**0**

votes

**0**answers

43 views

### Linking theorem, elliptic pde

I am trying to solve some linear system of the form
$$ \Delta \phi +p v(r)^{p-1} \psi = -f, \qquad \Delta \psi + qu(r)^{q-1} \phi =g $$ in $ \mathbb{R}^N$ where $ f,g$ are given and $ \phi,\psi$ are ...

**3**

votes

**2**answers

200 views

### Heat equation and evolution of number of critical points

Let $u_0$ be a smooth function on the unit sphere $S^1$ and assume that $u(t,x)$ is a smooth solution of the heat equation with initial data $u(0,x)=u_0(x)$. How one can apply the maximum principle to ...

**0**

votes

**0**answers

111 views

### Solving gradient of an especial heat equation

In my research I came up with a gradient of heat equation on a edge-weighted graph as:
\begin{equation*}
\nabla_w T_t(t,w) + T(t,w) . \nabla_w L_w + L_w . \nabla_w T(t,w) = 0
\end{equation*}
where ...

**10**

votes

**9**answers

3k views

### Textbooks for PDE between Strauss and Folland

Walter A. Strauss's Partial Differential Equations: An Introduction is a classic PDE textbook for the undergraduate students. While Folland's Introduction to Partial Differential Equations, is a nice ...

**1**

vote

**0**answers

72 views

### Frobenius-Perron operator leaving a function sub space invariant

I consider a linear subspace of $L^1(\mathbb R^n \to \mathbb R)$,
$$ \mathcal S := \{ h: L^1(\mathbb R^n \to \mathbb R) \; | \; \int_{H^{-1}(\{y\})} h(x) \, dS = 0 \text{ for all } y \in \mathbb R\} ...

**1**

vote

**1**answer

88 views

### Method of characteristics of a system of first order pdes

I asked the question on math.stackexchange.com, but didn't get any reply. So, I asked it again here. Any suggestion or hint is welcome, and thank you for your attention.
Consider the system of first ...

**0**

votes

**0**answers

15 views

### Existence to an advection equation with level-set constraints

Let $f: \mathbb R^n \to \mathbb R^n$ be a sufficiently smooth vector field and let $h: \mathbb R^n \to \mathbb R$ a scalar field. We consider the following advection equation
...

**2**

votes

**0**answers

47 views

### only solution to wave equation under certain restriction?

Suppose $u_1, u_2: \mathbb{R}^2 \rightarrow \mathbb{R}$ are two bounded solutions to the wave equation
\begin{equation}
\partial_t^2u_i = \partial_x^2u_i
\end{equation}
obeying the restriction
...

**1**

vote

**0**answers

63 views

### Existence of Green's functions for PDEs

Here is what I think I know: Given a symmetric linear differential operator $\mathcal{L}$ that is positive definite on a function space(/space of distributions) $\mathcal{H}$, we can find its inverse, ...

**2**

votes

**2**answers

131 views

### Some Questions from Reading on Wave Front Set from Hormander's Linear PDE Vol. 1

In Hormander's Linear PDE Vol. 1 (pg 252-253, before the definition of wave front set is introduced), Lemma $8.1.1$ says that if $\phi \in C_{0}^{\infty}$ and $v \in \mathcal{E}^{\prime}$, then ...

**0**

votes

**0**answers

76 views

### Non-trivial global solution for Dirichlet eigenvalue problem

Suppose $f:\mathbb{R}^2\to\mathbb{R}$ is smooth everywhere except a set of measure zero.(i.e. A set of area zero) and satisfies the equation
$\Delta f=\lambda f$ for some constant $\lambda$ off this ...

**1**

vote

**0**answers

129 views

### Existence of solution?

I am sorry if this question is not at the MO level. But I have not found a reference so I would like ask it here.
Follow this paper :http://www.math.ku.dk/~hugger/articles/CTAC2003.pdf
Let ...

**1**

vote

**1**answer

65 views

### Maximal minimum of Bessel functions

This comes from a scattering problem. Consider the usual non singular Bessel functions of the first kind, $J_n(x)$. It is known that their zeros are countable, and all zeros are distinct. My question ...

