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0:00 Existence of the Fractional Derivative 07:51 Existence and Uniqueness of the Weak Solution for the Time-Dependent Navier-Stokes Equation 11:48 Existence in 3D 19:30 Approximation of the Solution via Galerkin Method 28:01 The Way to Prove the Existence 30:54 A Priori Bounds 36:12 Estimate for the Time Derivative 58:57 H-gamma Estimate 01:11:11 Limit Process

0:00 Existence and Convergence of the Solution for the Non-Linear Problem in the Finite-Dimensional Case 04:32 Stationary Navier-Stokes Equation 10:15 Existence of Weak Solutions 16:08 The Usage of Brouwer Fixed Point Theorem 21:32 Limit of Solution Sequence 34:54 From Weak Convergence to Strong Convergence 39:23 Fractional Derivatives 52:49 Compact Embedding 1:06:45 Strong Convergence in V0 Space

0:00 Existence of Weak Solutions for the Wave Equation by a Galerkin Method 15:29 Proof by Galerkin Method 18:43 Approximate Solutions 29:02 A Priori Estimates 45:50 Going to the Limit 59:33 The Space of Solutions in Wave and Heat Equations 1:01:30 Uniqueness of the Solutions 1:11:01 Time-Dependent Incompressible Navier-Stokes Equations in a Weak Form

0:00 Finalization the Discussion of Linear Parabolic Equations 00:40 Maximum Principle 08:00 Stampacchia's Lemma 16:59 Gluing Together of Solutions 37:06 From Scalar Equations to Systems 47:05 Introduction to Wave Equations

0:00 Existence of Weak Solutions of the Energy Method 10:32 Proof of the Equivalence in L2 Norm 42:15 Approximation in H1.0 45:00 Generalizations 45:47 The Inhomogeneous Case 56:41 Inhomogeneous Boundary Conditions 01:05:42 Neumann Boundary Data 01:19:20 General Elliptic Operators 01:25:57 With General Boundary Conditions

0:00 Weak Convergence of Solutions of the Energy Method for the Heat Equation 05:34 Fundamental Theorem of Calculus for W1,1 11:48 Gelfand Triple 33:24 Lions-Aubin Theorem 47:14 Gronwall Inequality 59:00 Implicit Euler Method 1:05:46 A Priori Bound

0:00 Introduction 8:34 Energy Method for the Heat Equation 39:00 Bochner Spaces

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