Questions on "Partial Differential Equations", as opposed to "ordinary differential equations".

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How can I use J.-L. Lions’s principle of intermediate derivatives to prove continuity in time?

If $ u ∈ L^{\infty}_{loc}(0,T ,H^1(X))$, $u_{t} ∈ L^{\infty}_{loc}(0,T ,L^2(X))$ and $u_{tt} ∈ L^{\infty}_{loc}(0,T ,H^{-1}(X))$ How does one prove that $u_{t} ∈ C([0,T] ,L^2(X))$? I was told using ...
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1answer
16 views

In search of a necessary condition for completeness of some metric space with application to pde

$A$ is an operator. Consider a metric space $K$ (a function $f$ is in $K$ if and only if $Af$ is in $L^2$) where the metric between two functions $f$ and $g$ is defined as $\mu (f ,g) = \int_{R^3} ...
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26 views

Dissipation term in wave equation

If we're given a string with mass density $\rho$ in units $\frac{M}{L^3}$ with constant cross-section $A$, tension $T$ in units $\frac{F}{L^2}$, and whose length is $L$; and then we assume that the ...
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6 views

Linear (possibly underdetermined?) PDE

I'm currently trying to solve the following PDE: $\partial$$\mu_1$$(\theta_1,\theta_2)$$/$$\partial$$\theta_1$ + $\partial$$\mu_2$$(\theta_1,\theta_2)$$/$$\partial$$\theta_2$ $=$ ...
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1answer
24 views

Manipulating vorticity equation

We have $\omega = (0,0, \xi(x,y,t))$ and $\textbf u =(u(x,y,t),v(x,y,t),0)$ and that $$\frac{\partial \xi}{\partial t} +u \frac{\partial \xi}{\partial x} +v\frac{\partial \xi}{\partial y}=0$$ is a ...
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17 views

Use Duhamel's Principle to solve a PDE.

So I have a final and I do not understand how to use Duhamel's Principle. Can someone help me setup this problem on my practice test? The equation is $u_t-u_{xx}=tsin(x)$ $0<x<\pi$, $t>0$ ...
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1answer
31 views

Method of characteristics - heat convection

I have following PDE: \begin{align} v \cdot \frac{\partial T}{\partial x} + \frac{\partial T}{\partial t} &= k \cdot (T-T_0)\\ T(x,0) &= T_0\\ T(0,t) &= T_1 \end{align} It's first order ...
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1answer
22 views

Closedness of first order differential operator on $L^2(\Omega)$

I am considering the when the following first order differential operator is a closed operator $$Au=b(x)\dfrac{\partial u}{\partial x_i},$$ on $L^2(\Omega)$ with the domain $D(A)=H^1(\Omega)$. Here I ...
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1answer
26 views

Bound first order derivative by $L^2$ norm of elliptic elliptic operator

Consider an symmetric 2nd order differential operator on a bounded domain with smooth boundary $$A=-\sum_{i,j=1}^n \partial_j (a^{ij}(x)\partial_i)$$ be uniformly elliptic if there exists $C_0>0$ ...
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1answer
45 views

Consider the function : $L_{ij}=x_{i}\frac{\partial}{\partial x_j}-x_{j}\frac{\partial}{\partial x_i}$

Let $\Omega$ be a smooth bounded domain of $\mathbb{S}^n$, the unit $n$ sphere centered at the origin of $\mathbb{R}^{n+1}$, and consider the functions $$L_{ij}u=x_{i}\frac{\partial u}{\partial ...
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1answer
19 views

Clarification regarding heat kernel for Brownian motion on a manifold

Let $X$ be Brownian motion on a Riemannian manifold $M$ starting at $x\in M$, D a domain and $f$ a bounded continuous function on $D$. Define $\tau_D$ to be the first exit time of $X$ from $D$. ...
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1answer
32 views

Conclusions About Solution of Cauchy Problem

Let $u=u(x,t)$ be the solution of the Cauchy problem $$\frac{\partial u}{\partial t}+\left(\frac{\partial u}{\partial x}\right)^2=1, x\in\Bbb R, t>0$$ $$u(x,0)=-x^2$$ Then which of the ...
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37 views

Dirichlet Problem in Stochastic PDE Section of Probability Textbook.