**1**

vote

**1**answer

85 views

### Wave equation with linear coefficients

The following pde came up in a physics problem:
$$
(Cy+D)\frac{\partial^2 u}{\partial x^2}-(Ay+B)\frac{\partial u^2}{\partial y^2}-A\frac{\partial u}{\partial y} =f(x,y),
$$
A,B,C,D are fixed ...

**0**

votes

**0**answers

33 views

### evolution PDE in comlex domaim

i have got (an example for some other theorem) an existence theorem for such a type problem
$$u_t(t,z)=-u(t,z)+a(t)z^m\frac{\partial^N u(t,z)}{\partial z^N},\quad u(0,z)=\hat u(z)\in \mathcal ...

**1**

vote

**0**answers

49 views

### Uniform bounds for a coupled parabolic system of PDE (linear)

Let $V=H^1(\Omega)$ and $H=L^2(\Omega)$ where $\Omega$ is a compact Riemannian manifold. Define $W = \{ w \in L^2(0,T;V) : w_t \in L^2(0,T;V^*)\}$.
Consider the system, with $u^\epsilon, v^\epsilon ...

**1**

vote

**1**answer

135 views

### Estimates on evolution operator

Let's consider the following evolution operator in $\mathbb{R}^3$
$$S(t)=e^{(i+\delta)t\Delta }$$
How to get the following estimate
$$\Vert S(t)f\Vert_2\leq C_\varepsilon t^{-\frac{1}{4}}\Vert ...

**1**

vote

**0**answers

51 views

### References Request : Existence and Uniqueness for PDE which is “ALMOST (?)” Parabolic

I am doing a research using PDE which is a little different from the standard Parabolic type. The following is my case:
\begin{equation}
Lu = - \sum_{i, j = 1}^{N}a^{i,j}(x, t)u_{x_{i}x_{j}} + ...

**0**

votes

**1**answer

116 views

### Holder regularity for the heat potentials

First I apologize for my bad English and for any error: this is my first question.
I need some regularity results for the single and double layer heat potentials.
If $\Gamma(t,x)$ is the fundamental ...

**2**

votes

**2**answers

102 views

### First order pde with characteristics [closed]

Consider a first order pde of the type $$u_y+b(x)u_x=0$$ and suppose that the coefficient $b$ is not necessairly continuous (for instance with a jump in some point).
Is it still possible to apply in ...

**8**

votes

**1**answer

602 views

### Regularity of the Maxwell equations

As is well-known, the Maxwell equations can be phrased vectorially as,
\begin{align}
\nabla \cdot \mathbf E &= \frac{\rho_f}{\varepsilon}, &\text{Gauss's law,}\\\
\nabla \cdot \mathbf ...

**5**

votes

**0**answers

103 views

### Reference request: Optimal $L^p$-decay for nonhomogenous heat equation in $\mathbb R^d$

Let $u$ be a classical solution for the nonhomogeneous heat equation in $\mathbb R_+ \times\mathbb R^d$:
$$
\begin{cases}
\partial_tu(t,x)-\Delta u(t,x) = f(t,x), \\
u(0,x)=u_0(x).
\end{cases}
$$
...

**2**

votes

**0**answers

102 views

### Regularity of solution to Fokker Planck equation

Suppose that $\rho \in L^1(\mathbb{R}^n \times (0,T))$ for every $T < \infty$ is a weak solution of the PDE
\begin{align}
\partial_t\rho &= \Delta \rho + \text{div}(\rho\nabla\Psi(x))\\
\rho(t ...

**1**

vote

**0**answers

95 views

### A linear operator equation (PDE) with non-monotone term

I'm interested in the existence and/or uniqueness to the following problem. Let $V$ and $H$ be Hilbert spaces and $V \subset H \subset V^*$ form a Gelfand triple.
There is a linear operator $L:{D}(L) ...

