Questions about linear partial differential equations. Often used in combination with the top-level tag ap.analysis-of-pdes.
0
votes
1answer
67 views
Nodal sets under the heat flow
Let $u(t,X)$ be a smooth solution of the heat equation on $R^2$
$u_t=\Delta u,$
where $(t,X)\in R \times R^2$. Suppose $\lim_{t \rightarrow 0} u(t,x,y)=x^2-y^2$. Can we prove that the nodal set of ...
1
vote
0answers
16 views
Renormalization for Transport Equations with SBD velocity field
In the paper Traces and fine properties of a $BD$ class of vector fields and applications by Ambrosio, Crippa and Maniglia (to be found here)the authors prove a chain rule for vector fields $B\in ...
1
vote
1answer
69 views
Strong maximum principle for the heat equation in non-cylindrical domains
let $u(t,x)$ be a bounded smooth solution of the heat equation $u_t=\Delta u$, $(t,x) \in R \times R^2$, and let $V \subset (R \times R^2)$ be an open connected component of $\{(t,x) \in R \times R^2: ...
6
votes
1answer
123 views
Mountain Pass theorem for minimization problems with constraints
Let $I[u]$ be a functional on a (possibly infinite dimensional) Hilbert space. Then, under some conditions, the Mountain Pass theorem guarantees the existence of a saddle point (see ...
0
votes
0answers
54 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
2answers
217 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
0answers
119 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 ...
2
votes
1answer
127 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
0answers
19 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
0answers
48 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
0answers
66 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, ...
0
votes
0answers
83 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
0answers
130 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
1answer
72 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
1answer
86 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
0answers
33 views
evolution PDE in comlex domaim
i have got (an example for some other theorem) an existence theorem for such a type problem
$$u_t(t,z)=-u(t,z)+a(t)z^m\frac{\partial^N u(t,z)}{\partial z^N},\quad u(0,z)=\hat u(z)\in \mathcal ...
1
vote
0answers
75 views
Uniform bounds for a coupled parabolic system of PDE (linear)
Let $V=H^1(\Omega)$ and $H=L^2(\Omega)$ where $\Omega$ is a compact Riemannian manifold. Define $W = \{ w \in L^2(0,T;V) : w_t \in L^2(0,T;V^*)\}$.
Consider the system, with $u^\epsilon, v^\epsilon ...
1
vote
1answer
137 views
Estimates on evolution operator
Let's consider the following evolution operator in $\mathbb{R}^3$
$$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
0answers
54 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}} + ...
2
votes
2answers
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 ...
5
votes
0answers
106 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 $\mathbb R_+ \times\mathbb R^d$:
$$
\begin{cases}
\partial_tu(t,x)-\Delta u(t,x) = f(t,x), \\
u(0,x)=u_0(x).
\end{cases}
$$
...
0
votes
1answer
123 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
0answers
110 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
0answers
97 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
0answers
57 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
1answer
175 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 ...
2
votes
2answers
139 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
0answers
50 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
2answers
118 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
0answers
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
0answers
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
1answer
232 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
0answers
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
2answers
186 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
2answers
299 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
...
2
votes
1answer
127 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 ...
9
votes
1answer
361 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 ...
3
votes
0answers
57 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:
$$
dy(t,x) + y_x(t,x) + ...
1
vote
0answers
83 views
Limit Toward Discontinuous Point of Dirichlet Boundary Value
The question arises from a paper on Schwarz's domain decomposition method (click here).
We consider a bounded domain in $\mathbb{R}^2$ and a curve splits it into two, see the figure below.
Now we ...
2
votes
2answers
121 views
Let $\mathrm{div}\,(A\,\mathrm{grad}\,u) + b u = f$. Is $(A\,\mathrm{grad}\,u)$ weakly differentiable?
Let us consider the basic linear elliptic PDE
$$
\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
1answer
117 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 ...
0
votes
0answers
110 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
1answer
96 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
1answer
168 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 ...
5
votes
0answers
309 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
0answers
60 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
0answers
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 ...
2
votes
1answer
130 views
Extending a harmonic function in a ball to subharmonic in a larger ball
Consider the Laplace equation in a ball $B(r) \subset \mathbb{R}^n$ of radius $r$:
$$
\begin{cases}
-\Delta u &= 0, \quad \text {in} \quad B(r), \\
\ \ \ \ \ \, u&= g, \quad \text {in}\quad ...
1
vote
0answers
59 views
Parabolic partial differential equation, initial conditions
Let $U\subset\mathbb{R}^n$ be open bounded, $T>0$.
Given the parabolic PDE $$\partial_tf+a\partial_xf+b\partial_{xx}f = g \qquad (1)$$ I'm interested in the initial and boundary conditions that ...
0
votes
0answers
118 views
Solvable PDEs and their Green's functions
I have a class of PDEs of the form
$$
-\Box\phi(x)+\lambda\phi_0^2(x)\phi(x)=0
$$
with $\phi_0^2(x)=\sum_{n=-\infty}^\infty b_ne^{ip_n\cdot x}$. I know some exact solutions for them (see here and ...