I recently started learning about stochastic calculus and stochastic PDEs, and I don't know where to begin with some of the problems in my textbook. Any help with the following or a push in the right ...
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14 views

Modeling population density with PDE

If we know that the population density $u(x,t)$ in some lake varies as a function of $x>0$ and time $t$, where $x$ increases downwards with depth, and that the population diffuses with constant $D$ ...
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21 views

How to deal with exotic boundary conditions for Wave equation

I'm currently trying to solve a PDE, the problem is as follows: Solve the equation $\frac{\partial ^{2} u}{\partial t^{2} } = c^{2}\frac{\partial ^{2} u}{\partial x^{2} }$ on the semi-infinite ...
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1answer
86 views

How to integrate a PDE?

How does one integrate this PDE with respect to $x$? $$u\frac{\partial u}{\partial t}+ u^2\frac{\partial u}{\partial x}+u\frac{\partial^3 u}{\partial x^3}=0$$ My idea is to rewrite this equation as ...
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21 views

Writing PDE in the form of convervation law

What does one need to know in order to write $\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+\frac{\partial^3 u}{\partial x^3}=0$ in the form of a conservation law, which contains the ...
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1answer
65 views

Why are second order linear PDEs classified as either elliptic, hyperbolic or parabolic?

Is there a geometric interpretation of second order linear partial differential equations which explains why they are classified as either elliptic, hyperbolic or parabolic, or is this just a naming ...
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1answer
49 views

Proving surjectivity of Laplacian for the $L^{p}$ case, $1<p<\infty$

For $1<p<\infty$ and $\lambda>0$ I want to show that $\lambda-\Delta:W^{2,p}(\mathbb{R}^{n})\to L^{p}(\mathbb{R}^{n})$ is bijective. Injectivity is obvious since if we have $\lambda-\Delta ...
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13 views

Average value of harmonic function

I am trying to find the average value of a harmonic function over a sphere. I know that the average value will be at the center of a sphere by the Mean value property over spheres. However I was ...
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7 views

Relatively compact Family of Polynomials

Can someone please help me to solve this problem: For $r\in \mathbb{N}$ ,we note : $E_r$={$P\in \mathbb{R}[x_1,x_2,...,x_d] $ such that $deg P\le r$ } And for $P\in E_r$ we define the function ...
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1answer
10 views

Show that $\int r\cdot n ds$ equals three time the volume of $\omega$.

Let $\Omega$ be an open region in $\mathbb{R}^3$ with surface $∂\Omega$ on every point $P$ of which the unit outward pointing normal $n = n(P)$ is well defined and smoothly varying. Let $r = (x, y, ...
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1answer
37 views

Homogeneous Wave Equation with Dirichlet Conditions

Let $u(x,t)$ a solution of $u_{xx} = \frac{1}{c^2}u_{tt}, a<x<b, t>0.$ The integral energy $u$ is given by $E(t)=\int^{b}_{a}[u_x(x,t)^2+\frac{1}{c^2}u_t(x,t)^2]dx, t>0$. (i) Show that ...
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1answer
42 views

How do fixed point arguments for PDE work?

My understanding (which I'm not altogether sure of) is that a a "fixed point" argument often used in PDE goes something like this: If we have some PDE like $$u=F(u),$$ we consider a sequence of ...
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1answer
26 views

Can someone help me solve a first order pde?

I am having trouble and I am unsure how to solve this PDE. $x\frac{\partial v}{\partial x} +y\frac{\partial v}{\partial y} = 2xy(x^2-y^2)$ I know you will use method of characteristics, but I am being ...
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20 views

Witten Laplacian problem-show that $\lim\limits_{n \rightarrow +\infty} ||f_nu||=0$

Can someone help me please to correct my answer of this problem: We consider the Witten Laplacian with domain $C^∞_0(R^d)$. We know that this operator is essentially self adjoint,positive and with ...
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37 views

Is there a nontrivial solution to this wave equation boundary value problem?