**2**

votes

**0**answers

55 views

### Solve a PDE related to free boundary problem

I would like to solve the following system for my problem:
$$\max\Big(\frac{1}{2}u_{ss}+u_l\delta(s-s_0), F(l)-\lambda(s)-u(s,l)\Big)=0.$$
where $u=u(s,l): R\times R_+\to R$ is the unknown function ...

**3**

votes

**1**answer

169 views

### Uniqueness of weak solutions of a heat equation

Let $M$ be a smooth compact closed manifold.
Let $u \in H^1(0,T;H^{-1}(M)) \cap L^2(0,T;H^1(M))$ be a solution of
$$u_t - \Delta u - u = 0$$
$$u(0)=u(T)$$
satisfying $\int_M u(t) = 0$ for all $t$. Is ...

**8**

votes

**1**answer

187 views

### Failure of Fredholm property of elliptic PDE systems

Roughly speaking, a PDE operator satisfies the Fredholm property if its principal symbol is elliptic and the information provided on the boundary satisfies the Shapiro-Lopatinskii condition.
What can ...

**0**

votes

**0**answers

49 views

### Finding gradient of an optimization

I am trying to find the gradient of the following optimization problem and then add to objective, but I got some trouble in computing. Could you please help me?
Assume that we have an optimization ...

**-2**

votes

**2**answers

116 views

### Lack of parabolicity of PDE due to invariancy under diffeomorphisms? [closed]

Let a nonlinear differential equation is invariant under all diffeomorphisms, then we get lack of parabolicity?

**0**

votes

**0**answers

45 views

### $L^\infty$ estimate for a fourth order (hyperbolic) equation

Consider the following fourth order equation
$$u_{tt}+u_t= d\Delta u-\Delta^2u+f,$$
with Dirichlet or Navier boundary conditions, that is on $\partial\Omega$, we assume that
...

**1**

vote

**0**answers

46 views

### Finite element convergence rates for mixed problems [closed]

I've coded up a Stokes Flow problem using finite elements and am in the process of verifying that it works. I'm just not sure what convergence rate I should be expecting as I globally refine the mesh.
...

**0**

votes

**1**answer

215 views

### When is separation of variables an acceptable assumption to solve a PDE?

We know that one of the classical methods for solving some PDEs is the method of separation of variables. It works for known types of PDEs and many examples of physical phenomena are successfully ...

**0**

votes

**0**answers

83 views

### Estimate for an integral of a function of the solution to a PDE

Let $\Omega \in \mathbb{R}^3$ be a bounded smooth domain. Assume that smooth functions $\sigma_1,\sigma_2$ satisfy $\sigma_1-\sigma_2 \in C_0^\infty(\Omega)$ and
$\lambda\leq \sigma_1, \sigma_2 \leq ...

**4**

votes

**2**answers

182 views

### Solution to Schrödinger equation

I asked this question already on stackexchange, but I did not get any resonance at all, so maybe anybody here can give me a few hints about my problem.
My goal is to solve this PDE for $f:[-1,1] ...

**4**

votes

**2**answers

266 views

### Analytic solution of a system of linear, hyperbolic, first order, partial differential equations

In a try to solve a physical problem, I've faced a system of first-order partial differential equations of the form
...

**9**

votes

**1**answer

357 views

### Special Second-Order PDE

Let $\Phi$ be a given smooth function on a neighborhood of zero in $\mathbb{R}^n$ with
$$\Phi(0) = 0, ~~~~D \Phi(0) = 0, ~~~~ D^2\Phi(0) >0,$$
the latter meaning that the Hessian is positive ...

**2**

votes

**1**answer

122 views

### Heat transfer: boundary conditions with fluid velocity

The following equation is considered:
$$
\frac{\partial u}{\partial t} - a\Delta u + \mathbf v \cdot \nabla u = f.
$$
I have difficulties in formulating boundary conditions for this equation.
If ...

**3**

votes

**0**answers

56 views

### How can one do change of variables for solutions to a staochastic partial differential equation?

isHow can one do change of variables for solutions to a staochastic partial differential equation? For example, let us consider the following stochastic transport equation:
$$
dy(t,x) + y_x(t,x) + ...

