A novel integral transform approach to solving partial differential - PowerPoint PPT Presentation

A novel integral transform approach to solving partial differential equations in the curved space-times Karen Yagdjian University of Texas Rio Grande Valley Microlocal and Global Analysis, Interactions with Geometry Colloquium in honor of Professor Schulze’s 75th birthday University of Potsdam, March 4-8, 2019 Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

The Integral Transform: purpose and structure The purpose: target problem (partial differential equations) The structure: The function subject to transformation The kernel function Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

Outline The Target Equations Motivation. Gas Dynamics. The Expanding Universe From Duhamel’s Principle to Integral Transform The Kernel of Integral Transform Applications The Klein-Gordon Equation in the de Sitter Space-time Maximum principle for hyperbolic equations Estimates for solution Huygens’ Principle. Semilinear equation in the de Sitter space-time Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

The Target Equation: x ∈ Ω ⊆ R n . ∂ 2 t u − a 2 ( t ) A ( x , ∂ x ) u − M 2 u = f , t ∈ (0 , T ) , Here M ∈ C and � a α ( x ) ∂ α A ( x , ∂ x ) = x , | α |≤ m ∂ α x = ∂ α 1 x 1 · · · ∂ α n x n , | α | = α 1 + . . . + α n The Goal : Explicit representation for the solutions of that equation The Tool : The new integral transform Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

Why this equation? x ∈ Ω ⊆ R n . ∂ 2 t u − a 2 ( t ) A ( x , ∂ x ) u − M 2 u = f , t ∈ (0 , T ) , Here M ∈ C and � a α ( x ) ∂ α A ( x , ∂ x ) = x | α |≤ m Equations of Gas Dynamics Equations of Physics in Expanding Universe Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

Gas Dynamics Tricomi equation (Chaplygin’1909, Tricomi’1923): x ∈ Ω ⊆ R n . ∂ 2 t u − t ∆ u = f , t ∈ R , The equation representing in hodograph variables a steady transonic flow (flight) of ideal gas. The small disturbance equations for the perturbation velocity potential of a near sonic uniform flow of dense gases (Kluwick, Tarkenton, Cramer’93) x ∈ Ω ⊆ R n . ∂ 2 t u − t 3 ∆ u = f , t ∈ R , Here ∆ u = ∂ 2 u + ∂ 2 u + · · · + ∂ 2 u ∂ x 2 ∂ x 2 ∂ x 2 n 1 2 Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

Einstein’s Equations with Cosmological Term, 1917 The metric (tensor) g µν = g µν ( x 0 , x 1 , x 2 , x 3 ) , where µ, ν = 0 , 1 , 2 , 3 R µν − 1 2 g µν R = 8 π GT µν − Λ g µν R µν is the Ricci tensor R = g µν R µν Scalar curvature Energy-momentum tensor T µν Λ is the cosmological constant Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

The de Sitter space-time The line element in the spatially flat de Sitter space-time has the form − c 2 dt 2 + e 2 Ht ( dx 2 + dy 2 + dz 2 ) , ds 2 = − c 2  0 0 0  e 2 Ht 0 0 0   g ik =  e 2 Ht  0 0 0   e 2 Ht 0 0 0 c is the speed of light, H is the Hubble constant. We set c = 1 and H = 1. ds 2 = − dt 2 + a 2 sc ( t ) d σ 2 , where a sc ( t ) = e Ht is the scale factor. Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

Big Bang and evolution of the Universe scale factor e t 150 de Sitter model 100 Einstein - de Sitter spacetime ( matter dominated universe ) scale factor t 2 / 3 radiation dominated universe scale factor t 50 Time Big Bang 1 2 3 4 5 Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

The Covariant Wave and Klein-Gordon Equations The covariant wave equation 1 ∂ �� | g ( x ) | g ik ( x ) ∂ψ � = f . � ∂ x i ∂ x k | g ( x ) | The covariant Klein-Gordon Equation 1 ∂ �� | g ( x ) | g ik ( x ) ∂ψ � − m 2 ψ = f , ∂ x i ∂ x k � | g ( x ) | where | g ( x ) | := | det( g ik ( x )) | and x = ( x 0 , x 1 , x 2 , x 3 ) ∈ R 4 , x 0 = t . The Einstein’s summation notation convention is used. Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