Is there a nontrivial solution to the problem $\begin{cases}u_{xx}-u_{tt}=0\\u(0,t)=u_x(L,t)=0\end{cases}$ for $0<x<L,\ \ t>0$ maybe it should be $u_x(0,t)$ instead of $u(0,t)$, ...
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37 views

Finding the coefficients of $a^2\frac{\partial ^4u}{\partial x^4}+\frac{\partial ^2u}{\partial t^2}=0$

Solving using the substitution $U=XT$ yields $X=c_1\cosh(\alpha x)+c_2\sinh(\alpha x)+c_3\cos(\alpha x)+c_4\sin(\alpha x)$ and $T=c_5\cos(\alpha^2at)+c_6\sin(\alpha^2at)$ From the boundary conditions ...
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1answer
44 views

Passing from Classical Formulation to Weak Forulation on PDEs. Integration by parts in $n$ dimensions?

I am reading about the Variational Method for solving PDEs and ODEs. The process to pass from a classical solution to a weak solution is pretty clear when we are dealing with ODEs, by using ...
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18 views

Can the constant be a complex number in wave equation

Solving the Wave equation $u_{tt}=u_{xx}$ If I apply separation of variables, with $u(x,t)=f(x)g(t)$, then we find $$\frac{f''(x)}{f(x)}=\frac{g''(t)}{g(t)}=k$$ Can $k$ be a complex number ...
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1answer
23 views

Methods of Characteristics

I have a problem solving the ODE associated with the question, any help will be greatly appreciated. Use method of characteristics to solve the problem $(x-y)\dfrac{\partial u}{\partial ...
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1answer
22 views

Solving PDE of type $x u_x + y u_y = f(x, y)$

How do I solve the differential equation of the type $$ x u_x + y u_y = f(x, y) $$ For example, let $f(x, y) = xy$. Using following method, I found $F(x,y) = F(x/y)$ is solution to homogeneous ...
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1answer
17 views

Nonhomogeneous Diffusion Equation with Nonhomogeneous Dirichlet Boundary Condition

How to solve: \begin{cases} u_t - ku_{xx} = g(x,t) \hspace{2cm} k>0, x>0, t>0 \\ u(0,t) = f(t) \hspace{3cm} t>0 \\ u(x,0) = h(x) \hspace{2.8cm} x>0 \end{cases} I know how to solve ...
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55 views

Solving 2D Laplacian eigenvalue problem with non-standard Dirichlet boundary condition

I have to solve the following eigenvalue problem, i.e. find eigenvalues and eigenfunctions (some of you will notice that this is the Schrödinger equation): $$-\frac{\hbar^2}{2m}\left( ...
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1answer
34 views

A corollary of Li-Yau-Hamilton estimate

Picture below is from the Hamilton's The Harnack estimate for the Ricci flow .How to get the corollary 1.2 by Theorem 1.1 ? It seemly be not immediately and hard to compute. Maybe just because I am ...
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1answer
25 views

Separation of variables for $tu_t = u_{xx} + 2u$

Separate the variables for the equation $$tu_t = u_{xx} + 2u$$ with the boundary conditions $u(0,t) = u(π,t) = 0$. Show that there are an infinite number of solutions which satisfy the initial ...
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32 views

Options on Futures Black-Sholes

I am taking the Financial Risk Management course, and the topic now is "Variations on the Black-Scholes Model". I am following Paul Wilmott's "The Mathematics of Financial Derivatives: A Student ...
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76 views

Recast the scalar SPDE $du_t(Φ_t(x))=f_t(Φ_t(x))dt+∇ u_t(Φ_t(x))⋅ξ_t(Φ_t(x))dW_t$ into a SDE in an infinite dimensional function space.