**1**

vote

**0**answers

81 views

### Limit Toward Discontinuous Point of Dirichlet Boundary Value

The question arises from a paper on Schwarz's domain decomposition method (click here).
We consider a bounded domain in $\mathbb{R}^2$ and a curve splits it into two, see the figure below.
Now we ...

**2**

votes

**2**answers

119 views

### Let $\mathrm{div}\,(A\,\mathrm{grad}\,u) + b u = f$. Is $(A\,\mathrm{grad}\,u)$ weakly differentiable?

Let us consider the basic linear elliptic PDE
$$
\mathrm{div} (A\,\mathrm{grad}\,u) + bu = f,
$$
with $f\in L^p,$ $A,b$ uniformly bounded. Do we have, for a weak solution $u\in W^{1,p}(\Omega')$,
$$
...

**1**

vote

**1**answer

113 views

### Existence of the solution of a linear parabolic pde

Good day!
Let $V = H^1(\Omega)$, $\Omega \subset \mathbb R^3$.
Consider the linear parabolic equation $y' + Ay = f$ where $f \in L^q(0,T;V')$, $y \in W = \{y \in L^p(0,T;V) \colon dy/dt \in ...

**5**

votes

**0**answers

288 views

### Feynman-Kac theorem: probabilistic proof of existence of solution to parabolic PDE

Friedman (in his book: PDEs of Parabolic Type) shows how to construct a solution to the Cauchy problem
$$
\partial_t u(t,x) = b(x) \partial_x u(t,x) + \frac{1}{2} \sigma(x)^2 \partial_{x,x} u(t,x)
$$
...

**0**

votes

**0**answers

108 views

### positive eigenfunction on complete Riemannian manifold

Let $(M^n,g)$ be a complete(non-compact) Riemannian manifold. Consider the positive solution to the equation
$$
\Delta u = u
$$
where $\Delta=\nabla_i \nabla_i$ is negative semi-definite. Is there ...

**0**

votes

**1**answer

94 views

### Harmonic extension in a ball $B(x, r) \subset \mathbb R^n$

I have recently been trying to understand the theory regarding harmonic extensions in $\mathbb R^n$. I have, however, had some difficulties to find the kind of results I am looking for. For that ...

**2**

votes

**1**answer

166 views

### Tempered distribution solution to a simple PDE

Let's consider the following PDE in $\mathbb R^d$ :
$$\frac{\partial^d u}{\partial x_1...\partial x_d}=f$$
where $f$ is a tempered distribution with support in $\mathbb R^d_+$. There is a result by ...

**6**

votes

**2**answers

268 views

### Uniform bound on the eigenfunctions of the Laplacian

Is it possibly to have $L_\infty$ bounds on the eigenfunctions of the Laplacian operator on bounded regular domains with Dirichlet condition? I found several papers by Sogge but these are pretty ...

**2**

votes

**1**answer

154 views

### The centralizer of Lienard equation

Consider the lienard vector field $\cases{
x'=y -F(x) \\
y'=-x }
$ in $\mathbb{R}^{2}$, where $F$ is a polynomial fuction with $F(0)=0$. Assume that $Y$ is a smooth vector field globally defined ...

**1**

vote

**0**answers

58 views

### Parabolic partial differential equation, initial conditions

Let $U\subset\mathbb{R}^n$ be open bounded, $T>0$.
Given the parabolic PDE $$\partial_tf+a\partial_xf+b\partial_{xx}f = g \qquad (1)$$ I'm interested in the initial and boundary conditions that ...

**0**

votes

**0**answers

57 views

### Why is it impossible to reduce a linear PDE of the second order in more than two independent variables to canonical form globally

It is known that in the case of more than two independent variables, it is usually not possible (especially in the case of PDE with the variable coefficients) to reduce a linear partial differential ...

**2**

votes

**0**answers

95 views

### Algebraic methods in pde [duplicate]

I'm finding myself with a linear, very symetric system of first orders pde with polynomial coefficients. Wandering on the web, i learnt there is some nice alebraic way to deal with it involving ...