Waves in the Universe (Cosmological Models) The (non-covariant) wave equation in the radiation dominated universe: u tt − t − 1 A ( x , ∂ x ) u = f . The wave equation in the Einstein-de Sitter space-time (matter dominated universe). The covariant d’Alambert’s operator, after the change ψ = t − 1 u of the unknown function, leads to u tt − t − 4 / 3 A ( x , ∂ x ) u = f . Here √ 1 − Kr 2 ∂ � 1 − Kr 2 ∂ u � r 2 � A ( x , ∂ x ) u = r 2 ∂ r ∂ r � ∂ � 2 1 ∂ � sin θ∂ u � 1 + + u , r 2 sin θ r 2 sin 2 θ ∂θ ∂θ ∂φ where K = − 1 , 0, or +1, for a hyperbolic, flat or spherical spatial geometry, respectively. Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

The Klein-Gordon Equation in Expanding Universe The metric g 00 = − 1, g 0 j = 0, g ij = e 2 t σ ij ( x ), i , j = 1 , 2 , . . . , n , Scale factor a sc ( t ) = e t (accelerating expansion). The covariant Klein-Gordon equation in the de Sitter space-time: ψ tt − e − 2 t A ( x , ∂ x ) ψ + n ψ t + m 2 ψ = f . Here m is a physical mass of the field (particle) while n �� � 1 ∂ | det σ ( x ) | σ ij ( x ) ∂ � A ( x , ∂ x ) ψ = ∂ x j ψ � ∂ x i | det σ ( x ) | i , j =1 If u = e nt / 2 ψ , then u tt − e − 2 t A ( x , ∂ x ) u − M 2 u = f , where M 2 = n 2 4 − m 2 is curved (or effective) mass. This example includes equations in the metric with hyperbolic, flat or spherical spatial geometry. Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

The Klein-Gordon Equation of Self-Interacting Field in Expanding Universe In the spatially flat de Sitter universe the equation for the scalar field with mass m and potential function V is c 2 nH φ t − e − 2 Ht ∆ φ + m 2 c 2 c 2 φ tt + 1 1 h 2 φ = 1 c 2 V ′ ( φ ) . ∂ 2 Here x ∈ R n , t ∈ R , and ∆ is the Laplace operator, ∆ := � n j , j =1 ∂ x 2 � H = Λ / 3 is the Hubble constant, Λ is the cosmological constant. In the case of Higgs potential (Higgs boson) φ tt + 3 H φ t − e − 2 Ht c 2 ∆ φ = µ 2 φ − λφ 3 with λ > 0 and µ > 0 while n = 3. Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

The New Integral Transform Let f = f ( x , t ) be a given function of t ∈ (0 , T ), x ∈ Ω. Ω is a domain in R n , | α |≤ m a α ( x ) ∂ α A ( x , ∂ x ) = � x . The function w = w ( x , t ; b ) is a solution of the problem w tt − A ( x , ∂ x ) w = 0 , t ∈ (0 , T 1 ) , x ∈ Ω , w ( x , 0; b ) = f ( x , b ) , w t ( x , 0; b ) = 0 , x ∈ Ω , with the parameter b ∈ (0 , T ) and 0 < T 1 ≤ ∞ . We introduce the integral operator K : w �− → u , which maps function w = w ( x , t ; b ) into solution of the equation u tt − a 2 ( t ) A ( x , ∂ x ) u − M 2 u = f , t ∈ (0 , T ) , x ∈ Ω . Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with

The New Integral Transform The integral operator K : w �− → u is u ( x , t ) = K [ w ]( x , t ) � t � | φ ( t ) − φ ( b ) | := db K ( t ; r , b ; M ) w ( x , r ; b ) dr , x ∈ Ω , t ∈ (0 , T ) . 0 0 � t Here φ ( t ) = a ( τ ) d τ is a distance function produced by a = a ( t ), 0 M ∈ C is a constant. Integral transform is applicable to the distributions and fundamental solutions as well. In fact, u = u ( x , t ) takes initial values u ( x , 0) = 0 , u t ( x , 0) = 0 , x ∈ Ω . Karen Yagdjian (University of Texas RGV) A novel integral transform approach to solving partial differential equations in the curved space-times Microlocal and Global Analysis, Interactions with