Weak solution

A weak solution satisfies an integrated identity in which derivatives are transferred to test functions. It can therefore exist with fewer classical derivatives.

For a partial differential equation $F(x,u,Du,D^{2}u)=0$ whose operator is nonincreasing in the Hessian variable, a continuous viscosity subsolution satisfies $F(x_0,u(x_0),D\varphi (x_0),D^2\varphi (x_0))\le 0$ whenever a smooth test function $\varphi$ touches it from above. A viscosity supersolution satisfies the reverse inequality for tests touching from below. A viscosity solution satisfies both conditions. For $F=-\mathrm{tr}{D}^{2}u$, these conditions are respectively $\Delta \varphi \ge 0$ and $\Delta \varphi \le 0$. They make sense even when $u$ has no classical second derivatives.

Let a continuous function satisfy both viscosity inequalities for the Laplace operator. On any relatively compact ball, take its harmonic replacement $h$ with matching boundary values. If $u-h$ has a positive maximum, the function $h+\epsilon (R^2-|x-a{|}^2)$ still lies below $u$ somewhere for sufficiently small positive $\epsilon$, while matching it on the boundary. At an interior maximum of their difference it is a valid upper test, but its Laplacian is $-2n\epsilon <0$, a contradiction. The opposite perturbation excludes a negative minimum. Thus $u=h$ throughout the ball and is a smooth harmonic function. A locally defined test is extended near its touching point using a smooth cutoff function when global tests are required.

A weak-strong uniqueness principle says that a weak solution with the same data as a sufficiently regular classical solution agrees with that solution throughout its interval of regular existence, within the specified weak class. The conclusion is class-dependent: all bounded weak solutions of a prescribed $C^1$ linear transport field satisfy it, while nonlinear conservation laws require an admissibility condition such as an entropy solution condition.

For $u_{tt}+L(t)u=f$ with homogeneous Dirichlet boundary conditions, smooth uniformly elliptic principal coefficients, and initial data $({\psi }_0,{\psi }_1)\in H_0^1(U)\times L^2(U)$, a space-time $H^1$ weak solution has zero lateral trace, trace $u(0)={\psi }_0$, and satisfiesfor every space-time $H^1$ test function with zero lateral trace and $v(T)=0$. The velocity initial condition is encoded by the last term; an arbitrary space-time $H^1$ function need not have an $L^2$ trace of its time derivative. The equation itself gives $u_{tt}\in L^2(0,T;H^{-1}(U))$, so $u_t$ has a continuous $H^{-1}$ representative.

A weak formulation multiplies a differential equation by a test function, integrates, and uses integration by parts to move derivatives away from the unknown function. Natural boundary conditions appear in the resulting boundary terms.

For sufficiently regular coefficients and a function $u$, this boundary quantity is the outward flux appearing when a divergence-form principal operator is integrated by parts. Its zero condition is the natural Neumann boundary condition for the corresponding bilinear form. With only bounded coefficients and a Sobolev solution, the weak formulation can encode that natural condition without asserting that a pointwise boundary derivative exists.

The weak form of $(L^{\ast }-\lambda )u=f$ includes the same scalar shift as the differential equation. The unshifted identity $B^{\ast }[u,v]=(f,v)$ instead represents $L^{\ast }u=f$ and cannot imply the shifted equation for nonzero $\lambda$ unless $u=0$. On a smooth bounded domain choose any nonzero compactly supported smooth $u$ and take $L=-\Delta +1$, $f=Lu$ for a direct counterexample. For constant drift coefficients, integration by parts reverses the drift sign in $L^{\ast }$ without changing this spectral term.

The Galerkin method seeks an approximate solution in a finite-dimensional subspace and requires the equation's residual to be orthogonal to that subspace. Uniform energy estimates and compactness can then produce a weak solution as the subspaces become dense.

For a divergence-free drift $\mathbf{b}$ and a function $v$ with zero boundary Sobolev trace, integration by parts gives ${\int }_{\Omega }v\mathbf{b}\cdot \nabla v=\frac{1}{2}{\int }_{\partial \Omega }{v}^2\mathbf{b}\cdot \mathbf{n}-\frac{1}{2}{\int }_{\Omega }(\nabla \cdot \mathbf{b})v^2=0$. Consequently the advection matrix in a conforming Galerkin method has zero quadratic form; its symmetric part vanishes. Adding a diffusion stiffness matrix gives a nonpositive spatial generator and a nonincreasing L2 norm in the semidiscrete advection-diffusion equation. The same identity holds with a suitable zero normal-flux boundary term instead of zero trace.

The Petrov-Galerkin method chooses an approximate solution $u_h$ in a finite-dimensional trial vector space $V_h$ and tests a weak formulation against a possibly different test space $W_h$. For a bilinear form $a$ and a linear functional $\ell$, it imposes $a(u_h,v_h)=\ell (v_h)$ for every $v_h\in W_h$. Taking $W_h=V_h$ recovers the usual Galerkin method. Different test functions can alter stability and accuracy; the Galerkin orthogonality is then against the test space and need not be an orthogonal projection in an energy norm.

A Fourier truncation retains a finite set of boundary-compatible Fourier modes and projects the PDE onto their span by the Galerkin method. Nonlinear products can generate unretained harmonics, so their coefficients are discarded by projection rather than declared zero pointwise. Agreement with a perturbation expansion through cubic order requires retention of every quadratically forced mode that feeds back on the critical mode at cubic order.

// Destination: dynamical-systems.bigb

A useful version of the Aubin-Lions lemma concerns Banach spaces with embeddingswhere the first embedding is compact and the second continuous. For $1<p<\infty$ and $1<q\le \infty$, a family bounded in $L^p(0,T;X_0)$ whose distributional time derivatives are bounded in $L^q(0,T;X_1)$ is relatively compact in $L^p(0,T;X)$. The exponents and embedding hypotheses must be stated; an unspecified suitable derivative bound does not identify a version of the theorem.

For a bounded Lipschitz domain $\Omega$, Rellich-Kondrachov compactness theorem and continuous dual embeddings giveConsequently boundedness of $u_j$ in $L^2(0,T;H^1(\Omega ))$ and of ${\partial }_t{u}_j$ in $L^2(0,T;H^{-1}(\Omega ))$ yields a strongly convergent subsequence in $L^2(0,T;L^2(\Omega ))$. The spatial bound prevents fine-scale spatial oscillations, while the derivative bound controls time translations. This is why the lemma lets a Galerkin method pass to nonlinear terms, for which weak convergence alone is often insufficient. On an unbounded spatial domain, the displayed spatial compact embedding can fail through translations.

If $u\in L^{\infty }(0,T;H)$ and $u_t\in L^q(0,T;V^{\prime })$ for a dense continuous embedding $V\hookrightarrow H$ and some $q>1$, then $u$ has a representative in $C([0,T];H_{\mathrm{weak}})$. Indeed, its pairing with each element of a dense subset of $V$ is absolutely continuous, and the uniform $H$ bound extends continuity to every element of $H$.

If $u:[0,T]\to H$ is weakly continuous and an energy equality makes $t\mapsto \Vert u(t){\Vert }_H$ continuous, then $u$ is strongly continuous. This follows from the Radon-Riesz theorem applied whenever $t_n\to t$.