Let$^1$ $(\Omega,\mathcal A,\operatorname P)$ be a probability space $U$ be a separable Hilbert space $Q\in\mathfrak L(U)$ be nonnegative and symmetric operator on $U$ with finite trace ...
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37 views

Under the given conditions. Prove that $\lim_{t\to\infty} u(x, t) = 0 $, uniformly in x.

For any $(x, t)\in R^n × (0, +∞)$ let $ K(x, t) := (\frac{1}{4πt})^\frac{n}{2} e^-\frac{|x|^2}{4t} $ be fundamental solution of the heat equation (also called the heat kernel) and consider $u(x, t) = ...
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Proving functions are odd in cauchy wave problem

I am stuck on a problem right now that asks the following: $u_{tt}-c^2u_{xx} = 0$ $u(x,0) = g(x)$ $u_t(x,0) = h(x)$ $x$ goes from $-\infty < x < \infty$ I need to prove that if $g(x)$ and ...
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28 views

Fundamental solution matrix of a linear PDE

A vectorial function $\boldsymbol{f}(\boldsymbol{x})$, satisfies the following PDE $$ (\boldsymbol{c \cdot \nabla}) \boldsymbol{f}(\boldsymbol{x}) = \boldsymbol{A}(\boldsymbol{x}) ...
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1answer
12 views

Geometry of level sets of an harmonic function

Suppose you have an harmonic function on an exterior domain of $\mathbb{R}^n$, i.e., a function $u \colon \mathbb{R}^n \setminus \bar\Omega \to \mathbb{R}$, where $\Omega$ is a smooth and bounded open ...
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1answer
15 views

A question related to Laplace equation on pde.

Let $B_R(p)$ the open ball of radius R centered at p in $R^n$ and consider the following problem $∆u = 0 $ in $ R^n\setminus B¯_R(p)$ ,$u = c$ on $∂B_R(p)$, where c is a given constant. Find a ...
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2answers
33 views

What is a boundary condition for a PDE in a rectangular domain?

In the method of separation of variables, we need homogeneous BCs. For the elliptic pde with inhomogeneous BCs: $u_{xx}+u_{yy}=0$, with $0<x<a$ and $0<y<b$. With $u(x=0,y)=0$ and ...
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29 views

Why are PDEs with Hamiltonians usually solved on compact manifolds?

The title is self explaining: I see in a lot of literature that PDEs with some Hamiltonian structure in it are solved over a torus or some other compact manifold. Why is that? At least I now that it ...
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1answer
41 views

How to find the Green's function

Consider a domain $$D= \{(x,y) : x>0, y>0 \}$$ Let $\textbf x = (x,y)$ and $\xi =(\xi_x , \xi_y)$. Find the Green's function, $G(\textbf x , \xi)$ such that $$\nabla ^2 G=\delta (\textbf x - ...
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1answer
28 views

General Methods to Solve First-Order PDE

Question is as simple as: What are the different methods for solving a first-order PDE? I'm aware of nearly all forms of Method of Characteristics - Lagrange Method, Charpit's Method. I'm ...
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1answer
71 views

Operator $f \mapsto u(f)$ solution of non-homogeneous Laplace equation is compact and self-adjoint

Let $u : L^2_0(D) \to L^2_0(D): = \lbrace f \in L^2 : \int_D f = 0 \rbrace $ be the linear operator which associates $f$ to $u(f)$ the solution of $$ \begin{cases} \Delta u = f & \text{in } D \\ ...
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0answers
20 views

Taylor expansion in proof of weak maximum principle

Picture below is part of proof of weak maximum principle. On the red line ,I don't know how to use the Taylor expansion to get $-u''(x_0) \le 0$. I think the Taylor expansion of $u(x)$ at $x_0$ is $$ ...
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34 views

About Semiclassical Analysis and other

I read something about this theory. I honestly do not care to find out the link between quantum mechanics and general relativity, because it's too much for me. But I have seen that there are still ...