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This book discusses some aspects of the theory of partial differential equations from the viewpoint of probability theory. It is intended not only for specialists in partial differential equations or probability theory but also for specialists in asymptotic methods and in functional analysis. It is also of interest to physicists who use functional integrals in their research. The work contains results that have not previously appeared in book form, including research contributions of the author.

Partial differential equations --- Differential equations, Partial. --- Probabilities. --- Integration, Functional. --- Functional integration --- Functional analysis --- Integrals, Generalized --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk --- A priori estimate. --- Absolute continuity. --- Almost surely. --- Analytic continuation. --- Axiom. --- Big O notation. --- Boundary (topology). --- Boundary value problem. --- Bounded function. --- Calculation. --- Cauchy problem. --- Central limit theorem. --- Characteristic function (probability theory). --- Chebyshev's inequality. --- Coefficient. --- Comparison theorem. --- Continuous function (set theory). --- Continuous function. --- Convergence of random variables. --- Cylinder set. --- Degeneracy (mathematics). --- Derivative. --- Differential equation. --- Differential operator. --- Diffusion equation. --- Diffusion process. --- Dimension (vector space). --- Direct method in the calculus of variations. --- Dirichlet boundary condition. --- Dirichlet problem. --- Eigenfunction. --- Eigenvalues and eigenvectors. --- Elliptic operator. --- Elliptic partial differential equation. --- Equation. --- Existence theorem. --- Exponential function. --- Feynman–Kac formula. --- Fokker–Planck equation. --- Function space. --- Functional analysis. --- Fundamental solution. --- Gaussian measure. --- Girsanov theorem. --- Hessian matrix. --- Hölder condition. --- Independence (probability theory). --- Integral curve. --- Integral equation. --- Invariant measure. --- Iterated logarithm. --- Itô's lemma. --- Joint probability distribution. --- Laplace operator. --- Laplace's equation. --- Lebesgue measure. --- Limit (mathematics). --- Limit cycle. --- Limit point. --- Linear differential equation. --- Linear map. --- Lipschitz continuity. --- Markov chain. --- Markov process. --- Markov property. --- Maximum principle. --- Mean value theorem. --- Measure (mathematics). --- Modulus of continuity. --- Moment (mathematics). --- Monotonic function. --- Navier–Stokes equations. --- Nonlinear system. --- Ordinary differential equation. --- Parameter. --- Partial differential equation. --- Periodic function. --- Poisson kernel. --- Probabilistic method. --- Probability space. --- Probability theory. --- Probability. --- Random function. --- Regularization (mathematics). --- Schrödinger equation. --- Self-adjoint operator. --- Sign (mathematics). --- Simultaneous equations. --- Smoothness. --- State-space representation. --- Stochastic calculus. --- Stochastic differential equation. --- Stochastic. --- Support (mathematics). --- Theorem. --- Theory. --- Uniqueness theorem. --- Variable (mathematics). --- Weak convergence (Hilbert space). --- Wiener process.

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This book offers a survey of recent developments in the analysis of shock reflection-diffraction, a detailed presentation of original mathematical proofs of von Neumann's conjectures for potential flow, and a collection of related results and new techniques in the analysis of partial differential equations (PDEs), as well as a set of fundamental open problems for further development.Shock waves are fundamental in nature. They are governed by the Euler equations or their variants, generally in the form of nonlinear conservation laws-PDEs of divergence form. When a shock hits an obstacle, shock reflection-diffraction configurations take shape. To understand the fundamental issues involved, such as the structure and transition criteria of different configuration patterns, it is essential to establish the global existence, regularity, and structural stability of shock reflection-diffraction solutions. This involves dealing with several core difficulties in the analysis of nonlinear PDEs-mixed type, free boundaries, and corner singularities-that also arise in fundamental problems in diverse areas such as continuum mechanics, differential geometry, mathematical physics, and materials science. Presenting recently developed approaches and techniques, which will be useful for solving problems with similar difficulties, this book opens up new research opportunities.

Shock waves --- Von Neumann algebras. --- MATHEMATICS / Differential Equations / Partial. --- Algebras, Von Neumann --- Algebras, W --- Neumann algebras --- Rings of operators --- W*-algebras --- C*-algebras --- Hilbert space --- Shock (Mechanics) --- Waves --- Diffraction --- Diffraction. --- Mathematics. --- A priori estimate. --- Accuracy and precision. --- Algorithm. --- Andrew Majda. --- Attractor. --- Banach space. --- Bernhard Riemann. --- Big O notation. --- Boundary value problem. --- Bounded set (topological vector space). --- C0. --- Calculation. --- Cauchy problem. --- Coefficient. --- Computation. --- Computational fluid dynamics. --- Conjecture. --- Conservation law. --- Continuum mechanics. --- Convex function. --- Degeneracy (mathematics). --- Demetrios Christodoulou. --- Derivative. --- Dimension. --- Directional derivative. --- Dirichlet boundary condition. --- Dirichlet problem. --- Dissipation. --- Ellipse. --- Elliptic curve. --- Elliptic partial differential equation. --- Embedding problem. --- Equation solving. --- Equation. --- Estimation. --- Euler equations (fluid dynamics). --- Existential quantification. --- Fixed point (mathematics). --- Flow network. --- Fluid dynamics. --- Fluid mechanics. --- Free boundary problem. --- Function (mathematics). --- Function space. --- Fundamental class. --- Fundamental solution. --- Fundamental theorem. --- Hyperbolic partial differential equation. --- Initial value problem. --- Iteration. --- Laplace's equation. --- Linear equation. --- Linear programming. --- Linear space (geometry). --- Mach reflection. --- Mathematical analysis. --- Mathematical optimization. --- Mathematical physics. --- Mathematical problem. --- Mathematical proof. --- Mathematical theory. --- Mathematician. --- Melting. --- Monotonic function. --- Neumann boundary condition. --- Nonlinear system. --- Numerical analysis. --- Parameter space. --- Parameter. --- Partial derivative. --- Partial differential equation. --- Phase boundary. --- Phase transition. --- Potential flow. --- Pressure gradient. --- Quadratic function. --- Regularity theorem. --- Riemann problem. --- Scientific notation. --- Self-similarity. --- Special case. --- Specular reflection. --- Stefan problem. --- Structural stability. --- Subspace topology. --- Symmetrization. --- Theorem. --- Theory. --- Truncation error (numerical integration). --- Two-dimensional space. --- Unification (computer science). --- Variable (mathematics). --- Velocity potential. --- Vortex sheet. --- Vorticity. --- Wave equation. --- Weak convergence (Hilbert space). --- Weak solution.