follows that introduced by Shkoller et al in the setting of a compressible liquid-vacuum interface. Although already considered by Shkoller et al, we also make some brief comments on the case of a compressible liquid-vacuum interface, which is obtained from the vortex sheets problem by replacing one of the fluids by vacuum, where it is possible to obtain a structural stability result even without surface tension.